The At Home in College
Core Math Curriculum
Written by Steve Hinds
with Kevin Winkler, Christina Masciotti, and Wally Rosenthal
Fall 2012 edition
Sponsored by
The City University of New York Office of Collaborative Programs
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
2
CUNY Start College Transition Initiative Core Math Curriculum
Course Contents (Version A)
Class Contents N EA F Page
0 CUNY Start Core Math Initial Assessment 8
1
Number Puzzle 1
24
Core Math Expectations and the CUNY Start Math Binder
Uses of Parentheses
Signed Numbers 1: Addition
Best Buy Commissions
2
The Many Faces of Function Rules and Tables, part 1
38 Signed Numbers 2: Subtraction
Extra Practice #1
3
Review Extra Practice
50 The Many Faces of Function Rules and Tables, part 2
More Signed Numbers 2: Subtraction
4
Signed Numbers 3: Multiplication
59
Times Table Baseline Test
Expressions and Equations
Evaluating Expressions I
CUNY Start Math Textbook — Index of Topics and Vocabulary Index
Extra Practice #2
5
Review Extra Practice
77 Return Times Table Baseline Test and flashcard introduction
Evaluating Expressions II
6
Three Views of a Function
82 Signed Numbers 4: Division
Extra Practice #3
7
Review Extra Practice
94 Counting Cubes
Exponents 1
8
2xx
102
Terms and Expressions discussion
Rectangle ________ and Rectangle Perimeter with Variable Side Lengths
Terms and Expressions
Extra Practice #4
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
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Class Contents N EA F Page
9
Review Extra Practice
116
Creating Student Number Puzzles
Exponents 2
Connecting Function Solutions
Integers
Student Number Puzzles
10
Aftermath of Hurricane Katrina and A New Orleans Levee
125 Highest and Lowest Elevations in Selected Continents
Extra Practice #5
11
Review Extra Practice
Remaining Group Presentations on elevation activity
139
Multiplying Terms
How to Study for a Math Test
My Review Problems for CUNY Start Exam #1
AlgeCross I
12
Rectangle _____ and Rectangle Area with Variable Side Lengths
148 Combining and Multiplying Terms
Student- and Instructor-Generated Exam Practice (Extra Practice #6)
13
Review Exam Practice and student-devised practice
160 Times Table Test #2
CUNY Start Exam #1
Patterns in Functions and Their Graphs, part 1
14
Review Patterns in Functions and Their Graphs, part 1
171 Patterns in Functions and Their Graphs, part 2 and Discussion
Extra Practice #7
15
Returning CUNY Start Exam #1 and Reviewing selected problems
183 Exploring Squares
Introduction to Square Roots
Introduction to Factoring and Factors and Finding Factors
16
_______ Fractions
196
Writing Fractions in Higher Terms
Large Numbers of Eligible Comm. Coll. Students Do Not Apply for FA
Writing Fractions in Lowest Terms
Extra Practice #8
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Class Contents N EA F Page
17
Review Extra Practice
212 Dividing Terms
…for all 0x .
Expression Puzzles
18
Perfect and Non-Perfect Squares
219
More Squares and Perfect Squares
Introduction to Polynomials
Adding Polynomials
Extra Practice #9
19
Review Extra Practice
233 Rate of Change
Identifying and Interpreting Rate of Change
20
A Moment for Mental Math
242 Three Scenarios and the Distributive Property
Uses of the Distributive Property
Extra Practice #10
21
Review Extra Practice
255 Pencil Packs
The Distributive Property — Addition and Subtraction
Identifying and Interpreting Rate of Change from graphs
22
Subtracting Polynomials, part 1
265 Subtracting Polynomials, part 2
Rate of Change = _____________
Extra Practice #11
23
Review Extra Practice
281 Cube roots
Rectangle Area Revisited
AlgeCross II
24
Factoring Binomials I
291 Finding Linear Function Rules
Students Begin Exam Study
Extra Practice for CUNY Start Exam #2 (Extra Practice #12)
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
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Class Contents N EA F Page
25
Review Student-Devised Problems and Extra Practice for Exam #2
303
Introduction to Function Notation
Times Table Test #3
CUNY Start Exam #2
Using Function Notation
26
Review Comparing Function Graphs
318
Take a Hike!
An Alternate Method for Hiker Sharing
Group Exploration of the Mean I
Two NYC Internet Plans
27
Returning CUNY Start Exam #2 and Reviewing selected problems
329
Calculating and Explaining Means
Radicals with Fractions
Comparing Function Graphs
Extra Practice #13 (Review Problems)
28
Metro Movies and Reaching a Mean
345
Combining Like Roots
Functions and Parallel Lines
Group Exploration of the Mean II (Optional)
Extra Practice #14
29
Review Extra Practice
362 Factoring Binomials II
Benchmark Percents and Solving Problems with Percents I
Best Buy vs. The Digital Source
30
Systems of Equations
371 Dividing Polynomials I
Extra Practice #15
31
Review Extra Practice
389 Expression Puzzles II
10% and Problem-Solving with Percents II
Estimation in Percent Problem-Solving
32
Two Functions to Compare
402 Functions and Their y-Intercepts
Multiplying Binomials I
Extra Practice #16
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
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Class Contents N EA F Page
33
Review Extra Practice
417 More y-Intercept In and Out of Contexts
Multiplying Binomials II
34
Multiplying Binomials III
431 Multiplying Binomials IV
Extra Practice #17
35
Review Extra Practice
447
Multiplying Binomials, shortcut
Working Backwards
AlgeCross III
Student-Generated Exam Practice (Extra Practice #18)
36
Review Student-Generated Exam Practice (minimal)
455
Solving 1-Variable Equations I
Times Table Test #4
CUNY Start Exam #3
Solving 1-Variable Equations II
37
Review Solving 1-Variable Equations II
475 Factoring Trinomials I
Trinomial Puzzles I
Solving 1-Variable Equations III
38
Returning CUNY Start Exam #3 and Reviewing selected problems
483
Factoring Trinomials II
Fraction Slopes I
Fraction Slopes II – equations and tables
Extra Practice #19
39
Review Extra Practice
499 Factoring ―Trinomials‖ III
Fraction Slopes III – negative slopes
Trinomial Puzzles II
40
Dividing Polynomials II
510 The Zero Product Property
Solving 1-Variable Equations IV - Quadratic Equations
Extra Practice #20
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
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Class Contents N EA F Page
41
Review Extra Practice
524 Solving 1-Variable Equations V – Proportions
Fraction Slopes IV – equations and graphs
42
Fraction Slopes V – two functions to compare
534 Fraction Slopes VI – lowest terms slopes from two solutions
Fraction Slopes VII – systems
Extra Practice #21
43
Review Extra Practice
546 Multiplying a Square Root By Itself
FOILing with Identical Square Roots
Solving Equations VI - Decimal and Mixed Numbers
44
Strategies for Multiple-Choice Math Tests
555 Practice with Multiple-Choice Test-Taking Strategies
Problem Set for Phase One Final Exam
45 Problem Set for Phase One Final Exam – answer key
575 Review Solutions to Final Problem Set
46 CUNY Start Math Phase One Final Assessment 580
47 Review Phase One Final Exam and Prepare COMPASS Review Problems
597 Tips for Taking the CUNY Placement Exams
Appendix Contents N EA F Page
A CUNY Start Math Workshop Curriculum 602
B Phase 2 — Core Math Curriculum
C Phase 2 — Math 1 Extra Practice
D Phase 2 — Math 2 Mixed Problem Sets
E Phase 2 — Supplemental Math 1 Materials
F Phase 2 — Supplemental Math 2 Materials
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
8
This is not a test.
Put your name and the date on the top of this page.
You have already been accepted into the CUNY Start College Transition Initiative.
You will still be in the program even if you do not know how to answer any of the
problems that you will see on the next pages.
This assessment will help us learn a bit about what you already can do. It also
represents much of what you will learn in our course. If you do not know how to
do many of these problems, you should be excited to know that you will be able to
do many or all of these problems after you have taken our class.
You may not use a calculator of any kind for this assessment.
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
9
CUNY Start Core Math Initial Assessment
Calculators are not permitted.
1. Complete the missing values for the following function: 63 xy (2 points total)
2. Check ―Yes‖ or ―No‖ for each of the following. (2 points total)
3. Evaluate 1522 xx when 4x . (1 point)
4. Evaluate 22xy when 3x and 2y . (1 point)
x 2 4
y 30 21
Yes No
Is 5x an example of an equation?
Is 5x an example of a binomial?
Is 5x an example of a trinomial?
Is 5x an example of a polynomial?
Is 5x an example of an expression?
Is 5x an example of a function?
Is 5x an example of two like terms?
Is 5x an example of a GPA?
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
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5. )3(12 x is equivalent to which of the following? Circle one answer. (1 point)
6. )7)(2( xx is equivalent to which of the following? Circle one answer. (1 point)
7. Simplify as much as possible. (1 point each)
a. 25.
3 b.
0
5 c. )15.1)(4(
d. )7(10 e. )5)(1)(2( f. 65
g. 3
24
h. 49 i. 3 27
a. 312 x b. x36 c. 3612 x d. 48 e. x15
a. 1492 xx
b. 1492 xx
c. 9142 xx
d. 1492 xx
e. 142 x
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8. A chemical solution begins at a temperature of 12 Fahrenheit. After a reaction has occurred,
the solution has a temperature of 8 Fahrenheit. How much did the temperature change? (1 point)
9. Your friend owns a small electronics store. She asks you for mathematical help so
that she can determine the best prices for some of her products. After observing
store receipts and doing some calculations, you create the following function in
order to represent the number of walkie-talkie sets that she can expect to sell based
on the price that she charges.
802)( xxg
In the function that you created, x represents the price in dollars that she charges for one walkie-
talkie set, and g(x) represents the number of walkie-talkie sets that she is likely to sell at that price.
a. Calculate the following: g(10) =
(1 point)
b. Using the same function and its context (price and the
number of walkie-talkie sets sold), describe the
meaning of the numbers shown in the table at the right.
(1 point)
10. Consider the function 14)( xxf . Which of the following is a solution to the function? Circle
one answer. (1 point)
x g(x)
30 20
a. )0,0( b. )41,10( c. )1,4( d. )19,5( e. )6,25(
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11. 31 lies between which pair of integers? (1 point)
12. Simplify the following expression as much as possible: )42()73( xx
(1 point)
13. You have earned the following scores on the first three exams in your psychology course: 77, 80,
and 85. What score is needed on the fourth test in order to have a mean of 84 for all four tests?
(1 point)
14. 42 )(b is equivalent to which of the following? Circle one answer. (1 point)
15. The expression ))(4)(2( baa is equivalent to which of the following? Circle one answer.
(1 point)
a. 5 and 6 b. 10 and 11 c. 15 and 16 d. 25 and 36 e. None of these.
a. 6b b. 62b c. 8b d. 82b e. None of these.
a. ab16 b. ba216 c. ab8 d. ba28 e. ba28
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16. The expression )65()103( xx is equivalent to which of the following? Circle one answer.
(1 point)
17. Simplify 7
2142
x
xx when 7x . (1 point)
18. The expression 3
22
5
15
rs
sr is equivalent to which of the following when r and s are non-zero?
Circle one answer. (1 point)
19. Which of the following is NOT a factor of 24? Circle one answer. (1 point)
20. Write 28
8 in lowest terms. (1 point)
a. 42 x b. 42 x c. 162 x d. 162 x e. 168 x
a. rs3 b. rs3 c. s
r
3 d.
s
r
3 e.
s
r3
a. 1 b. 6 c. 10 d. 12 e. 24
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21. Calculate and simplify as much as possible the perimeter and area of the large figure made up of
the three rectangles. (1 point each)
Perimeter =
Area =
22. Fill in the missing term in order to create a true equation. (1 point)
62 ))(( xx
23. Factor each of the following. (1 point each)
a. 892 xx b. 1522 xx c. 162 x
24. Which of the following is equivalent to xx 62 ? Circle one answer. (1 point)
a. 6x b. 37x c. )6( xx d. 36x e. None of these
8
3
y
x
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25. Which of the following is a factor of 2213 aa ? Circle one answer. (1 point)
26. Which of the following is equivalent to 2)12( x ? Circle one answer. (1 point)
27. Simplify 328
205
x
x when 4x . (1 point)
a. 7 b. 2a c. 21 d. a7 e. None of these
a. 14 2 x
b. 122 2 xx
c. 142 2 xx
d. 144 2 xx
e. 244 2 xx
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
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28. Graph the line 13 xy . An appropriate
graph will include at least two function
solutions. (1 point)
29. Graph the function 12 xy . An appropriate
graph will include all solutions (except
decimal ones) that can be graphed on this
xy-grid. (1 point)
30. Which ordered pair is a solution to the following system of equations? Circle one answer.
(1 point)
63
34
xy
xy
a. (4, 13) b. (5, 21) c. (–3, –15) d. (9, 33) e. None of these
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
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31. A system of equations is shown below.
Graph the functions so that the system
solution is shown. Write any solution to
the system using an ordered pair and
circle your answer. (1 point)
13 xy 92 xy
32. Which pair of functions produces parallel lines when graphed? (1 point)
33. Use a check mark to indicate if each of the following functions is linear or non-linear.
(1 point total)
Linear Non-Linear
xy 2
1)( 2 xxf
123 xxxy
5)( xxh
a. 12 xy and 21 xy
b. 13
2 xy and 1
2
3 xy
c. 4 xy and 82 xy
d. 63
1 xy and 23 xy
e. None of these
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
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34. Identify the slope and y-intercept of this linear function: 15 xy
(1 point each)
slope ________ y-intercept ________
35. Identify the slope and y-intercept of the linear function that has
this table of values:
(1 point each)
slope ________ y-intercept ________
36. Identify the slope and y-intercept of the linear
function shown at the right: (1 point each)
slope ________
y-intercept ________
37. What is the slope of the line that passes through the points (4, 6) and (7, 4)? (1 point)
x y
0 –2
1 –1.75
2 –1.5
3 –1.25
4 –1
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38. Which one of the following functions has a slope of 6 and passes through the point (–2, –10)?
Circle one answer. (1 point)
a. 62 xy
b. 610 xy
c. 26 xy
d. 26 xy
e. 106 xy
39. Which linear function has an integer slope? (1 point)
a. 52 xy
b. 63
1 xy
c. 5.25
2 xy
d. 15.3 xy
e. 104
3 xy
40. Determine the function equation whose table of values is shown.
(1 point)
x y
4 7
5 12
6 17
7 22
8 27
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41. Solve for x. (1 point each)
a. 12444 x b. 16113 x
c. 64
6 x
42. Solve for x in the following: 26102 x . Circle one answer. (1 point)
43. Solve for x in the following: 02452 xx . Circle one answer. (1 point)
a. 5x or 13x b. 5x c. 6x d. 6x or 6x e. 4x or 4x
a. 5x or 24x
b. 3x or 8x
c. 3x or 8x
d. 8x
e. 24x
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44. Simplify each of the following as much as possible. (1 point each)
a. 3
8
1 b. )33()35(
45. After a jewel is added to a $180 necklace, the price is increased by 75%. What is the new price?
(1 point)
46. Housing prices in a small city dropped sharply in a recent economic downturn. Last year, an
apartment had a monthly rent of $2,000, but the monthly rent decreased by 30% this year. What is
the monthly rent for that apartment this year? (1 point)
47. A $1,400 investment increases in value by 10% in the first year and another 5% in the second
year. What is the value of the investment after the second year? (1 point)
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
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Scoring Guide for Initial Assessment
Follow the scoring instructions you see below. No partial credit should be awarded unless clearly
indicated in the scoring guide. These test papers should not be left with the students. These are not
high-stakes tests, but we want to protect the items so that we may adapt or re-use them later. Keep
these assessments in a safe place so that you can show them to your students at the end of the course in
comparison to their final assessment. CUNY placement tests are untimed. Try to allow your students
the time they need to do their best work.
1. Each correct input/output is worth .5 points.
2. Each correct response is worth .25 points.
3. 9
4. –24
5. c
6. b
7a. 12 7b. undefined 7c. 4.6
7d. 3 7e. –10 7f. –11
7g. 8 7h. 7 7i. –3
8. 20 degrees
9a. 60
9b. This item is difficult to score. Do not give any credit if a student simply restates the meaning of
the variables without mentioning the numbers — ―x represents the price in dollars and and g(x)
represents the number of walkie-talkie sets shat she will sell.‖ You are instead looking to see if
the student can appropriately attach the numbers to their context — ―If she sells the walkie-talkie
sets for $30, she will sell 20 sets.‖ Half-credit (.5) may be awarded if the student mechanically
attaches the numbers to the variables but does not really link them together — ―30 is the price. 20
is the number of sets sold.‖
10. d
11. a 12. 3x or )3(x 13. 94 14. c 15. e
16. a 17. 3x 18. e 19. c 20. 7
2
x 2 4 8 –9
y 12 –6 30 21
Yes No
Is 5x an example of an equation? x
Is 5x an example of a binomial? x
Is 5x an example of a trinomial? x
Is 5x an example of a polynomial? x
Is 5x an example of an expression? x
Is 5x an example of a function? x
Is 5x an example of two like terms? x
Is 5x an example of a GPA? x
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
23
21. Perimeter = 2222 yx Area = xyyx 38
22. 4x 23a. )8)(1( xx 23b. )3)(5( xx 23c. )4)(4( xx
24. c
25. e 26. d 27. 8
5
28. Students must include at least two correct solutions, draw a line that connects them, and include
arrows at either end to indicate it continues in both directions. Half credit for the solutions, and half
credit for a correct line with arrows.
29. Students must include all seven correct solutions that fit on the xy-grid, draw the curve that
connects them, and include arrows at either end to indicate it continues in both directions. Half-credit
for the solutions, and half credit for the correct curve with arrows.
30. d
31. A student earns full credit for graphing both lines, showing the intersection, and correctly identifies
the ordered pair. A student earns half-credit (.5) if he/she identifies the correct ordered pair (2, 5)
but does not correctly graph the lines or show the intersection.
32. e
33. Each correct response (or non-response) is
worth .25 points. The maximum score is 1
point.
34. slope = –5 y-intercept = 1
35. slope = .25 y-intercept = –2
36. slope = 2
3 or 1.5 y-intercept = 1 (If possible, please add arrows before giving the test.)
37. 3
2 (the negative sign may also be attached to the numerator or denominator)
38. c 39. a 40. 135 xy
41a. 8x 41b. 9x 41c. 9x 42. d 43. b
44a. 2
1 44b. 3212
45. $315 46. $1,400 47. $1,617
Total possible points = 67
Linear Non-Linear
xy 2 x
1)( 2 xxf x
123 xxxy x
5)( xxh x
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
24
CUNY Start Core Math Curriculum, Class 1
Class Contents N EA F
1
Number Puzzle 1
Core Math Expectations and the CUNY Start Math Binder
Uses of Parentheses
Signed Numbers 1: Addition
Best Buy Commissions
Number Puzzle 1
The Number Puzzles were adapted from College Preparatory Mathematics. Eventually, we will
use these puzzles so that students may practice combining, multiplying, and factoring expressions.
Even when limited to numbers, though, these are an engaging way that students may practice
factoring numbers and do decimal and signed number arithmetic.
This is a good warm-up for your students as they enter the room on the very first day of class.
Rather than sitting around in awkward silence for the bulk of the students to arrive, students can
get started doing math right away which sets a good tone. With written directions, they hopefully
will not need much attention from you in order to get started.
This and related examples (where no bases are given) foreshadow a skill
students will need near the end of this course when they factor trinomials.
Avoid mentioning any of this to your students. We hope they will discover
this connection themselves later on.
Many math teachers will solve this particular puzzle by focusing on the
product of 24 because that involves fewer pairs to test (4) than the 7 pairs
whose sum is 14. Teachers often make this decision very quickly (perhaps not realizing they are
even making a decision) while some students will opt to test the pairs whose sum is 14. Of course
there is nothing wrong with this method and students should not be discouraged when they
proceed in this way. In the class discussion of this example, though, it is important to ask students
how they solved the puzzle so that both methods can be described. Students themselves will
notice that one method includes fewer options.
Students can be weak in organizing their work/thinking, and so when you see a student who is
methodically writing down pairs of numbers that he/she is testing, praise that in front of the other
students. More typically, students will write down a pair and then erase it when it does not solve
the puzzle. Encourage students to keep a list and cross out the pairs that do not work, both to keep
track of what has already been tried, and also because it gives you and the student important
information on what was done later on. Math handouts covered in student work may not be
attractive from a student‘s point of view, but they are very valuable to the teacher who is curious
about understanding student thinking and to the student who may forget later on how to complete
an item.
x
+
36
2014
24
+
x
3
x
+
3
3
18
+
x
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
25
Most of our students are weak in decimal understanding and computation. When some students
have reached the decimal number puzzles, we recommend calling the class to join an all-group
discussion that begins with you writing $2.74 on the board and asking students to tell you what
they see.
Ask students if they can create this amount of money using only
dollars, dimes, and pennies. They will most likely tell you they used 2
dollars, 7 dimes and 4 pennies. (It is nice to push for other possibilities
as well.)
Direct students to begin taking notes on a separate sheet of notebook
paper at this point. Some students will not have notebook paper with
them. Bring some extra sheets to share with them, or gently ask fellow students to share a few
sheets with their classmates this time.
Point at each of the numbers in the original quantity, and
with the following questions, students should help you to
label the place values. See the figure.
This 2 is the number of what?
This 7 is the number of what?
This 4 is the number of what?
It is helpful for students to associate the tenths place as the number of dimes, and the hundredths
place as the number of pennies. Emphasizing the terms ―tenths‖ and ―hundredths‖ is not
important here. Teachers have lectured students many times over the years about the names of
these places, without successfully focusing student attention on their relative sizes.
Follow this up by asking about another example or two of money written as a decimal number,
such as $7.16 and/or $3.09.
Once students imagine decimal places as dimes and pennies, gauge their understanding with a
question or two such as the following. These comparisons confound students who have little
conceptual understanding of decimals, but when thought of as the number of dimes and pennies,
they become much easier.
Which is larger, .4 or .08? Why?
Which is larger, .6 or .60? Why?
The second of these two questions can be a real eye-opener for students. Many students have been
told without any meaningful justification that they can add a zero to .6 and it means the same
thing. Students may believe their teachers in these instances but belief should not be mistaken for
understanding. With the dimes/pennies formulation, .6 = .60 will make sense to some students for
the first time.
Reinforce, reinforce, reinforce. As you encounter decimal numbers in this and other activities, no
matter if one or two decimal places are shown, point to one of the numbers and ask ―What is this
the number of?‖
2 dollars 7 dimes
4 pennies
74.2$
74.2$
pennies
es
dollars
4
dim7
2
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
26
After this discussion of money and decimal
places, students may have enough understanding
to calculate the sum in the following examples.
The thing that might trip them up is the meaning
of 10 and 8. Because no decimal point is visible,
they may feel uncertain about the meaning.
Ask students if they have seen and understand
each of the following prices that might appear in a store. This will quickly remind them based on
their own experience that a number written without a decimal point can be imagined as a number
of dollars and that the decimal point is ―hiding‖ to the right when it is not shown.
This discussion should ensure that students are all capable of determining
the sums without a calculator. You might draw a few new examples on the
board to assess this. At this stage, only focus on the sum.
In order to determine the products for the decimal
puzzles in the second row, students can of course use
the traditional algorithm for multiplying decimals. The
only trouble is many students struggle with these
procedures (especially placement of the decimal point)
and have little or no ability to judge the reasonableness
of their answer. This problem is an ideal place to assist
students in deepening their understanding of
multiplication so that they have alternatives to the
standard algorithm and so they have a ―reasonableness
check‖ for any method they choose to use. This discussion of multiplication is a critical one
because related ideas will appear in several sections of this course. It cannot be skipped.
Ask your students to do the following multiplication:
53
They will be able to tell you the product is 15, but then we must ask them — ―Why is it 15?‖
10.5
x
+
8 1.5
+
x
30
x
+
3119
60
+
x
Handbags
$5
Handbags
$5.00
3 .15
+
x
10.5
x
+
8 1.5
+
x
30
x
+
3119
60
+
x
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
27
Our goal in this discussion is to relate multiplication to adding groups. Once a student describes
the multiplication in this way, have them add it to their notes:
53
Three groups of five = 15
Of course it is important to also have students articulate that 53 may also be interpreted as ―five
groups of three.‖
We can now combine two ideas developed in this activity to determine the product of numbers
involving decimals — translating decimals into money and the idea of multiplication as groups.
5.18
Eight groups of $1.50
A student now has an alternative to the traditional algorithm. Virtually all students can add these
amounts quickly and accurately.
$1.50 $1.50 $1.50 $1.50
$1.50 $1.50 $1.50 $1.50
The money/groups formulation will give students who struggle with the standard algorithm an
alternative, but this thinking also helps students who prefer that more traditional approach. When
a student sticks to the standard algorithm, there will be a time when the decimal point must be
placed. The student will have four choices:
.120 1.20 12.0 120
A student who has ―eight groups of $1.50‖ echoing in her head, even if she does not do the
addition, will be more likely to catch an error in decimal placement because three of the four
possibilities are so obviously wrong.
There are other ways to imagine this same multiplication problem, including ―one-and-one-half
groups of eight.‖ Based on the time you have with your students, consider exploring this
alternative as well. And yes, not all decimal multiplication problems can be accurately solved in
this way (think 6.105.6 ) but estimation techniques can often be used in these situations to
continue to give students a good idea of a reasonable result.
You must carefully introduce the dimes/pennies and multiplication-as-groups formulations here
because these ideas are going to reappear throughout the course.
Core Math Expectations and the CUNY Start Math Binder
Carefully discuss Class Expectations or your version of this handout. In addition to trying to
prepare students to pass CUNY‘s math placement exams, we hope to enable students to use the
language and reasoning of mathematicians. We believe that math and math learning can be
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
28
enjoyable. We hope that this will be the case when we emphasize students‘ understanding and not
simply a series of math rules to memorize.
Notice that ―attendance‖ and ―homework‖ appear under the topic ―supporting one another.‖ This
is because strong attendance and homework by all students lead to a more successful classroom.
When students miss class or fail to complete their homework, it holds up the rest who do not need
to see an idea re-introduced. A student who misses class may benefit from the generosity of a
teacher who agrees to meet separately to review missed material, but even in this case, the
important conversation between the full class of students and teacher has been lost forever.
Extra Practice will generally be assigned in every other class session. It is designed to give both
the student and the teacher the opportunity to find out how well the student has mastered recent
objectives and retained older ideas. When possible, we will begin extra practice in class. We
hope that students will not have too too much work to complete outside of class.
Occasionally, ―challenge problems‖ appear at the end of the extra practice. We do not expect all
students to be able to solve these problems. Solutions to challenge problems will generally appear
upside-down at the end of the homework.
All of our work (in and out of class) will be collected in the CUNY Start math binder. This
includes loose-leaf paper for class notes (that students need to provide) and all handouts from the
instructor. Give out the binders to students (if you have not already) and guide them to date and
organize the handouts that they have already received in this first class. CUNY Start students are
going to build their own math textbook. Students should keep every piece of paper and every
handout should be hole-punched, dated, and logged in date order. No papers from other classes
should appear in the CUNY Start math binder. You might consider ―grading‖ math binders
periodically in the course for completeness and orderliness. This can prepare students for science
courses at CUNY where grades on lab notebooks are a part of the overall course grade.
Have extra pencils available to show students that pencils are the writing utensils we will use
throughout the course.
Calculators will occasionally be used in the class, but the vast majority of CUNY Start math is to
be done without a calculator. When calculators are to be used, they will be mentioned specifically
in a problem. Calculators are now permitted on the CUNY math placement exams, but we have
heard that the math scores have actually declined since calculator use was allowed. Most of our
students need to develop better number sense and better ability to do basic arithmetic on their own.
Uses of Parentheses
Your students should return to their notes. Ask them the following:
What can parentheses indicate in math? (or in math problems?)
As they give ideas, encourage them to give a mathematical example as well as describe the use in
words. The first three uses (at least) should make it on to the board and into their notes. The
fourth is not critical now, but if it comes up, that is fine. We will speak much more about ordered
pairs later.
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
29
1. Grouping — we use parentheses to group numbers as a way of influencing the order of
operations. Ex.: )32(7 , )36(10 , or )14(20
2. Multiplication — we use parentheses to indicate multiplication. Ex.: )3)(5( or )3(4
3. Separation — we use parentheses to separate a number‘s sign from an operation. Ex. )2(3
4. Ordered pair — we use parentheses to indicate the coordinates of an ordered pair. Ex.: (2, 3)
Signed Numbers 1: Addition
Fluency with signed numbers will be very important in College Transition Math. Because our
students first learned about signed numbers in various classrooms and countries with different
methods and different educational interruptions, it is impossible for us to try and reinforce each
student‘s signed number arithmetic using the method they each first learned. Our only real choice
is to carve a method of thinking about signed numbers that we will use in this course. Certainly,
students can also use other methods. We can tell them that mathematicians need to be flexible
enough to think about mathematical ideas in more than one way. We ask here that they develop a
way of talking about signed number addition using the idea of ―money held‖ and ―debts‖.
―Money held‖ and ―debts‖ will be used as
the primary device for building conceptual
understanding around adding signed
numbers. Lead a short discussion with the
whole group in order to introduce this
language to the students. See the example.
Occasionally take the opportunity to point
to the parentheses and ask students what they indicate (in this instance, separation).
Put a few more examples on the board and ask students to explain the sum using the language of
money and debts. Make sure to include one example with a positive and a negative, an example
with two negatives, and a third with two positives.
When you first introduce an example with two negatives such as )9(3 , at least one of your
students will probably object to a result of –12 with the following:
―Wait — isn’t negative and negative a positive?‖
Even if no student raises this question, one or more of your students are probably thinking it. Do
not try to discuss or teach multiplication rules here. We are going to look at those much more
carefully later. For now, gently tell the student that he/she is remembering a rule for a different
type of problem and to not rely here on any rule but instead to focus on what makes sense using
money held and debts. When they do this (one $3 debt and another $9 debt), it will make sense
that the result is negative.
1)2(3
$3 you hold
a new $2 debt
$1 left after paying the debt
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
30
Do not lecture about signed number addition rules here. We do ask students to make their own
generalization in the handout, but in general, our focus is on them using the money context as a
way of imagining and calculating these sums. This money context gives students more conceptual
understanding than rules such as ―for same signs, add and keep the sign.‖ Those mechanical
procedures are easily jumbled in students‘ minds and then misused.
As students are working on Signed Numbers 1: Addition, circulate and ask as many students as
possible to describe a problem using the language of money held and debts. Gradually, decimals
appear in the examples. Hopefully the work done with the Number Puzzles will lead to strong
student work here. Ask questions to reinforce what each digit represents in terms of money.
In the last part of the front, after getting examples from students, ask if anyone knows a name for
these pairs of numbers. After discussing, have students write ―opposites‖ by these pairs on the
handout. What do these problems show us about opposites? (Opposites always add up to zero.)
The problem asking students to generalize about the sum of two negatives is important in several
respects. It is the first time we are asking students to write about their reasoning. We provide
enough lines and space that students should realize we are looking for more than a one-word
answer here. Give students time in pairs to discuss and respond thoughtfully to this. If you treat
this problem superficially and permit them to write little or nothing, you are signaling to them that
writing will not be valued in the course. That is not the tone we want to set at the outset. Some
students will write about a specific example they provide and in this instance you should press
them to respond as generally as the prompt. A model response will look something like this:
No, the sum of two negative numbers is never positive. When a person has one debt and
then takes on another debt, he/she will be further in debt. Debts are always negative.
At some point, it can be useful and clarifying for you and your students to get in the habit of
saying ―negative two‖ for the number –2, rather than ―minus two‖. Especially when we work with
subtraction in the next class it will help to clarify when we are talking about a sign versus an
operation. Certainly in the public schools it is currently more in fashion to refer to the sign of –2
strictly as ―negative‖ and not as ―minus‖ as was more common a few decades ago. Some
professors may still say ―minus two‖, though, and so an explicit conversation about this is
probably a good idea.
Best Buy Commissions
It is fine to have calculators around for this activity. Once students have completed the table,
discuss their solution methods. For the trickier ones where the weekly pay is given, some students
will work backwards using inverse operations to determine the missing numbers of cameras while
others will use trial and error and perhaps use information already in the table to guide their trial
and error. Highlight all of these legitimate solution methods in order to set the tone early on that
there is often more than one correct way to solve a problem. If you tell students they should use
inverse operations, they may believe you, but it will not make sense to some of them. Working
with inverse operations is not the objective of this activity, and so you do not want to get off track
here. The important thing is for all students to be able to complete the table with or without the
calculator using whatever method works for them. Once you have discussed their solution
methods, turn the sheet over and have them recopy the weekly pay amounts for the appropriate
inputs. They should leave the center column blank until you are ready to move forward together.
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
31
Starting with 2 cameras sold (not 0 or 1 because they are a bit special), ask a student to remind you
how they calculated the weekly pay. Write the calculation in the center column. Also do this for
10 cameras and 16 cameras. It does not
matter if they choose to start with the 150,
except that it will introduce an order of
operations issue. Once you have an
expression in the middle column, try to
remain consistent. Ask students to
describe, in general, what you do to the
number of cameras to determine the
weekly pay. Once a general pattern has
been described, ask if it holds when the
number of cameras is 0 or 1. Some will
not realize the multiplication still applies
in the case of 0 and 1, and it is important
for them to see that they still fit the
pattern. Complete the table in this way.
Once you have discussed the pattern and written ―X 18 then +150” in the table in the second row,
ask your students if they know what words mathematicians use to describe our ―starting‖ numbers,
the ―ending‖ numbers, or the repeated set of operations that we use to get from one to the other.
When they or you introduce these terms (you can use ―function‖ or ―function rule‖), add the titles
that you see in the first row of the table. We do not need a formal definition of function here —
please do not mention the vertical line test or formal characteristics such as ―each and every input
is associated with exactly one output‖. Our goal here is to give a friendly idea of a function as a
set of starting numbers (inputs) that all go through the same consistent rule to produce a set of
ending numbers (outputs). See the completed table below as an example.
Inputs Function Rule Outputs
Digital Cameras You
Sell in One Week X 18 then +150
Your
Weekly Pay
0 150180 150
1 150181 168
2 150182 186
10 1501810 330
16 1501816 438
Do not introduce variables or equations at this stage. These are going to be phased in across
subsequent classes.
Digital Cameras You
Sell in One Week
Your
Weekly Pay
0 150180 150
1 150181 168
2 150182 186
10 1501810 330
16 1501816 438
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Number Puzzle 1
Adapted from College Preparatory Mathematics
The Number Puzzle always includes four circles. The circles on the left
and right are our two ―base‖ numbers. The top circle is for the product of
the two base numbers. The bottom circle is for the sum of the two base
numbers.
In each Number Puzzle, you are given two numbers. Your task is to
figure out the missing numbers. Good luck!
x
+
36
2014
24
+
x
3
x
+
3
3
18
+
x
10.5
x
+
8 1.5
+
x
30
x
+
3119
60
+
x
9
7.5
7.5
x
+
50
22.5
+
xx
+
15
6.5
4
+
xChallenge! Challenge!
6
+
x
3
base
number base
number
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33
CUNY Start Math Class Expectations
1. Introductions
2. Our objectives
The language and reasoning of mathematicians
More than formulas and rules
CUNY placement tests
3. Supporting one another
Attendance
Homework
Extra Practice
Check-in
Challenge problems
Working together
4. Our materials.
The CUNY Start math binder
3-hole punched notebook paper for notes
Pencils
Calculators
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34
Dim sum is the name of a Chinese cuisine that
includes a wide range of small dishes served
alongside Chinese tea. Dim sum has nothing to
do with mathematics.
Signed Numbers 1: Addition
Calculate each sum, or fill in the blank to make a true equation.
a. )2(4 b. 54
c. )4(3 d. 35
e. )4(7 f. 6___1 g. 5)2(1 h. 1)3(4
i. 96100___ j. )50(100 k. 65.2 l. )5(20
Provide different pairs of numbers that satisfy each equation. At least one pair must include a decimal
number.
a. 0________ b. 0________ c. 0________
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Create three of your own signed number addition problems, and then solve them. At least one of your
problems must include a decimal. Challenge yourself!
a. b. c.
Can the sum of two negative numbers ever be positive? Defend your answer with at least one example
and a written explanation using the language of ―money‖ and ―debts.‖
Calculate each sum, or fill in the blank to make a true equation.
a. )14(10 b. )2(000,1 c. 408 d. )40(4
e. )2.7(2.7 f. 10)40(___ g. 5.25.4 h. 5.325.6
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36
Best Buy Commissions
You take a job at Best Buy selling digital cameras. Your base pay is $150 per
week. For each digital camera that you sell, you earn an additional $18.
Complete the table.
Digital Cameras You
Sell in One Week Your Weekly Pay
0
1
2
10
16
$366
$492
$546
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
37
Digital Cameras You
Sell in One Week
Your Weekly Pay
0
1
2
10
16
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
38
CUNY Start Core Math Curriculum, Class 2
Class Contents N EA F
2
The Many Faces of Function Rules and Tables, part 1
Signed Numbers 2: Subtraction
Extra Practice
The Many Faces of Function Rules and Tables, part 1
This activity is designed to transition students from function rules written in words to function
rules written as equations. Even before this switch, though, there is an opportunity on the front
side of the handout to do a little instruction on inverse operations. Remember, though, an
understanding of inverse operations is not necessary here and we do not want to give students the
impression that inverse operations are the ―right‖ way to do these problems.
The best place to do this instruction is in the second table where the function rule is ―Multiply by 4
then add 6.‖ Where students need to calculate the input that will produce an output of 86, you
should not simply tell a student that they should subtract and divide. That will make no sense to
many students who will be looking at the rule. Instead, try the following:
Instructor: What is the rule?
Student: Multiply by 4 then add 6.
Instructor: What is the last thing you do to an input number, according to the rule?
Student: Add 6.
Instructor: Now look at that output of 86. Just before we added 6, what was it?
Student: 80.
Instructor: So, what do you think my input should be to get me to 80?
You have assisted the student in working backwards part of the way only by asking questions and
without making any statements. This is a terrific way to give students additional tools to solve
these problems that will reappear throughout the course. Notice that the instructor did not use the
word ―subtraction‖ and certainly did not say ―inverse operations‖.
It will be up to you whether or not you think students will be able to make the transition between
side one and side two without an all-class discussion. A discussion before they move on may be
best, though it can be difficult to keep the quickest ones from moving forward. Once the function
rules are presented as equations, it means we are now also using variables (even though they are
written as words here). It is not necessary for us to define or even use the word ―variable‖ here.
Students may be puzzled why the rule is shown with ―Output‖ appearing first. Of course we are
thinking about the most conventional way of expressing functions, which is to begin with an
isolated output variable. If this confuses students, help them to think about how a function works
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
39
— we typically start with input numbers. When we are
inputting a number, then, we should focus on the part of the
equation that says ―Input‖. If it helps, you might use your
hand to cover the left side of the equation so that they can see
the rule in a way that is more similar to the earlier non-
equation examples.
Again, determining missing inputs can be done using a
variety of methods. Do not prioritize or privilege the
―working backwards‖ method modeled above. It is more important that students use a method
that they feel comfortable with than use a teacher‘s method that they do not understand.
Signed Numbers 2: Subtraction
As an opening to this discussion, we need clarity on what subtraction really means. Write the
following on the board:
58 What do you see?
In the discussion that follows, students need to articulate that 8 is a sort of ―starting amount‖ and 5
is the amount we are subtracting from it. If it helps, ask students for a more ―friendly‖ way of
describing subtraction (as they probably first did when they were very young). If you then ask
which number we are ―taking away‖, it will be clearer to them.
Now, thinking about these numbers in terms of money as we did in the last lesson,
would you say subtracting (taking away) 5 makes you better or worse off?
Students need adequate practice identifying the number that is being subtracted.
What are we subtracting in each of the following? Do not try to do the subtraction.
14 83 )2(5 )9(7
For these examples, you have an important opportunity to have students distinguish between
negative signs and subtraction symbols.
―Are we subtracting 3 in the second example?‖
―No, we are starting with –3 or a $3 debt. We are subtracting 8.‖
What did you say we are subtracting or taking away in the last example? (If necessary, cover the
–7 in the last example with your hand.) How can we imagine the –9? Does it seem like taking
away a $9 debt makes you better or worse off? Why?
This last comment can be an effort to casually plant the idea in students‘ minds that subtracting a
positive number will make you worse off (when thinking of money), and perhaps subtly prepare
students for the idea that subtracting a negative number makes you better off (not an idea they
normally accept easily).
Output = Input x 3 + 10
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
40
After that opening conversation, our critical objective is to help students connect subtraction to
addition. Write the following items on the board and ask students to copy and simplify them in
their notes — in the same order. They should leave a few lines between the first and second rows.
58 110 5.29
)5(8 )1(10 )5.2(9
Before talking about any patterns, establish what number is being subtracted in each case. After
this is done, and you have simplified each, look at the work on the board and ask them ―What do
you see? What’s going on here?‖
358 9110 5.65.29
3)5(8 9)1(10 5.6)5.2(9
Do not rush things here, but press students for clear, accurate statements. Rather than accept a
student comment such as ―It‘s the same,‖ push for precision about what is the same for each pair
and what is different. Look for statements such as ―We start with 8 in both problems, and we end
up with 3 in both problems,‖ or ―In both cases, we are taking away 5.‖ This last comment is tricky
but good. The upper problem can be described as ―taking away‖ because it is subtraction. In the
lower example, though, we have addition. Still, students‘ understanding of –5 as a debt can help
them see this also is a form of taking away, even though it is addition.
Ultimately, it will help if students can see that ―subtracting 5 is the same as adding –5‖, and
similarly with each pair. Our goal is to reach the following conclusion:
Subtracting a number is the same as adding its opposite.
Be prepared for student frustration in this discussion. Most students are accustomed to being told
what the important math relationships are, and are not accustomed to being asked to look and
describe patterns themselves. Push for clarification and confirmation from other students.
A pedagogical goal here is not to see how helpful we can be, but to see how little help we can give
while students reach this conclusion themselves.
Once the conclusion has been stated and written in student notes, give your students additional
subtraction problems and ask them to transform them into addition problems and calculate the
answers. Use a good mixture of situations, such as:
72 72 27 )7(2 )7(2
When reviewing these examples with your students, keep pushing for clarity on what is being
subtracted, which symbols are operations, and which are signs.
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
41
The example of 27 can be useful because students can do the calculation as subtraction to
verify that their conversion to addition worked.
Some of your students arrive in your class with a method for subtracting signed numbers. One
popular method is ―keep, change, change‖. For an expression such as 105 , this is used in the
following way — ―keep‖ the 5, ―change‖ the operation to addition, and ―change‖ the 10 to –10.
Another common method is ―multiply the signs‖. An expression such as )7(2 is written as
72 because when you ―multiply the two negatives‖, the result should be ―positive‖.
These are examples of signed number rules that often do not carry much meaning for students. In
these and some other methods, signs and operations are equated. Of course signs and operations
are related, but they are not exactly the same thing. Multiplying signs is also odd when, really, it
only makes sense to multiply numbers. The second example also requires students to know signed
number multiplication rules before doing subtraction, which is an atypical lesson order. The
biggest problem with these rules/tricks, though, is that they are often misused by students. For
students who are not excellent in their own signed number subtraction, we should gently insist that
they use the CUNY Start approach. Even for students who use those rules correctly, though, it is
important that they learn and can use this method because it reveals the critical link between
subtraction and addition. It will make them understand why ―keep, change, change‖ works.
Depending on your goals and how your students reacted to the ―worse off/better off‖ notion in the
introduction to subtraction, you might have them look back at an expression (after simplifying it)
and ask whether they would be better or worse off after subtracting. For example:
72
How much are we subtracting? What does that mean in terms of money? Are we taking away
money or taking away debt? (We are taking away $7.) Is that good or bad for you? (You are
going to end up worse off after the subtraction.) Where did you start? (With a $2 debt.) Where
did you end up? (With a $9 debt.) Did you indeed end up worse off than you started?
)7(2
How much are we subtracting? What does that mean in terms of money? Are we taking away
money or taking away debt? (We are taking away –7, or taking away a $7 debt.) Is that a good
thing or a bad thing? (You are going to end up better off after the subtraction — I like it when
people get rid of debts for me.) Where did you start? (Having $2.) According to our work here,
where did you end up? (Having $9.) Did you indeed end up better off than you started?
If students can have this much understanding before they begin the calculation, they will be in a
position to assess the reasonableness of their work. If we only have them mechanically turn
subtraction problems into addition problems, we may not put them in a position to judge the
reasonableness of their answer. However, we do not want to lead students to try to figure out the
answer to all subtraction problems in terms of money. For some problems, that will be more
confusing than helpful for many students.
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
42
For example, )7(2 is difficult to think about as subtraction. If you only owe $2, how can we
take away a $7 debt? Similarly, for )7(2 , if you start out having $2, how can we take away a
$7 debt? Where is the debt to be taken away? We should not emphasize thinking of subtraction in
terms of money, but instead just use a money context to check if an answer is reasonable in terms
of our ―better off/worse off‖ formulation.
The most common error in these subtraction problems occurs when a student changes subtraction
to addition but does not change the number subtracted to its opposite. For example, 72
becomes 72 . When you see this, ask your students ―Is subtracting 7 the same as adding 7?‖ or
―Is taking away $7 the same thing as giving you $7?‖ Again the example of 27 can be helpful
here, or you could also pose a new pair of examples like 520 and 520 and ask if they are the
same or equal, and follow up by asking if subtracting 5 is the same as adding 5.
Extra Practice #1
Take a moment to identify problem #5 as an important part of the homework. Point out that it
relates to Best Buy Commissions from the first class. Encourage them to explore this problem
deeply — it is more complex than it first appears. Reiterate to them that you expect a written
solution that includes supporting calculations, a table, and maybe even a graph on a separate sheet
of notebook paper. It will not be acceptable for them to try and cram their ideas on the homework
paper. You can tell them that you will be collecting their work and giving them a written response
in the following class session.
This is really a system of equations problem where the better workplace depends on the number of
cameras you expect to sell. The comparison can be shown using tables of values, with a graph,
but can also be done in a purely text-based response. Watch out for student tables that only use
the inputs given in the Best Buy problem (0, 1, 2, 10, and 16). Emphasize specificity in student
writing. The goal here is for students to make clear arguments that are based on the data. A
student could decide one is clearly the better option (and not conclude that ―it depends‖) if they
defend their assertion. For example, a former student wrote ―I am a very introverted person and
wouldn’t be good at selling so I would prefer working at the Digital Source where my guaranteed
pay is higher.‖
Do not introduce systems of equations, system solutions, slope, or intercepts at this stage. All of
these concepts will be carefully developed in the course and the Best Buy and Digital Source
problems will reappear several times as a way of illustrating those concepts.
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
43
The Many Faces of Function Rules and Tables, part 1
1.
2.
Input Rule: Multiply by 20 then subtract 5. Output
1
2
5
75
135
Input Rule: Multiply by 4 then add 6. Output
0
2
8
86
16
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
44
This function rule is the same as
the following:
Multiply by 3 then add 10.
3.
4.
5. Output = Input x 5 + (–10)
Input Output = Input x 3 + 10 Output
0
10
12
61
121
Input Output = Input x 2 + .75 Output
1
2
3.5
4.5
30.75
11.75
Input 0 1 2 5 100
Output 90 25
Original material under copyright, 2009 CUNY College Transition Initiative Core Math Curriculum
45
Signed Numbers 2: Subtraction
1. In each case, what are we subtracting? (Do not do the actual
subtraction.)
a. )2(4 b. 24
c. 78 d. 100x
e. a3 f. 10b g. )10(83
2. Complete any missing numbers in order to create true equations. Include parentheses where
appropriate to separate operations and signs.
a. )6(8___8 b. ____121012
c. ____2)3(2 d. 1824___24
e. )5.6(10___10 f. ____383
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46
3. Rewrite each of the following using only addition and then simplify.
a. 104 b. 54 c. )4(3
d. 35 e. )4(7 f. 24
g. 5)2(1 h. 1)3(4 i. )100(4
j. 65.8 k. )5.4(25.6 l. 9982
m. 5.18 n. 5.18 o. )5.1(8
p. )5.1(8
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47
Extra Practice #1
1. Do not use a calculator. Look at your class notes if you need to be reminded of the various ways
that we may do these types of decimal calculations.
2. 3.
Input Output = Input + (–3) Output
–2
0
5
8
–10
15
Input Output = Input + 8 Output
–14
–1
–2.5
5
12.25
–20
32.253.5
x
+
18
1.5
+
xx
+
15.57
12
+
x
.25
2
4
x
+
64
20
+
xx
+
7.2512
36
+
x
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48
4.
5. A store called The Digital Source is trying to lure you away from
working at Best Buy (see the activity Best Buy Commissions from
class). The manager at The Digital Source comes to you with the
following proposal:
―If you come and work for me, I will pay you according to the following function.
2008 InputOutput
In this function, the input is the number of digital cameras that you sell in one week, and the
output is your weekly pay in dollars.‖
Should you take this job offer? Use a separate sheet of paper to develop your response. This
page will be collected in the next math class. Include your calculations and any supporting
information (such as a table) that you feel will clarify your reasoning. Assume that the person
reading your paragraphs has not seen any information about the two stores before.
6. Rewrite each of the following using addition and then simplify. Do not use a calculator.
a. 123 b. 123 c. )12(3 d. )12(3
e. )5.4(7 f. 2100 g. 25.62 h. )500,1(000,1
Input Output = Input x Itself + (–5) Output
0
1
6
95
59
76
The Digital Source
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7. Complete each of the following in order to create a true equation.
a. 6___3 b. 9)4(___
c. 5.12___8 d. 5___0
8. Create true equations using one negative number and one positive number.
a. 12______ b. 5.4______
Challenge
Frederick is working on a number puzzle and discovers that the product of the two
base numbers is exactly twice as large as the sum of those same base numbers.
Assuming that the base numbers are whole numbers (no negatives), can you figure out
any pairs of base numbers that Frederick could have been looking at?
x
+
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CUNY Start Core Math Curriculum, Class 3
Class Contents N EA F
3
Review Extra Practice
The Many Faces of Function Rules and Tables, part 2
More Signed Numbers 2: Subtrraction
Review Extra Practice
Review the Number Puzzles. The first one is straightforward. Write the second
puzzle (pictured at the right) on the board. If the first student who explains this
uses traditional algorithms, look also to highlight a student whose solution
method uses money. For the product, press students to find at least three ways
to do this calculation. Show these clearly on the board so that students may
compare them. What do your students think about the relative strengths and/or
weaknesses of the different methods? Encourage students to record these
methods in their notes, and not to simply cram them in the homework handout.
Method #1 – the traditional algorithm. Method #2 – ―Twelve groups of $3.50.‖
Method #3 – ―Three-and-one-half groups of 12‖
The third Number Puzzle can also be solved in all three of these ways, though there are going to
be some uncertain students if you say ―one-quarter of a group of 32.‖ It will likely be easier to
think of 32 quarters. How many do we need for $1? How many $1 groups will we have?
32.253.5
x
+
18
1.5
+
xx
+
15.57
12
+
x
42.0
1
70350
3.5 x 12
7x6=$42
or, add the 3s and then add pai rs of .50
$7
$3.50 $3.50$3.50 $3.50$3.50 $3.50$3.50 $3.50$3.50 $3.50$3.50 $3.50
one-half of 12
42
12 12 12 + 6
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The fourth Number Puzzle also deserves a whole-group conversation as a
follow-up to the three methods shown above. See if students can describe what
we need to do here using not only traditional algorithms, but also our idea of
grouping.
Method #1 – Divide 1.5 into 18. Very difficult for students — you can write it on the board as
185.1 , but we do not recommend spending class time going through this calculation. Either
students already have this skill, or they should be encouraged to use one of the other methods.
Even for students who can do the long division, an understanding of the alternatives deepens their
number sense and will help them judge the reasonableness of their answer. If they know from the
outset that we can also think of this as the number of groups of $1.50 that make $18, they will
know that something is wrong when they finish the long division and have a result of 1.2 or 120.
Method #2 – ―How many groups of $1.50 will it take to have a total of $18?‖ This is manageable,
especially if students simply write $1.50 over and over. They should naturally start to group them
into $3 chunks. Watch out that they conclude the number of $1.50 groups and not $3 groups.
Method #3 – ―One-and-one-half groups of what number makes 18?‖ This is a good opportunity
for students to use trial-and-error. How about 10? No? Should my next guess be higher or
lower? Why?
When you review one or more of the first few Extra Practice sets with your students, have a
discussion with them about what they might want to do and write when homework (or any other
handout) is being reviewed in class. Ask them what they can do or write that will help them use
the handout to study in the future. We want to be sure that students correct any errors on these
problems, but it would be nice if some students recognize the benefits of leaving their errors on
the page as well for studying purposes.
Ideally they will also discuss marking particular problems that they may want to look back at later.
Which problems would you mark? How would you mark them? They might suggest many ways
to mark problems — highlighting, circling the problem, drawing a star, putting an arrow in the
margin, etc. The method of marking problems does not matter, but it would be useful for them to
think about marking problems that were particularly challenging, interesting, or new in some way.
We want to encourage students to develop these kinds of helpful academic habits, especially early
in the semester.
You will probably want to collect students‘ writing for The Digital Source without discussing their
ideas as a class. We recommend that you type a written response for each student. Usually you
will need to push them to clarify their writing and reasoning. One common feature of student
work is to consider only the same inputs that were used in the table for the original Best Buy
problem. This will prevent them from honing in on the precise instances where it would or would
not be more beneficial to work at one of the stores.
When you return their papers with your responses, you could decide to discuss the problem as a
group, or you could maintain the two-way communication you have created with each individual
32.253.5
x
+
18
1.5
+
xx
+
15.57
12
+
x
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52
student. Students who submit incomplete or imprecise work on this problem, and maybe all
students, should be asked to do and turn in a revision within the next two or three classes. A final
draft of a response to this problem is an ideal item to require as part of a math portfolio.
The Many Faces of Function Rules and Tables, part 2
You may want to copy this as a single 3-page handout that includes Heavenly Jalapeño Cheddar
or keep Heavenly Jalapeño Cheddar as a separate handout from the first double-sided sheet.
In the handout, ask your students what they suspect x and y represent in the first function. (If you
think that it will be difficult to keep students focused on the discussion if they have the handout,
put part of the first table on the board and discuss it before giving out the handout.) After they
have named them, write ―Input‖ and ―Output‖ in the appropriate cells above the variables. An
important goal here is to decipher the new function notation by translating it into our input/output
format. They have already identified y. Decoding 2x is next. One of your students will likely
recall that the ―missing‖ operation indicates multiplication. Ask why we do not write 2xx if the
operation is multiplication so that students see and articulate why our symbol for multiplication
must change. In addition to 2x, you may ask the students how else 2 times the input might be
written. Possibilities include ))(2( x and x2 .
Your students should lead you to write the full rule using our input/output formulation
(Output = 2 x Input + 4 or Output = Input x 2 + 4). If both of these surface, ensure that students
accept that both result in the same output. If it is needed, discuss the following equation to decide:
Insist for the rest of the problems that students not only fill in missing inputs/outputs, but that they
also translate each function rule into input/output format. Discourage them from using calculators,
especially on #5. If they need it, remind them of the concept of multiplication as groups.
Continue encouraging students to use any method they like to determine missing inputs and
outputs. Some of these problems may even be more challenging when inverse operations are
used, such as in #3.
After the second page of this handout, students should no longer use the x symbol to represent
multiplication in this course. It will only be used as a variable. Multiplication should be shown
using a dot, parentheses, or implied in expressions such as 2xy. Some students are sloppy in
writing their multiplication dots (sometimes making them look more like decimal points). Make
certain that these are written carefully.
In problems such as Heavenly Jalapeno Cheddar,
students will frequently skip the text and go
straight to filling out the table of values using the
function equation. Insist they read the description,
and encourage them to label the meaning of the
variables in the function equation. See the model.
This is an important part of focusing them on how
?
123321
profits pounds of cheese sold
8005.2 cp
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53
to make sense of functions in realistic contexts. With the function labeled, it makes it easier to go
back and interpret the results from their table.
Students should write as clearly as possible in this problem using complete sentences. Take time
to have students read and evaluate each others‘ descriptions in a and b. It is worth trying to elicit
the idea that the values in the table do not mean that these things actually happened or will
definitely happen. They only reveal what the result would be if (or when) a particular number of
pounds of cheese is sold. If students do not come to this realization easily, you could ask if the
table indicates that the company sold (or will sell) 1,000 pounds of cheese in the first month, 800
pounds in the second month, and so on. (When is that first month?)
More Signed Numbers 2: Subtraction
Before giving students the handout, put pairs of numbers like these on the board.
5 –8 Which is a larger number? Why?
–3 –10 Which is a larger number? Why?
–15 0 Which is a larger number? Why?
This conversation will be important before considering Henrietta‘s claim from the handout.
Because we have not been emphasizing the number line, it would be strange for us now to say that
numbers are larger as you move to the right on the number line. We can return to a money-based
description here instead. Students will not be confused when comparing positive numbers. For 5
and –8, they should be able to see that you are ―better off‖ having $5 than owing $8 and so 5 is
larger (better) than –8. With that formulation, they should be able to say that –3 is larger (better)
than –10 because you are closer to getting out of debt. Thinking of the numbers in terms of
temperature could also help to confirm which number is larger/greater/higher in each pair.
Encourage students to work in pairs or groups as they consider Henrietta‘s claim and the rest of
the handout.
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54
The Many Faces of Function Rules and Tables, part 2
1.
2. 15 xy
3. )20(10 xy
x 42 xy y
0
1
2
24
34
x y
0
1
2
3
56
101
x 0 1 2 7 11
y 60 30
Translate this function rule into words:
Output =
Translate this function rule into words:
Output =
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4. 34 xy
5. 475. xy
6. 12010 cp
x y
4
7
45
33
1
x y
0
1
2
5
10
12.25
c p
0
1
2
10
.5
270
620
Translate this function rule into words:
=
Translate this function rule into words:
=
Translate this function rule into words:
=
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56
Heavenly Jalapeño Cheddar
Your company makes cheese — a new kind of spicy cheese that has hot peppers
inside. Each month, you produce 1,000 pounds of this cheese, which costs you the
same amount regardless of how much of it you manage to sell. The profit that your
company earns depends on how much of it you sell compared with the production
costs. Profits can be measured using the following function:
8005.2 cp
where c is the number of pounds of spicy cheese sold, and p is the profit in dollars.
Complete the following table of values for this function.
Try to do the calculations without using a calculator.
(Think of groups!)
a. When the value of c is 400, the value of p should be 200. Explain what this means using the
context of the problem (cheese, profit, etc.).
b. When the value of c is 200, the value of p should be –300. Explain what this means in the context
of the problem (cheese, profit, etc.).
c 8005.2 cp p
1,000
800
600
400
200
0
More Signed Numbers 2: Subtraction
1. Henrietta makes the following claim:
I start with a positive number. Then I subtract a second number from it.
The result is larger than my starting number.
What do you think of Henrietta‘s claim? Discuss it with a partner. Write your ideas and sample
calculations below.
Henrietta was a student in the spring
2007 College Transition Math class.
She finished her Associate‘s Degree
at BMCC and now is working on
her Bachelor‘s Degree at Temple
University in Philadelphia,
Pennsylvania.
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58
2. For each row, compare the quantities in Column A and Column B, and then check one of the boxes to the right.
Column A Column B
The quantity in Column A
is larger than the quantity
in Column B.
The quantity in Column B
is larger than the quantity
in Column A.
The quantities in Column A
and Column B are equal.
It cannot be determined
if one is larger or if
they are the same.
3.5 3.50
–2 –8
0 –.7
–4 –4.2
3. Rewrite each of the following using addition and then simplify. Do not use a calculator.
a. 1230 b. 1230 c. )12(30 d. )12(30
Challenge!
Simplify the following. )102()153(
CUNY Start Core Math Curriculum, Class 4
Class Contents N EA F
4
Signed Numbers 3: Multiplication
Times Table Baseline Test
Expressions and Equations
Evaluating Expressions I
Extra Practice
Signed Numbers 3: Multiplication
Begin with student notes. Have your students write the following: 63
Students may know the product, but it is useful to have students review and verbalize why
1863 . Three times six can indicate ―three groups of six‖ )666( or ―six groups of three‖
)333333( , both of which total 18.
Now write a new problem on the board: )5(2
Can one of you describe this in terms of groups? There are two groups of $5 debts.
10)5(5)5(2
Give another example that reverses the order such as 47 and seek the same sort of description.
You may need to ask ―What’s the fewest number of groups that we can have of something?‖ as a
way of moving them away from trying to think about a negative number of groups.
Now it is time for the class to arrive at a rule for multiplying one positive and one negative. In
pairs, you can ask them to consider and complete the following statement:
Multiplying one positive number and one negative number
will produce a ____________ number because…
This is a very important conversation, and their writing after the word ―because‖ is more important
than the earlier missing word. As your students discuss this in pairs, ask them to think in terms of
the previous discussion. They may not think to do additional examples when stuck, and if this is
the case, you can give them this idea. As they look at additional examples, push them to describe
to each other, and to you, how to think of the example using groups. Here is a model response:
Multiplying one positive number and one negative number will produce a
negative number because collecting groups of debts will always result in a debt.
Allow a few students to read their sentences to the whole class to wrap this discussion up.
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60
Draw this function table on the board and have students copy
it in their notes:
Give students time to complete the outputs for the given
inputs. Encourage them to write the calculation in the middle
column so that they will have some evidence later of the work
that we did here. Talk about how the outputs are changing.
Are they getting larger or smaller? –15 is a smaller number
than –10 because it is as if you have less money — a worse
debt. So, as the inputs get smaller, the outputs are growing.
Label these changes in the way
that you see below.
Now it is time to continue the pattern. What do you think
the next few inputs will be if we continue the pattern? What
do you think the next few outputs will be if we continue the
pattern? I want you to ignore the function and simply
follow the patterns that we described.
Okay, now we are going to look at the function again.
What does the function tell us has to happen when the input
is –1? –2? Let’s fill in this middle column.
Looking at the results, ask students what appears to be true
when multiplying certain signed numbers.
xy 5
x y
3
2
1
0
xy 5
x y
3 35 15
2 25 10
1 15 5
0 05 0
xy 5
x y
3 35 15
2 25 10
1 15 5
0 05 0
–1 5
–2 10
–3 15
xy 5
x y
3 35 15
2 25 10
1 15 5
0 05 0
–1 )1(5 5
–2 )2(5 10
–3 )3(5 15
Inputs are
decreasing by 1.
Outputs are
increasing by 5.
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61
This last statement must be clearly written in student notes.
This is not a proof. Nevertheless this inductive strategy uses our work with functions and the
conclusion we made earlier regarding the product of a negative and positive. This is more
justification for ―negative times negative equals positive‖ than students almost ever receive.
It is time for the handout, Signed Numbers 3: Multiplication. Note that students will need to
follow the correct order of operations in order to simplify some of these expressions. We have
intentionally not included a separate section on defining the order of operations in this curriculum.
Students have been through math teacher descriptions of the order of operations several times in
their lives. We will treat it when it comes up in other work we are doing. Look out for this when
you are observing student work on the handout, and in discussions of the handout.
Times Table Baseline Test
We typically do not emphasize speed in our activities. This is one instance, however, where we
push students to move quickly in order to build fluency. An appropriate introduction to this test
and use of the flashcards will help to keep students calm about it. Here is a sample introduction:
I rely on my skill with times tables every day. I make quick calculations in my head and estimate
all the time drawing on this knowledge. Improving your speed and accuracy with the times tables
will help you in your everyday life as well as to solve math problems.
Normally we do not press you to finish problems within a time limit. Your placement tests at
CUNY are untimed. Still, we want your fluency with the times tables to improve, and the only way
to measure this is to time you. You are not graded on this. We will score your papers so that you
can measure your improvement, but you are not being compared to other students. Your goal is to
improve from where you are right now. After doing a times table assessment today, we will give
you cards to help you practice. In a few weeks, we will ask you to do another times table test.
Keep these pages face down until I tell you to turn them over. Write your name on the back. You
will have 90 seconds to do as many as you can. Work quickly but carefully. When I tell you to
stop, please put your pencil down and turn the paper back over. Remember, you are not being
graded on this. We are scoring them only to help you measure your progress.
Keep a record of the times table test scores using the log so that we can look for evidence of class-
wide improvement over the semester. Return their graded tests in the next class.
xy 5
x y
3 35 15
2 25 10
1 15 5
0 05 0
–1 )1(5 5
–2 )2(5 10
–3 )3(5 15
Inputs are
decreasing by 1.
Outputs are
increasing by 5.
Multiplying two negative
numbers produces a
positive number.
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62
Expressions and Equations
We are re-visiting this early problem in order to distinguish between expressions and equations.
Give students the handout Expressions and Equations. Have your students complete the function
table and record the function rule using x/y notation. Then have students complete the two tables
at the bottom of their paper with the information that you see below.
Examples of Equations
15018 xy
1220 x
55
12+6=18
03072 xx
What do students notice that make equations different from expressions? Include the important
difference in student notes — that equations include equal signs and expressions do not.
In student notes (and no longer on the handout), have students write the expression 15018 x .
Why is this an expression and not an equation? When we work with expressions, we occasionally
want to calculate the value of the expression — this is called ―evaluating‖ an expression.
Example: Evaluate 15018 x when 4x .
Have a student walk you through the substitution and calculations: 150418
When they feel they have a result, push them to explain their answer.
When 4x , the value of 15018 x is 222. How is this related to our work with functions?
Likely (and without you suggesting it), a student will recognize the
Best Buy function here and remark that your weekly pay is $222 when
(or if) you sell four cameras. This gives you the opening to relate this
expression to the function 15018 xy . This is certainly one of the
reasons why students are asked to evaluate expressions — to improve
their ability to work with functions. In this instance, the task of
evaluating the expression when 4x is not any different than the
process of completing the table shown at the right.
Evaluating Expressions I
If students correctly calculate the negative output for problem c and try to make sense of it, they
should be puzzled. This can lead to a lively discussion with some creative interpretations and
speculation. It also gives us the opportunity to point out that functions can sometimes — but not
always — help us to explain relationships in the world. Some inputs will not make sense here.
The fewest number of chirps per minute is certainly 0. The trouble here may be that crickets will
not likely survive at 20 degrees Fahrenheit.
Examples of Expressions
15018 x
122 x
x4
8
yx 57
15018 xy
x Y
4
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Introduction to the CUNY Start Math Textbook Index of Topics and Vocabulary Index
If you do not have time to introduce the vocabulary index (and possibly the optional index of
topics) at this point, you can do it in a short math class or in the next full-length class session.
This can happen any time from the second day to early in the second week, depending on when it
best fits in your class time.
Each blank index should be copied as a two-sided handout. Tell students to keep the index (or
both indexes if you are giving out both) in the back of their math binder.
Tell students that they will be using the vocabulary index throughout the semester to record
descriptions, examples, and sometimes definitions of important math terminology. Ask students if
there are any new math vocabulary words that we have discussed so far this semester.
Depending on when you do this activity, students may come up with ―opposites‖, ―expression‖,
―equation‖, and ―evaluate‖. As students say these words, put them on the board and tell them to
add those words to their vocabulary index. Ask what they could write as an example of each.
Ask what they could write as a description. It does not have to be a formal definition —
something as short and informal as ―does not have an equal sign‖ is fine for ―expression‖ — and
different descriptions are possible as long as they are accurate. Ask for the date and the page
number where these words can be found in their binders, and have students enter that information.
The four words mentioned above may be the only essential vocabulary words encountered so far,
but other possibilities include ―sum‖, ―product‖, ―function‖, and ―context‖. Students may even
suggest words like ―parentheses‖ or ―base pay‖, but we are generally looking for math words —
words that have meanings specific to math.
We want students to keep adding to their vocabulary index on their own, but we need to follow up
for most students to maintain it and use it effectively. You may want to discuss what words
belong in the vocabulary index once a week. Ask for student ideas. We suggest designating some
words as essential and required, while other words (like ―sum‖, ―product‖, and ―context‖) can be
optional. If you have a little time left at the end of a class session, that can be a good time for
working on the vocabulary index. Have students share their descriptions and examples. If you
only list and assign words to go in the index, some students‘ indexes will be just a list of words. If
a strong, quick student finishes a handout early, you can ask the student to update her index.
One or two classes before each test, around the time of the AlgeCross puzzles, check on the index,
eliciting a list of the essential words that should be in everybody‘s index. At least the first couple
of times that you do this, list the words on the board with the dates next to them. When you do a
binder check (such as during a test), check students‘ vocabulary indexes.
The ―Index of Topics‖ is an optional additional index that is not focused on vocabulary, but
instead serves to help students find notes and handouts on specific topics. You can decide whether
you want your students to make such an index or not. Again, remember that if you give it out but
never reinforce it, many students will not maintain this index well enough for it to be helpful.
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64
Signed Numbers 3: Multiplication
1. Write an addition problem that is equivalent to each of the
following multiplication problems.
a. 192
b. )5)(4(
c. x5
d. The product of 3 and 100 .
2. Write a multiplication problem that is equivalent to each of the following addition problems.
a. 77777
b. yyyyyy
c. )6()6()6()6()6()6()6(6
3. Find each product.
a. )3(4 b. )12)(3( c. )5)(2( d. 75
Well-known multipliers.
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65
4. Simplify.
a. 10)4(3 b. )8()4)(2( c. )64(3 d. )1)(1(4
5. Complete any missing inputs or outputs for the following function:
6. Simplify.
a. )1)(1(
b. )1)(1)(1(
c. )1)(1)(1)(1(
d. )1)(1)(1)(1)(1)(1)(1)(1)(1)(1)(1)(1)(1)(1)(1)(1)(1)(1)(1)(1(
7. What patterns can you identify in the results from the last problem? Consider all four parts of
problem #6 — a, b, c, and d.
x 0 4 5 –2
y 6 –24 0 42
186 xy
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Times Table Baseline Test (Test #1)
54 1211 53 123 104 95
96 87 119 65 1110 33
64 42 114 85 66 116
107 113 52 128 1210 32
109 105 94 62 63 124
55 99 75 108 72 98
73 125 83 76 1212 82
106 93 126 77 92 97
103 129 127 102 84 115
118 44 112 74 88 117
1010 122 22 1111 43 86
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67
Times Table Test Scores
Student Baseline Test 1 Test 2
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68
Expressions and Equations
You take a job at Best Buy selling digital cameras. Your base pay is $150 per week.
For each digital camera that you sell, you earn an additional $18.
Complete the table.
Digital Cameras You
Sell in One Week
Your Weekly Pay
0
1
2
3
4
Examples of Equations
Examples of Expressions
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Evaluating Expressions I
Field crickets chirp faster or more slowly based on the temperature.
The following function is a pretty good measure of the number of
chirps per minute depending on the temperature:
1504 tc
where t represents the temperature in degrees Fahrenheit and c
represents the number of cricket chirps per minute.
a. Alongside the function equation, label the meaning of the
variables c and t.
b. Evaluate 1504 t when 70t .
c. When 60t , the value of 1504 t is 90. Explain what this means using the function context
(temperature and chirps).
d. Evaluate 1504 t when 20t . Explain what this means in the context of this function
(temperature and chirps).
e. You are a scientist studying field crickets. You recorded some chirping but forgot to record that
day‘s temperature. Using the recording, you are able to determine that there are 170 chirps per
minute. What was the temperature at the time of the recording?
In mid to late summer, male
crickets begin chirping. The
character of their chirp provides
an indication of their past and
present health. Females evaluate
their chirps and move towards the
one they prefer. When the male
senses a female is near, he will
produce a softer courting sound.
After mating, the female will
search for a place to lay her eggs,
preferably in warm, damp soil.
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CUNY Start Math Textbook — Vocabulary Index
Vocabulary Word Example and/or Description Date &/or
Page #
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Vocabulary Word Example and/or Description Date &/or
Page #
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CUNY Start Math Textbook —Index of Topics
Math Topic Example Date &/or
Page #
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73
Math Topic Example Date &/or
Page #
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Extra Practice #2
1. Complete missing inputs and outputs for the function tables below.
2. At Bronx Best Rentals, the following function is used to calculate the
rental charge for one day:
5410. mc
where m represents the number of miles that you drive in the day, and
c represents the rental cost for the day in dollars.
a. Evaluate 5410. m when 80m .
b. When 120m , the value of 5410. m is 66. Explain what this means using the problem
context (miles and cost).
35 xy
x y
0
2
.5
.25
48
10.5
)12(6 xy
x y
4
7
2.5
0
–6
6
60
24 xy
x y
0
2
.5
3.25
46
8
Bronx Best Rentals
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3. One type of scale that measures weight is called a spring
scale. A spring hangs from a fixed position. A weight
can be attached at the bottom (see the hook on the
example pictured at the right). The weight causes the
spring to stretch, and the new length of the spring tells
you how heavy the weight is.
A function can describe the length of the spring based on
the amount of weight. The function equation for a certain
spring scale is
105.0 xy
where x is the weight in grams and y is the length of the
spring in centimeters.
a. Complete the table of values for this function.
x 2 5 8
y 17
b. What is the length of the spring when no weight is attached?
c. Evaluate 105.0 x when 7x . Explain what this means using the function context (weight
and length).
Problem adapted from Elementary Algebra
by Harold Jacobs.
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4. Calculate any missing inputs and outputs.
5. Rewrite each of the following using addition and then simplify. Do not use a calculator.
a. 85 b. 85 c. )8(5 d. )8(5
Challenge
x 63 xy y
0
–2
–4
.5
–24
–30
–2.25
7.5
13.5
-4.75
-7.5
x
++
x
10
22.75
+
x
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CUNY Start Core Math Curriculum, Class 5
Review Extra Practice
In Bronx Best Rentals and similar problems, review the written responses to questions that ask
students to describe the meaning of an input and output in the context. Some students will
misinterpret the problem and will simply describe their calculation rather than explain, ―When you
drive 120 miles, the rental charge is $66.‖
Problem #3b is tricky because students have to realize that ―no weight‖ means the input is 0.
Return Times Table Baseline Test and flashcard introduction
Return the tests and give students the Times Table Log so that they can record the products that
they missed on the test. Probably they should not record ones they did not even do because they
ran out of time. Also give students their flashcards and demonstrate how they work.
Pair students and give them a few minutes to quiz each other. Ask them to record any products
they miss in their Log. After they work a bit in pairs, ask students to use the cards alone. It is a
good idea to do just one math fact on a card at a time (and then go to the next card) because when
you turn it over to check the result, you may unintentionally see other products nearby. If you use
the cards in a stack and move from one to the next while you work your way around the outside
numbers, you should get a good mixture of problems.
As an alternative or in addition to the Log, you might suggest that students make tick marks on the
circles that correspond to products that they miss on the flashcards. This will help them when
looking at the cards to know which ones they have missed in the past.
Move around and ensure that students are working on the facts that they need to master.
Sometimes they stick with ones that they know. Try to give them a good 15 minutes to do this.
Remind students to use these cards, as well as the Log and/or the tick marks. They will not be
mentioned again in these notes until the next times table test.
Evaluating Expressions II
In their notes, students should record the following two problems and calculate the answers:
Evaluate 155 x when 4x . Evaluate 12x .
In this second example, there is not enough information to evaluate the expression. We must have
the value of x.
Class Contents N EA F
5
Review Extra Practice
Return Times Table Baseline Test and flashcard introduction
Evaluating Expressions II
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In this (second) expression, what does x represent — is it a color, an animal, a number,
something else? If it represents a number, then what kind of number can it be? Positive?
Decimal? Negative? Zero?
The point that we want to make is that in these sorts of ‗abstract‘ expressions, we must imagine
all possibilities unless we are told otherwise. In other words, x or another variable can represent
any number, including a negative number or zero. This might be a good time to differentiate
between algebra that is ―abstract‖ and algebra that is ‗in context‘. When we were working with
the function 15018 xy for the pay at Best Buy, could x be any kind of number (positive,
negative, decimal, zero)? How about when we are working only with the expression 15018 x ?
In this next example, some students in the class will likely be able to describe the operations
involved so you do not have to:
Evaluate bc5 when 2b and 3c .
(If a student is puzzled about how this could be connected to functions, you could consider showing
them that functions in some cases may rely on more than one input variable, such as:
605.3 lwc
In this instance, l could represent the length of a rectangular floor, w could represent the width, and c
could represent the cost of installing new flooring. You might challenge students to rewrite this
function using a single input variable — a for area in place of lw for example.)
Before moving to the handout, you need to do one final example with students:
Evaluate x when 10x .
What does the negative sign do to the value of x? You can also ask about what x represents. (A
number that can be positive, negative, or zero.) Putting the negative sign in front of x does not
indicate that ―x is negative‖. This is tricky. x already can be negative. By the end of this
discussion, the sign should ultimately be interpreted (ideally by students) as follows:
x means ―the opposite of x‖.
Students are now ready to begin the handout. When you reconvene after students have done that
work, leave time to discuss the last problem. Before discussing student thinking, it can be
interesting to poll the students and record the results on the board.
Allow students with differing thoughts to explain their
reasoning. It may be important for students to consider
various potential values of x for this to make sense to
them. If you can work with the class to create a table and
include number examples, it may help to clarify student
thinking.
x -x
Any number
(+ , - , or 0)
The opposite
of
the number x.
x is larger -x is larger They are equal. ?
Times Table Log
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Evaluating Expressions II
1. Evaluate x6 when 10x . 2. Evaluate )12(x when 8x .
3. Evaluate yx 2 when 5x and when 12y . 4. Evaluate 153 x when 2x .
5. Evaluate 1542 ba when 2a and 1b . 6. Evaluate t62 when 50t .
7. Evaluate )60(3 x when 20x . 8. Evaluate )8(5 y when 3y .
9. Evaluate 5.123 x when 5.5x . 10. Evaluate )6(4 a when 25.1a .
11. Evaluate cb 24 when 5.2b and 10c .
12. Evaluate ab2 when 10a and 3b .
13. Evaluate 44 rs when 3r and 2s .
14. Evaluate yx 2 when 10x and 5.4y .
15. Evaluate )( yx when 7x and 5.2y .
16. Which is larger, x or –x? Explain your reasoning. Careful!
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CUNY Start Core Math Curriculum, Class 6
Class Contents N EA F
6
Three Views of a Function
Signed Numbers 4: Division
Extra Practice
Three Views of a Function
We have a few objectives in this activity — to emphasize the connection between the function
equation, table, and graph, and to develop strong student understanding of what ‗function
solutions‘ are and how they can appear in a table, as ordered pairs, or as points on a graph.
Have a large sheet of 1‖ graph paper or dry-erase xy-grid for use in this activity. Draw the
function rule box, table, and add the graph on the board exactly as they appear on the handout.
Students should add 32 xy (or some other elementary function) to the rule box and complete
the appropriate outputs for the given inputs. Students should leave the last two rows and the right-
hand column blank for now.
The function equation gives us a very general understanding of the functional relationship — if
you ask students what they need to do to any input, the equation indicates that we should multiply
by 2 and then add 3. The table of values is a narrower view of the function as it gives particular
examples of input/output pairs.
Add ―Solution‖ as the heading for the final column in the table, and discuss this ―friendly‖
definition with your students. Of course they should record this definition.
A function solution is an input/output pair that fits the function rule.
We have another way of recording solutions — inquire whether any students know how to do this.
If they do not recall, it is okay for us to simply write the first solution as an ordered pair — this is
an example of customary notation that students cannot discover on their own. In addition to
describing this as a function solution, it can also be called an ―ordered pair‖ because they are
ordered (input then output) and they are a pair (two of them). After students complete the table,
provide them (or have them provide each other) with a few additional ordered pairs to have them
determine if they are also function solutions. Also ask for a solution or two not already shown.
Students have undoubtedly graphed points in prior math classes, but it is uncommon for them to
understand these points as representations of inputs and outputs for a particular function. Do as
much teaching as you feel is needed for the plotting of individual points. You might bring the
class up to the large graph you have provided on the board to focus their attention on individual
points. Before providing students with the word ―origin‖, see if anyone else knows it. For tips on
assisting students with point-plotting, see the Functions Rule lesson set. Once you have plotted
the points from the table on the graph, point to a few other points on the graph (ones that would
and would not be solutions), and ask students to describe what this point ―is saying‖ and whether
or not it represents a solution to the function.
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You should have two slots left for inputs. Add –1 and –2, and give students time to calculate the
outputs. Reminding students that –1 and –2 are the opposites of 1 and 2 may help them
understand why we move in the opposite direction (to the left) to graph those input values.
Students may comment on the orientation of the points along a line and ask if they should
―connect the dots‖. For now, we will focus only on plotting individual solutions and will discuss
―the line‖ more fully in a subsequent lesson.
Following discussion of Three Views, have students complete and discuss the back of the page.
Signed Numbers 4: Division
We will not emphasize the traditional long division algorithm here. We will use two methods that
emphasize the meaning of division, help students to do decimal division, illustrate signed number
relationships for division, and illustrate the tricky situations involving division and zero.
Write the following expression on the board: 2
8.
What do you see? (What operation do you see?) Are there any other ways to write this division?
82 28
These are both correct ways of re-writing the first expression, but these division symbols do not
typically appear in college math classrooms. In algebra classes, the convention is to write
division in fraction form.
Ask students to do the division. Once there is agreement that the result is 4, connect this to the
next discussion by asking students why the result is 4. Their response(s) will lead you to one or
both of the following ways of thinking about division.
Starting at the Bottom
One way to think about this division is to start at the bottom (denominator, if you like) and to say
to ourselves, (write this down)
―How many groups of 2 does it take to make 8?
or ―How many groups of $2 does it take (or do you need) to make $8?‖
Write the following division problem in your notes: 25.
5.
Can anyone help me to write a description of this division using the language of groups that we
just spoke about? Once you have written this on the board and in student notes, reach agreement
on the result with your students.
Write these other examples and do the same in your notes. Make sure you write the description of
what we are asking ourselves.
5.
6
05.
45.
5.
25.1 (challenge!)
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Okay, now that we have looked at some decimal examples, we are going to look back at some
special whole number examples.
Write a description of the division in this case, and then use that to do the division:
10
0
In this case, we imagine the number of groups of 10 that are needed to make 0. If it helps, put a
$10 bill your hand, ask your students what the goal is ($0), and then ask how many of these $10
bills are needed to reach your goal (none or 0 times).
010
0
Write this last example in your notes. How can we describe this division problem?
0
6
In this case, we are asking how many groups of 0 or $0 it takes (or are needed) to make 6, or $6.
If you need to, show them $0 in your empty hands. By taking your empty hands and seeming to
drop the amount on the table again and again, you can ask them how many times you need to do
this in order to have $6. It will never happen.
Mathematicians say that0
6 is undefined. We cannot divide by 0 because you cannot make 6 out of
groups of 0. In other words, you cannot make something out of nothing.
How about this one? Write a description of the division using the language of groups and
money/debts.
6
24
How many $6 debts do we need to make a total of $24 in debts? 4. The result is 4. What do you
notice about the signs? Dividing two negatives makes a positive. Does that remind of you
anything else we have studied? It is the same sign relationship as multiplication.
Starting at the Top
Now we are going to switch gears and think about division in a different way. Consider the
following division problem:
2
40
There are really two important ways to describe this division. What is the way we were doing this
before? How many groups of 2 do we need to make 40? We started at the bottom in that case.
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There is another way to think about this division — If we start at the top with 40 and split it into 2
equal parts, how much is each part?
Try this ―Starting at the top‖ method with the following: 3
75.3
Now try ―Starting at the top‖ with this next example: 3
18
If we start with an $18 debt and split it into 3 equal parts, how much is each part? –6 or a $6
debt. What do you notice about the signs? Dividing a negative by a positive produces a negative
number (same sign relationship as multiplication).
Neither of our approaches here works well with something like 3
18
. Remember that in the
instance of talking about 47 , it did not really make sense to talk about –7 groups. Our
discussion, then, is a bit simplistic. We really need a separate discussion of how
3
18
3
18
3
18
, but it probably should not happen here. This division work will already take
quite a bit of time, and a change of pace from these detailed arithmetic discussions is needed. We
will discuss this later in the course. If a student asks about 3
18
, compliment him/her on the astute
question and tell him/her that we will consider this issue later in the course — we promise.
It has been the case that the multiplication and division rules for signed numbers appear to be the
same. Write these on the board and ask your students to do the calculations.
One-half of 20. )20)(5(. 220
What is going on here?
The first expression in words is a friendly way of writing the second multiplication example.
Students will conclude that all three have the same result. Can any student describe what
operations appear to be the same and why?
Multiplying by one-half (or taking one-half of something) is the same as dividing it by 2.
In fact, all multiplication can be re-written as division, and vice versa. (Does that remind you of
anything?) If multiplication can be re-written as division, then the signed number rules for
multiplication must also apply to division. You could ask students to verbalize what we learned
about signed number multiplication and then extend that to division. Then go to the division
handout.
Extra Practice #3
The Three Views of a Function
x y
0
1
2
3
4
Funct ion:
y
x
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Consider this graph where some solutions of
a function have been plotted.
a. Identify 3 of the solutions using ordered
pairs.
b. Complete the table of values for the solutions shown on the previous graph.
c. Can you identify the function equation?
d. Provide an ordered pair that is not a solution to the function equation. Explain why your ordered
pair is not a function solution.
x –4 –2 0 2 4
y
x
y
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Signed Numbers 4: Division
1. Divide. Try using our grouping
strategies, and do not use a calculator.
a. 2
14
b. 75.
6 c.
2
6.1 d.
2.
6.1
e. 3
12
f.
10
0 g.
25.
5.2
2. Consider the following question: ―How many groups of 30 cents do we need to make $5.10?‖
a. Write a division problem that could represent this question.
b. What is the answer to the question?
Mitosis is the process of cell division.
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3. Divide. Try using our grouping strategies and never use a calculator.
a. 5.1
3 b.
3
15 c.
3
5.1
d. 5
25
e.
4
20
f.
1.
1
4. Consider the following: ―Three children collect $12.45 from their lemonade stand. How much
should each child take home?‖
Write and solve a division problem that would correspond to this question.
5. Evaluate 5
)10(
x when 25x . 6. Evaluate
4
ab when 8a and 2b .
7. Calculate missing inputs and outputs.
x 624
x
y y
2
–3
–4
7
5
.5
0
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Extra Practice #3
1. Complete the missing outputs and solutions in the table of values. Graph the solutions.
2. Evaluate xy3 when 4x and 2y . 3. Evaluate 102 x when 4x .
4. Evaluate x53 when 4x . 5. Evaluate xx 4 when 2x .
22 xy
x y solution
5
4
3
2
1
0
–1
–2
–3
–4
–5
x
y
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6. Evaluate cb 34 when 10b and 100c . 7. Evaluate 5
3
x when 10x .
8. Evaluate 2
20 y
x when 10x and 12y . 9. Evaluate )75.(x when
2
1x .
10.
8.5
20
12
-28
-4 3
x
++
xx
+
10
2.5
+
x
16
-8 12
36-10
3
x
++
xx
+
-20
.25
+
x
-11 0
-100-24
24
x
++
xx
+
30
201
2
+
x
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11. The Department of Motor Vehicles in a nearby state has created a new
system of traffic fines for drivers who receive tickets for reckless driving.
Under the new program, someone caught driving recklessly will receive a
fine, but that fine can be reduced depending on how many hours the
person attends ―good driving‖ classes. A function that describes this new
system is shown below:
36040 xy
where x represents the number of hours of ―good driving‖ classes
attended, and y represents the ultimate fine in dollars.
a. Complete the table of values for this function.
b. When x equals 6, y equals 120. Describe the meaning of these
values using the context of the problem.
c. Your friend just got a ticket for reckless driving, and she does not understand how this new
system works. Describe it to her using everyday language (do not use words such as function,
inputs, or outputs). Assume that she cannot see the table of values or equation.
x y
0
1
2
3
4
5
6
7
8
9
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12. Evaluate 25.
x when 3x .
13. Evaluate 416
a when 4a .
Challenge
Try to make change for a dollar using exactly 50 coins.
Is there more than one way?
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CUNY Start Core Math Curriculum, Class 7
Class Contents N EA F
7
Review Extra Practice
Counting Cubes
Exponents 1
Review Extra Practice #3
Ask students to describe problem #9 in terms of money. Many students will be able to describe ½
as $.50, and then the addition can be done with decimals and money.
Counting Cubes
Rather than starting by defining the meaning of an exponent, this functions activity will give your
students a chance to reveal their prior knowledge if they have it. It also can be used to help them
understand (for many, for the first time) why we can say ―five squared‖ in place of ―five to the
second power‖ and ―two cubed‖ in place of ―two to the third power.‖
You will need a set of 1-inch foam cubes and a single ruler.
This should be an interesting challenge for your students. Before giving out the handout, take 36
one-inch cubes and arrange the three figures on a central table. It is best to label them ―figure 1‖,
―figure 2‖, etc. using index cards in order to correspond to the handout. Ask your students to
come around and look at what you have done.
Ask your students to describe what they see. You will probably hear ―cube‖ and ―square‖ among
other utterances. Get these words on the board as students use them, and try to have students
clarify their meaning. Be clear that these figures are not ―squares‖ — squares are flat and only
exist on or in flat surfaces.
Each cube has six square ―faces‖. You may introduce the word ―edge‖ to describe the segments
that connect the faces. In trying to describe the pattern, students may say ―one, two, three.‖ If this
happens, interrogate them to get at more precise statements. It would be good to have a ruler there
so that they can make some statements about length. And while it would be good to have a
conversation about area and volume, it is probably too time-consuming for us at this moment.
After the discussion, give students the handout. We suggest that you lay it face-up on the desk
and not remark on the questions on the back. Students may solve problem #1 using a variety of
methods, and we would not want to indicate to them at the outset that they should use a function.
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Some students will try to draw the figure, but this is very difficult to do on a flat surface. Others
may try and build the figure, and you should certainly offer the cubes if they wish to do so. The
only problem is there are not enough of them to build the figure once, let alone in each group.
This false offering gently forces them to search for another solution method.
Frustration can set in with this problem, and a reasonable suggestion you can make is to ask
students to try and solve an easier problem — the number of cubes needed in the fourth figure.
There can also be a lot of confusion within a group because students are trying to articulate their
ideas without being able to build the 8th
figure. The fourth figure allows them to do this.
Typically, once they continue the pattern for one more figure, they will be able to extend that
reasoning to the 8th
figure.
When students solve the first problem and move on to creating a table and rule, there are a few
possible answers:
xxxy xxy 2 3xy
The middle function is actually quite common because students may count the cubes by first
figuring out the number on the top layer and then multiplying by the number of layers. xx 2
flows directly from that thinking.
Encourage the different function equations firstly because they are all correct, but more
importantly because they allow students to demonstrate exponent notation.
Exponents 1
In their notes, students should write the following expression: 25 . Review and write the different
ways of verbalizing this expression — ―five to the second power‖ and ―five squared.‖ Do not
permit students to refer to terms such as 2x as ―x two‖. Gently insist on the correct vocabulary:
I think I know what you mean, but what is the way that mathematicians say that?‖
After students have recorded the verbalizations, elicit and record the meaning using multiplication.
5552
Emphasize how this expression is different from 25 . When you ask students in subsequent
situations to tell you what an expression such as 25 means using multiplication, watch out for
ELLs who may say ―five, two times‖. This can be common, and we must nudge them towards
more precise language, especially when it is likely that they will be misunderstood.
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It can be important for your students to see why we use the language ―five squared‖ in addition to
―five to the second power‖. Starting with a segment whose length is 5 units, we can build a square
using that segment for each side. The resulting area may be calculated by multiplying 5 by itself.
We have ―squared‖ the length, giving us 25 square units of area.
In their notes, have your students write the following expression: 32 .
Review the different ways of verbalizing this expression — ―two to the third power‖ and ―two
cubed.‖ After recording the verbalizations, ask about and record the meaning using multiplication.
22223
We work re-writing exponents using multiplication and vice versa because these connections are
more important for the algebra work we are going to do than knowing that 823 . Yes, students
should know that also, but it is not enough for us.
Compare 32 to 32 to distinguish between repeated multiplication and repeated addition.
We can also illustrate why we say ―two cubed‖. Starting with a length of two units, we can build
a cube using that length for each edge. The resulting volume (number of cubes) can be calculated
by multiplying 2 by 2 (the number of cubes in one layer) and then by 2 again (the number of
layers). Holding up ―figure 2‖ from Counting Cubes can be very helpful here.
Length = 2 cm
Volume = number of cubes = 2x2x2 = 8 cm
3
1 cm
Length = 5 cm
Area = number of squares = 5x5 = 25 cm
2
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97
In their notes, have your students write the following expression: 43 .
How would we say this? You may let students try to come up with a short way (like ―squared‖ or
―cubed‖) before telling them that there is no short version for exponents besides ―squared‖ and
―cubed‖.
Your students are ready to move on to the handout Exponents 1.
Students will often skip text that describes the function context and focus only on the calculations
necessary to complete the table. As you walk around the room, ask students what they think of the
Greeks‘ view about falling objects when compared with Galileo‘s view. This will force them to
go back and read the text. Even though the text basically states that Galileo disproved the Greeks,
students may disagree with Galileo and trust their intuition that says the Greeks were correct. This
might create a healthy debate in your classroom. You could poll students about who they believe.
Then you could drop objects of varying weights at the same time as an experiment.
Watch out for order of operations issues with this first function.
On the second page of the handout, notice that all of the items in problem #4 allow multiple
solutions. Students who finish the handout quickly should be redirected to look for more solutions
to a and b because they often do not think of the solutions involving negatives.
The same is true in #5, the final function table. The missing inputs could be –3 or 3 and –8 or 8.
Try not to give that away with too large a hint.
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Counting Cubes
1. Determine the number of cubes that will be included in the 8th
figure. Be prepared to explain how
your group determined your answer.
Figure 1 Figure 2 Figure 3
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2. Create a table of values where the input is the figure number (1, 2, 3, etc.), and the output is the
number of small cubes needed to make that figure. Label the table clearly.
3. Determine a function rule that describes the relationship between inputs and outputs.
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Exponents 1
This problem was adapted from Mathematics: A Human Endeavor by Harold Jacobs
If it takes 16 seconds for a rock to hit the bottom of a well, can we conclude anything about how deep
the well is? Would a heavy rock and a lighter rock hit the bottom in the same amount of time?
The Greeks thought that a heavy rock would hit bottom first, but they were wrong. A seventeenth-
century Italian scientist, Galileo, discovered that the speed at which an object falls does not depend on
its weight. The two rocks would hit the bottom at the same time.
Galileo knew that the distance traveled depends on time.
Here is a function that shows the distance in feet traveled
by a rock (d) based on the time it falls in seconds (t).
2)4( td
1. Complete the table of values.
2. Think about the well described in the cartoon above.
Can you calculate its depth?
Time in Seconds Distance in Feet
0
1
2
3
4
5
6
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3. Re-write each of the following using an exponent.
a. 555 b. )2)(2)(2)(2)(2( c. 100
d. 25 e. aaaa f. 8
g. 1,000
4. How many different ways can you re-write each of the following using an exponent?
a. 16 b. 64 c. 1
5. Use the following function to complete any missing inputs or outputs: 42 xy
x 0 1 –2 –5
y 13 68
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CUNY Start Core Math Curriculum, Class 8
Class Contents N EA F
8
2xx
Terms and Expressions discussion
Rectangle ________ and Rectangle Perimeter with Variable Side Lengths
Terms and Expressions
Extra Practice
2xx
Write this problem on the board:
Consider 2xx
Is this equation always true, sometimes true, or never true?
Discuss with a partner and write an explanation of your answer.
Students should record this in their notes. Assign students to discuss this problem in pairs. As
you wander and observe student conversations, look out for pairs that may be assigning different
values for x on each side of the equation. If this happens, or if there is disagreement in a group
about the issue, it is okay simply to tell them that within one expression, equation, or diagram, a
variable (letter) appearing more than once must represent the same number. This is a convention
that is not ‗discoverable‘ by our students.
After students have had a few minutes to discuss this, call the class
together for a whole-group conversation. You might find it helpful to
record their numeric examples in a small chart such as the following:
Once you have arrived at a consensus on this problem, move directly on
to the next activity. This conclusion will hopefully reappear later in the
class session. Do not warn them of this.
Terms and Expressions Discussion
This is a lengthy whole-group discussion that requires important student note-taking. You will
move from this discussion directly into the work with rectangles before giving students the
handout titled Terms and Expressions.
We are going to introduce some new vocabulary. Write this down.
x7
x 2x
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While this can be described as an expression, we also refer to this as a term. Terms typically have
two parts.
x7
Write a few more terms on the board and ask that students identify the ―number part‖ and the
―variable part‖ for each. Include terms such as: b4 , a (number part = 1), 12xy (variable part =
xy), 8 (no variable part), and x .
Because of your discussion of the number part in the term ―a‖, your students may be in a good
position to conjecture that the number part in x would be –1. Exploring this a bit further will
help our students understand why x was earlier described as ―the opposite of x.‖ We really
needed the students to know the signed number rules for multiplication before doing this.
Create the table on the board and ask your students to
complete the missing values. Can your students
generalize from this information? If it helps, have them
create more examples.
Multiplying a number by –1
turns it into its opposite.
On the board and in student notes, record the following on the left side of your board space:
Evaluate xx 52 when 4x .
Give them time to complete the task and without erasing that previous example, add this second
example to the right and ask your students to evaluate:
Evaluate xx 52 when 4x . Evaluate x7 when 4x .
Just by staring at the board, you might pause to see if any student wants to make a comment about
what has happened. If not, ask them what they notice. Probably, a student will say ―they are the
same.‖ Push for clarification. The expressions are different in that one has two terms and the
other has one term, but they have the same value when 4x .
Keeping the original statements on the board (and erasing the work below), add this new
information and ask students to comment:
Evaluate xx 52 when 4x . Evaluate x7 when 4x .
when x=8. when x=8.
Is it surprising or not that the expressions also have the same value when 8x ? Do they have
the same value for any other values of x?
x The opposite
of x. x1
5
12
–6
number part variable part
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Some students may see quickly that the expressions will always have the same value. Other
students may need encouragement to test more cases, including different kinds of cases.
How about when x is a negative number? How about when x is zero?
Once students have tested a variety of cases, you should try and draw out a reason for this
sameness. If no students explain this in terms of ―groups‖, encourage this idea — ask how each
term could be described in terms of ―groups‖. The first expression has 2 groups of a number x and
then 5 groups of the same number x. This will always be the same as 7 groups of the number x.
If these expressions mean the same thing,
which one would you rather use when 5x ? Why?
We like working with expressions that have fewer terms because they can require fewer
calculations.
When we add two terms to make a single term, we call this ―combining‖ terms.
Do not write this entire sentence on the board because we will refine this definition soon. You
may want to write merely ―combining terms‖ on the board.
Please combine the following:
xx 144 aa 30 xx 82
Use the language of ―groups‖ to review student responses here. We are trying to solidify the
reasoning that because we are adding groups of the number x (or other variable), we can simply
express the total number of groups. If students feel they need it for xx 82 , you could evaluate
xx 82 and x6 with a few values of x to reassure them that they are the same. You could also
seek a description similar to our descriptions of money and debts. It is as if you owe 2 groups of
some number x, and you have 8 groups of that same number x. You will have 6 groups of x left.
When you think of combining, what operation(s) do you think of? Addition, of course. This is
correct, but what about the expression shown below?
xx 59
Ask students if they can describe this expression in terms of groups. We start with 9 groups of
some number x and take away 5 groups of x. What do we have left? Can we re-write this?
Also ask about the relationship between addition and subtraction. Students may recall that we can
re-write subtraction as addition. How can we re-write xx 59 using addition? So can we still
combine these terms? Combining terms, then, can involve either addition or subtraction. Write:
Combining terms means adding or subtracting two (or more) terms to make a single term.
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It is time to introduce some other types of expressions. Write the following on the board:
124 x
How many terms do you see in this expression? Do you think that we can re-write it as x16 ? Why
or why not? Give them some time to think about this. Try and steer the discussion to a verbal
description of what 4x indicates in terms of groups. This is ―four groups of a number x‖. We are
not adding an additional twelve groups of x, so it is not appropriate to combine them.
yx 34
Do you think that we can re-write this as 7xy? Why or why not? What about as 7x? Or 7y? We
do not know the value of x or y, and they could easily be different numbers. Four groups of one
and three groups of the other are not necessarily the same as seven groups of the former.
What can we conclude about the instances where we are allowed to combine terms? (They must
have the same variable part.) You may need to have them give you additional examples where
you can and cannot combine terms for them to be able to articulate it.
Finally then, this conclusion needs to go in their notes, along with some new vocabulary:
We may combine terms (using addition or subtraction) when they have the same variable part.
Terms that have the same variable part are called “like terms.”
Give a few more examples:
ww 95 yy 5.24 sr 23 tt 92
When you review student work, try to reinforce the new vocabulary. Also, highlight connections
between combining terms and adding/subtracting signed numbers. For example:
When considering tt 92 , we can think about it like how we would think about 92 .
7
)9(2
92
t
tt
tt
7
)9(2
92
How about this one? 225 xx
It is has been a while since the opening conversation of x and 2x , but students may be able to tap
this earlier discussion in order to conclude that the variable parts here are not necessarily the same.
(Are they never equal?) We cannot combine these terms. They are not like terms.
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So, what do you think we can do in an instance such as the following? yxyx 245
Tell your students that it is conventional (but not required) to write expressions with terms in
alphabetical order based on their ―variable parts‖. It is also conventional if there is a constant term
to put that after other terms. What is conventional should not be confused with what is correct.
When we re-write an expression with fewer terms,
we also call this “simplifying” the expression.
Now take a few minutes to look back in your notes from the past few class periods.
Can you give me an example of an instance where we have already simplified expressions?
Students may point to Exponents 1 or even the work adding/subtracting signed numbers among
other examples.
Do not give students the Terms and Expressions handout yet. We need to do some work with
rectangle perimeter first.
Rectangle _________
The key here is to have rulers close at hand so time is not wasted fumbling around. Distribute a
ruler to each student.
Ask students to use the inch-side of the ruler to
measure and label the sides of the rectangle.
(You may wish to emphasize that we are
measuring ―lengths‖ because we are measuring
the distance between 2 points.) You should have
a sketch of the rectangle on the board as well.
Ask students to determine the length when you begin at point C and travel through point D ending
at point B. Eventually, ask for the length beginning at point C and traveling all the way back
around to point C. When it is determined that this length is 16 inches, ask your students what this
measurement is called — the length around the edge of a flat figure — it is the perimeter. Have
them add this to the title of the handout. Show the calculation (the addition of the sides) below the
figure and ask them to add this to their notes. We want them to be certain about how to calculate
perimeters with numbers before we throw in variable side lengths.
Rectangle Perimeter with Variable Side Lengths
Sketch the first rectangle and corresponding table of values on the board. Talk a bit about what is
shown.
What does it mean that there is an x in the diagram?
Can we say anything about the value of x? Do you think it is larger or smaller than 8?
6 in
2 in2 in
6 inC D
BA
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This is a good place to caution students about figures that use variables and that do not include the
clarification ―drawn to scale‖. While it appears that the side labeled x is longer than the adjacent
side, we cannot assume this. We do not know the value of x — it could be larger or smaller than
8. Certainly, it does not make sense for it to be 0 or negative or we would not have a rectangle.
It makes sense to say something about units here. Before, we were careful to write ―inches‖
beside our measurements of lengths. In many algebraic settings, we leave off the units and they
are ―understood‖.
Provide the headings and lengths for the table of values and
ask students to calculate the appropriate perimeters. Perhaps
they will recognize that it behaves like a function — perhaps
not. In either case, you do not need to dwell on it.
When students reach the length described as x, question them.
What does it mean that I have put x here?
Does that mean that we cannot determine the perimeter?
If they are having trouble…
What did you do to determine the perimeter when the length was 10?
What about when the length was 6?
So what can you do now that the length is x? Can you use x this time? What happens?
Can you simplify that?
Sketch the second rectangle on the board and its
corresponding table of values including the headings
and inputs.
Clarify that the ―large rectangle‖ refers to the two
smaller ones combined.
Once we have finished this problem we are ready for the handout Terms and Expressions. Look
out for problem #4, where we connect context to our work combining terms. Ask students what
the simplified version of the function, particularly the term 35.25x, means in this context.
Extra Practice #4
Point out problem #10, where students are asked to creating their own number puzzles that others
in the class will complete. Draw the diagrams on the board to emphasize that the left-hand figure
should be the puzzle as students would see it (with two circles filled in), and the right-hand puzzle
is the solution (with all four circles filled in).
Length x Rectangle Perimeter
10 6
25 x
Length f Length g Large Rectangle
Perimeter
8 5
9 6
2.5 4
f g
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Rectangle ______________
A B
C D
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Rectangle Perimeter with Variable Side Lengths
x
8
g12
f
Terms and Expressions
You need to rent a car for an out-of-town trip. Brooklyn Rent-A-Car has
three separate charges that make up the total cost — there is a flat fee, a
rental charge for each day, and an insurance charge for each day. A
function that represents the rental charges is shown below:
4275.65.28 xxy
where x represents the number of days you rent the car, and y represents the rental cost in dollars.
1. Complete the table of values for this
function. You may use a calculator.
2. What do you think the flat fee amount is? Explain how you decided your answer.
3. What do you think the insurance charge per day is? Explain how you decided your answer.
4. Is there a way to re-write the car rental function using fewer terms? If so, do so.
x y solution
1
2
3
4
5
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5. Calculate the perimeter of the largest figure. All four-sided figures
are rectangles.
Simplify the following if it is possible to do so.
6. xyx 52 7. baba )2()6(4
8. rsr 687 9. bbb 104
10. 124102 xx 11. 11410 rpp
12. 1253 xx 13. xx 42 2
14. 22 7xxyx 15. 6263 222 baba
x
y
x
12
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Extra Practice #4
1. Calculate the perimeter of the largest figure. All
four-sided figures are rectangles.
2. Evaluate 23x when 5x . 3. Evaluate 2xx when 10x .
4. Divide without using a calculator.
a. 2
18 b.
75.
3 c.
3
45.6
x
y6
x
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5. Simplify the following if it is possible to do so.
a. xx 102 b. xx 102 c. yx 102
d. yxyx 3547 2 e. 22 524 bbb f. aaa 24
6. Complete any missing inputs or outputs.
32 xy
x y
0
1
–1
50
–7
3.5
13 xy
x y
0
1
–1
3
–2
–63 Challenge!
7. Re-write the following in a simpler way either using multiplication or an exponent — whichever is appropriate.
a. )6)(6)(6)(6( b. xxx c. )5()5()5()5()5(5 d. (2a)(2a)(2a)(2a)
8. For each row, compare the quantities in Column A and Column B and then check one of the boxes to the right.
Column A Column B
The quantity in Column
A is larger than the
quantity in Column B
The quantity in Column
B is larger than the
quantity in Column A
The quantities in
Column A and Column
B are equal
It cannot be determined
if one is larger or if they
are equal.
10 x
–5 0
y –2
a b
.25 ¼
–16 –4
x 2x
* 9. Evaluate 5.32 x when 2
1x .
10. Create your own Number Puzzle that your classmates may solve. On the left, record your puzzle
with only two numbers filled in. On the right, record the solution to your puzzle. Don‘t let any of
your classmates see your puzzle!
Challenge
Sketch and label the lengths of a rectangle whose perimeter is 1312 ba . There is more than one
solution to this challenge, and so you will need to confirm your drawing with your College Transition
Math teacher.
x
+
x
+
Record your puzzle here. Only two numbers should
be filled in.
Record your solution here. All four numbers should
be filled in.
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CUNY Start Core Math Curriculum, Class 9
Class Contents N EA F
9
Review Extra Practice (and making Student Number Puzzles handout)
Exponents 2
Connecting Function Solutions
Integers
Student Number Puzzles
Review Extra Practice
Problem #2 can be contrasted with Galileo‘s function in class #7. In the division problems,
problem #4b is easiest if you ―start from the bottom‖. Problem #4c is tricky, but can be done
relatively easily by ―starting at the top‖. Try to find students who applied those approaches.
In problem #7d, students could have 4)2( a or 416a . The latter requires skills in multiplying terms
that we have not studied yet. Hold this discussion off until we treat it more fully.
Problem #8 should provoke some discussion. Take time with the last example, comparing x and
2x. Students may conclude it cannot be determined because they thought of 0 as a possible value
of x. Ask if, aside from 0, 2x is always larger than x. When x is negative, 2x is actually smaller.
Problem #9 is challenging, but ))((21
21 can be thought of as half a group of $.50.
A blank sheet of number puzzles is included so that you may transfer student problems from the
extra practice to a handout. It is ideal if you can do this by hand during the class period and get
copies made to give to the students as extra practice. Check the student puzzles for accuracy, if
possible with the students at the beginning of class. We should try to be sure that the problems are
―doable‖ by our students and do not involve more than 2 decimal places. Once you have them all,
create a two-sided handout with everyone‘s puzzle, including the student‘s name beside his or her
puzzle. Hopefully, a cooperating teacher or tutor can help you with this.
Exponents 2
Ask your students to record the following in their notes and decide whether it is a true or false
equation:
4444444 23
Give students time to consider the question you have asked them. Discourage student call-outs.
When you sense that a good number of them have thoughts about this, manage a discussion.
Allow students who disagree to explain their thinking. Once the class agrees that it is a true
equation, ask how the right side may be re-written more simply.
You might ask why students commonly believe that 623 444 .
?
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Some students may say that it is possible to add the exponents. You can accept that, but follow
that by saying that this method of writing out the multiplication is important because it shows the
real mathematics. We will refer to this writing of the ―hidden‖ multiplication as ―expansion‖.
(There are also other kinds of expansion — for example, writing 3x as x+x+x. So we should
probably not suggest that expansion means writing out the multiplication indicated by exponents.)
Gently insist that students show all expansions when they get to the problems on the handout.
It is time to question students some more. Write the following on the board, off to the side, and
give them time to consider and discuss it.
xxxxx 4
It may help students to think of 4x as ―four copies of x‖ (as long as they remember that the
operation is multiplication). Leaving the previous item on the board, add the following, and give
your students time to consider and discuss it.
222242 5555)5(
We hope that students might connect this to the previous example, possibly describing it as ―four
copies of 25 ‖. Once there is agreement on the true equation, have a student complete the
expansion and the resulting simplification. If a student wants to talk about the rule that they
remember, that is fine. You should continue to emphasize the expansion. In problems such as
letter g in the handout, you can gradually allow students to forgo the expansion, but they should
still be able to describe to you what the expansion would look like.
Connecting Function Solutions
Draw and label the following horizontal segment on the board.
After students have recorded this in their notes, divide the segment
in the following way and ask your students to do the same.
Ask students to identify in their notes the quantity at each of the
three tick marks. Give them time before asking for oral responses so that all students have a
chance to think about this.
In the discussion of this divided segment, ask students to describe what they did — in the order
that they did it. For many students, they would first decide on the value of the middle mark before
considering the others. Below are two possible correct ways of labeling the diagram. Make sure
that students record at least these two options in their notes.
To help students to understand these quantities, you can ask them how many sections there are in
the divided segment. There are four sections. We are dividing one whole into four parts. We can
imagine, for example, that we are dividing $1 into four parts. Each part will be one quarter. We
can see the quarters in both the fraction form and in decimal form. For the discussion that
follows, we will focus on the decimal representation.
?
?
10
10
.25 .75.50 13
4
1
4
1
2
0 1
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118
y
x
We recommend creating a large version of the graph from
Connecting Function Solutions and bringing all the
students up to the board for this discussion. Clarify that
we are only looking at one section of the graph. It may
help to have a grid with all four quadrants nearby for a
comparison.
Ask your students to add the function xy 2 to the space
in the table of values. Your students may calculate the
outputs for the given 5 inputs and plot those function
solutions on the graph. They should leave the other spaces
blank in the table. What does each point represent?
Ask students what they notice about the pattern or orientation of the points on the graph.
Ask your students to focus their attention on the
segment between 0 and 1 on the x-axis. Have them
divide this segment into four parts in the way we
demonstrated earlier. See the graph. This gives you
another chance to ask students about the quantities that
you have created.
Students should place .5 in the input column of the table
of values and calculate the appropriate output. Two
groups of $0.50 makes $1 or 1. The appropriate output
is 1. Talk with your students about how this function
solution should be graphed. Invite a student to graph
the solution at the board.
When students agree on the location of the previous solution, include .25 and .75 as inputs in the
table. Discuss the outputs as well as the location of the function solutions on the graph.
Your students should recognize that decimal function solutions still lie along the straight line
created by the whole-number solutions.
Have we found and graphed all of the function
solutions for the inputs between 0 and 1?
No. We could divide the segment between 0 and 1 into
many smaller pieces, and we would still find a function
solution for each that would lie along the same line. It
is impossible to show them all. Instead of trying to plot
all of them, we often connect solutions with a line.
Draw a line segment only from (0, 0) to (4, 8).
Does this show all of the solutions to this function?
No, there are solutions beyond the segment in both
directions. A student will probably be able to suggest
adding arrows to indicate that the function solutions
continue in both directions.
y
x
y
x
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119
Emphasize the idea that points along the line are function solutions, and points off the line are not
function solutions. This is a very important idea, and it is true when we use decimal as well as
whole-number inputs. You could point to a place along the line like (2.5, 5) and ask about it.
Students can test if it is a function solution. You could also point to somewhere not on the line
and ask about it. You could also ask students if this function has a context or is abstract, and
whether the input could be any type of number. This could be contrasted with functions where
only some inputs make sense because of a real-world context.
Once your students have completed the function tables and graphs on the back side of this
handout, question them about the meaning of individual points they plotted, the line, and locations
that are not on the line. You may say that the line and arrows make the graph complete because
they indicate all possible function solutions.
Integers
This conversation requires some brief note-
taking by your students. Record the
following on the board.
In the subsequent discussion, you can ask which students think they know the difference. For
students who think they know, do not ask them to name the difference but instead ask them to
provide an example of an integer or a non-integer. If they give a valid example, add that number
to the other examples. You can keep doing this to confirm student understanding among many
students. It can be helpful for them to add this more general set of integer examples to their notes:
If you want, you can be more formal about this discussion and first define ―whole numbers‖,
which are 0, 1, 2, 3… In this context, integers can be defined as ―Whole numbers and their
opposites.‖ (We have seen students who, when asked what an integer is later in the semester, said
―Whole numbers and their opposites‖. So be careful that students understand the concept and are
not just memorizing a definition in a rote manner.)
As a way of connecting this discussion to the preceding activity, bring your students back to the
large drawing of the function from Connecting Function Solutions and ask them ―Which points
have integer inputs and outputs?‖ In general, students and instructors in this and subsequent
courses will emphasize integer solutions, but it should be remembered that the non-integer
solutions are shown using the line.
Student Number Puzzles
Try to get them started on these. Unfinished ones can be considered extra practice.
Integers Non-integers
Integers …–3, –2, –1, 0, 1, 2, 3,…
3 –8
0
123 –46
3.8 .2
–5.12
4
3
3
12
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120
Student Number Puzzles
Fill in the missing circles for each Number Puzzle.
x
+
x
+
x
+
x
+
x
+
x
+
x
+
x
+
x
+
x
+
x
+
x
+
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x
+
x
+
x
+
x
+
x
+
x
+
x
+
x
+
x
+
x
+
x
+
x
+
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122
Exponents 2
Re-write each of the following using only one exponent.
a. 662 b. 52 77 c. ))()(( 22 aaa
d. 43)6( e. ))(( 3xx f. 32 )(t
g. 312 bb h. xxx 3 i. 33)12(
Connecting Function Solutions
Use pencil!
x y
0
1
2
3
4
y
x
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124
Complete the missing outputs and solutions in the table of values. Then graph the solutions.
Complete the missing outputs and solutions in the table of values. Then graph the solutions.
82 xy
x y solution
7
6
5
4
3
2
1
0
–1
4 xy
x y solution
–4
–3
–2
–1
0
1
2
3
4
5
x
y
x
y
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CUNY Start Core Math Curriculum, Class 10
Aftermath of Hurricane Katrina, A New Orleans Levee
Begin this discussion by reviewing the photographs of New Orleans and Gulfport taken shortly
after Hurricane Katrina struck the Gulf Coast.
Why was New Orleans so vulnerable to flooding during a hurricane? Some students may have an idea that New Orleans lies below sea level, but that may not make
much sense to students who are not familiar with the concept of elevation. Once words like sea
level, levees (mounds of soil, clay, and/or sometimes sandbags), and elevation have come up,
direct them to the diagram titled A New Orleans Levee. You should put the same drawing on the
board. Discuss the diagram one label at a time, starting with sea level, then New Orleans
elevation, and finally the elevation of the top and bottom of a post that is sunk into the levee.
Students should complete their diagram along with you. Add a definition of elevation on the same
handout — the distance above or below sea level. Any measure of elevation cannot simply be
reported as a number. It must include both a number and a direction.
The post reinforces the levee — it is not a wall that makes the levee higher. These posts are in
place to keep the levee in place. Several New Orleans levees were breached during Hurricane
Katrina — not because the water was higher than the levee, but because large sections of the
levees collapsed.
Class Contents N EA F
10
Aftermath of Hurricane Katrina and A New Orleans Levee
Highest and Lowest Elevations in Selected Continents
Extra Practice
Top of post—16 ft above sea level.
Bottom of post—26 ft below sea level.
Sea level.
New Orleans – 8 feet
below sea level. Levee
(clay and sand)
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The engineers need to order lots of these stabilizing posts. What is the length of a post?
This should provoke an interesting discussion. (Some may think that the length is 10 feet.)
Students should be encouraged to come to the board to demonstrate their arguments. Once
students agree that the post is 42 feet long, add it to all drawings.
You should also state that the total distance of 42 feet can be considered the ―difference‖ between
the two elevations. When we say ―difference‖, we want to know how different they are — in this
case, how far apart they are. Students should write the following on their papers.
The difference between the two elevations is a measure of how far
it is from the top of the post to the bottom of the post.
In the diagram, add ―Difference between the elevation of top and bottom of post = 42 feet‖.
Clarify that this means the same thing as saying that the top of the post is 42 feet higher than the
bottom of the post.
Some students have been conditioned to the idea that ―difference‖ always means subtraction, and
they may object to the idea that the sum of 25 feet and 17 feet could be a difference in this context.
If your students are not too upset about this, you can move on to the next handout now. If enough
students do not want to accept this as a difference, you can present another example:
Draw a horizontal line representing the ground (or sea level if you want to stick to elevation
relative to sea level) and a vertical flagpole starting from the ground. Label the top of the flagpole
as 29 feet (above the ground or sea level), and label a place lower on the pole as 8 feet above. Ask
about the difference between these two elevations. Discuss what operation you would use to
calculate the difference. They will likely agree to subtract and end up with a difference of 21 feet.
Top of post — 16 ft above sea level.
Bottom of post — 26 ft below sea level.
Sea level.
New Orleans — 8 feet
below sea level. Levee
(clay and sand)
Total distance = 42 feet.
(Length of post)
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Students may be glad to call this a difference because they subtracted, but still not be willing to
consider 42 feet a difference in the levee example. Ask if the 21 feet could be described as the
―distance‖ between those places on the flagpole. We want our students to realize that ―difference‖
and ―distance‖ mean the same thing in this context. It may help to frame difference as ―how far
apart two things (or quantities) are‖. If Maria has $9 and Winston has $5, the difference between
those amounts is how far apart they are, or how different they are. Similarly, the difference
between two elevations is how far apart they are — the distance between their elevations.
Some students may realize that the reason these elevation examples involve different operations is
that with the levee, one elevations is above sea level and the other below, while with the flagpole,
both elevations are above sea level. This is not essential, but that visual comparison can help
students understand the idea of difference and accept that it may not always involve subtraction.
Highest and Lowest Elevations in Selected Continents
We suggest that you copy the first page as a single-sided handout, go over #1 first, and then hand
out a double-sided sheet with the remaining problems. Calculators are needed for this activity.
In considering problem #1 together and in subsequent problems, insist that students create a
drawing that shows the given information. This time, though, you will show them how to make a
simpler diagram than was used in the levee handout. Ask what they think you should draw first.
Students should always begin with a horizontal line representing ‗sea level‘. Then ask how to
show the information for this problem. An appropriate drawing for the first problem is shown.
sea level
Following what we did in the previous activity, and not really focusing on what the problem asks,
we will add a ―Distance‖ or ―Elevation difference‖ label.
19,340 ft above sea level
512 ft below sea level
sea level
19,340 ft above sea level
512 ft below sea level
Distance =
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A few of your students are still probably going to be upset when they discover that the word
―difference‖ appears in these problems, but at the same time addition is sometimes a perfectly
appropriate operation to use. Students can be highly conditioned to equate difference with
subtraction. You can remind them that subtraction can always be re-written as addition, and so
these operations are always linked. Also try to keep them focused on creating an accurate diagram
and then to do what operation makes sense. The diagram method will make it easier (though not
easy!) for students to do these problems, while a formal signed number approach is actually more
complex (partly because tricky absolute value issues are involved).
It would be nice to clarify that this difference of 19,852 feet means that Mt. Kilimanjaro is 19,852
feet higher than the shore of Lake Assal (or the shore of Lake Assal is lower…). You could try to
do this by asking how else we could describe this difference of 19,852 feet.
After you have agreed on the solution to problem #1, form your students into groups. Each group
should work together in order to solve the remaining five problems. Drawings must be made for
every problem, including a line showing and labeling the ―distance‖ or ―difference in elevation‖,
even if they do not feel that they need it to solve the problem.
Once the groups have been working for a while, give each group one sheet of newsprint so that
they may display one of the problems that you select. Insist they do their large drawing in pencil
first, and give them markers later so they will be visible in demonstrations to the class.
If you have a few strong students who are able to solve these problems using signed-number
arithmetic, that is fine. You should still insist that they make drawings. If more time is needed,
students can finish these drawings in the next class session and present them before the class.
If you had more than one group preparing to present some of the problems, to save a bit of time in
a class with many students/groups, you might have just one of the groups that did a particular
problem present their work. The drawings from other groups who focused on the same problem
can then be compared and the groups that did not present can be asked to confirm or challenge the
work of the presenting group.
Extra Practice #5
Aftermath of Hurricane Katrina
New Orleans, Louisiana
Gulfport, Mississippi
A New Orleans Levee
Highest and Lowest Elevations in Selected Continents
The elevation of any place on Earth is measured in terms
of how far it is above sea level or below sea level. The
following table shows the highest mountain and the
lowest point not covered by water on each continent, not
including Antarctica.
1. What is the difference in elevation between the highest and lowest points in Africa?
Continent Highest Point Elevation at
Highest Point Lowest Point
Elevation at
Lowest Point
Africa Mt. Kilimanjaro,
Tanzania
19,340 ft above
sea level
Lake Assal shore,
Djibouti
512 ft below
sea level
Asia Mt. Everest,
Nepal/Tibet/China
29,028 ft above
sea level
Dead Sea shore,
Jordan/Israel
1,371 ft below
sea level
Australia Mt. Kosciusko 7,310 ft above
sea level Lake Eyre shore
52 ft below
sea level
Europe Mt. El‘brus,
Russia/Georgia
18,506 ft above
sea level
Caspian Sea shore,
Russia/Iran/Azerbaijan
92 ft below
sea level
North America Mt. McKinley
(Denali), Alaska
20,320 ft above
sea level
Death Valley,
California
282 ft below
sea level
South America Mt. Aconcagua,
Argentina
22,841 ft above
sea level
Laguna del Carbón shore,
Argentina
344 ft below
sea level
Mt. Kilimanjaro, Tanzania
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2. What is the highest point of all the six continents? What is the lowest point not covered by water?
What is the difference in elevation between these two points?
3. The lowest point in the continental United States is in Death Valley, California. It is 282 feet
below sea level. The highest point in the continental United States is Mt. Whitney, California.
Mt. Whitney is 14,776 feet higher than Death Valley. How high is Mt. Whitney?
4. The lowest point in Vermont is at the shore of Lake Champlain. It is 95 feet above sea level. The
highest point in Vermont is Mount Mansfield at 4,393 feet above sea level. What is the difference
in elevation between these two points?
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5. The deepest point in the Atlantic Ocean is Milwaukee Deep, in the Puerto Rico Trench. The
highest mountain near Milwaukee Deep is Pico Duarte, in the Dominican Republic. Pico Duarte
is 3,098 meters above sea level. Milwaukee Deep is 11,703 meters lower than the top of Pico
Duarte. What is the elevation of Milwaukee Deep?
6. The highest point in New York State is Mount Marcy, which is 5,344 feet above sea level. The
lowest point in the Lincoln Tunnel is 93 feet below sea level. What is the difference in elevation
between these two points?
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Temperature Data in Three Cities
Miami, Florida
New York City, New York
Arctic Village, Alaska
Temperatures measured in degrees Fahrenheit.
Friday Saturday Sunday Monday Tuesday Wednesday Thursday
High 72 78 78 69 69 74 76
Low 60 71 58 54 57 62 69
Temperatures measured in degrees Fahrenheit.
Friday Saturday Sunday Monday Tuesday Wednesday Thursday
High 20 42 39 28 30 34 42
Low 17 33 23 21 23 30 33
Temperatures measured in degrees Fahrenheit.
Friday Saturday Sunday Monday Tuesday Wednesday Thursday
High 3 8 –8 0 2 –5 15
Low –10 –4 –19 –7 –11 –13 –2
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Extra Practice #5
Measuring Temperature Change in Three Cities
In order to respond to questions #1 – #3, use the data provided on the page titled
Temperature Data in Three Cities. If it helps, use the thermometer provided at the right.
1. In Miami, what was the difference between the high and low temperatures on Friday?
2. In Arctic Village, what was the difference between the high and low temperatures on
Friday?
3. On Tuesday, which city had the widest range in temperature? Provide evidence for
your answer.
4. In Fairbanks, Alaska, Sunday‘s temperature at 6:00 a.m. was F8 . By noon, the
temperature had risen 14 degrees. What was the temperature at noon?
5. In Chicago, Illinois, Monday‘s temperature at 3 p.m. was F11 . If it dropped to a
low of F3 that evening, how many degrees did the temperature drop?
-20
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
48
50
52
54
56
58
60
62
64
66
68
70
72
74
76
78
80
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6. Evaluate 2)10( a when 2a . 7. Evaluate 22 ba when 1a and .4b
8. Re-write each of the following using only one exponent.
a. 42 aa b. 42 )(a c. ))()()(( 432 bbbb d. 125
9. Simplify the following if it is possible to do so.
a. xyx 42 b. aa 24 2 c. qpqp 16842
d. cba 24 e. 4102 22 bb f. 20212 b
10. Re-write the following using addition and then simplify.
a. 5.1410 b. 5.1410 c. )5.14(10 d. )5.14(10
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11. Use the following function to complete any missing inputs and outputs: 83 xy
12. Manhattan Masters Pizza
A function that describes the price of pizzas at a Manhattan pizzeria is
the following:
5.1109. 2 xxy
where x represents the diameter of the pizza in inches, and y is its price in dollars.
a. Using a calculator, complete the missing values in the table.
b. Explain one of your solutions using the context of the problem.
x 3 –1 0 –3
y 0 –16 992
Diameter in Inches 12 16
Price in Dollars
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13. Create an expression that fits all of the following conditions:
a. There are exactly four terms.
b. Exactly two of the terms are like terms.
c. At least one term is an integer.
14. Simplify the expression that you created in the previous problem.
Super Challenge
What is the function rule that matches this table of values?
x 3 5 7 9
y 13 31 57 91
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CUNY Start Core Math Curriculum, Class 11
Class Contents N EA F
11
Review Extra Practice
Remaining Group Presentations on elevation activity
Multiplying Terms
How to Study for a Math Test
My Review Problems for CUNY Start Exam #1
AlgeCross I
Review Extra Practice
To review the temperature problems, it may help for you to have created a similar thermometer
using 1‖ graph paper that students can surround and discuss. Creating this could be a good
assignment for a tutor in a preceding class.
You may want to ask several students to put their responses to #13 (and to #14 beside it) on the
board so that their classmates can decide if they fit all the conditions.
Search for a student who can define ‗diameter‘ when you discuss Manhattan Master’s Pizza, as
this vocabulary word will appear in the AlgeCross puzzle at the end of class.
Remaining Group Presentations on Elevation Activity
Multiplying Terms
Students should write the following possible equation in their notes:
243324 . Do you believe this is a true equation? Why?
How about 432324 ? What do these equations demonstrate?
They demonstrate that when multiplying a series of numbers, the order does not affect the product.
It is more important that they understand the concept than that they know that it is called the
―commutative property‖.
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Students should write the following in their notes:
xx 52
What do you see? It is good to establish how many terms there are here, and that we are not
combining them. We are multiplying them. Some students may already want to start multiplying
them based on their memory of how to do so. Stop them by asking them all to focus on the
following discussion. Give all students time to think about this before allowing students to speak.
Is the following a true or a false equation? xxxx 5252
Ask for justification in student responses. Once that has been judged a true equation, ask if we
can alter the right side in the following manner. Some students will hopefully connect this step to
our conversation about 243324 .
xx
xxxx
52
5252
Ask if we can multiply 2 and 5, and then how we can re-write xx . Someone will likely now
know to write this as 210x . What have we done? This question helps us see if they can connect
their vocabulary to this new situation. Ideally, we question them so that they articulate that we are
multiplying two terms, the result is one term, and we have therefore simplified the expression.
It may also be helpful to make the comparison of evaluating equivalent expressions as we did with
combining like terms. For example, you could have students evaluate xx 52 and 210x for the
same value of x, or you could ask students to do some version of the following, and then compare.
Evaluate )3)(( xx when 2x . Evaluate 23x when 2x .
It might also be interesting to do this comparison when 2x , or to ask students to determine if
the expressions will have the same value when x is a negative number.
Ask your students to simplify the following if they are able to do so:
)2)(4( dc
Once students have had a chance to consider this example, ask what makes it different from the
previous example. We are still multiplying two terms, but in this case they are not like terms.
Some may believe, thinking along the lines of the lesson on combining terms, that nothing can be
done. Others may reorganize the numbers and variables and find a way to put them back together.
This leads the class to a very important distinction.
Like terms are necessary to combine terms, but not to multiply them.
You (or another student) may confirm the convention for your students that multiple variables in
the ―variable part‖ of a term are typically written in alphabetical order.
Unlike with combining terms, we have a few problems for students to use for practice right away
before moving on to rectangle area.
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How to Study for a Math Test
Announce (if you have not already) that CUNY Start Math Exam #1 is two classes away. Write
this question on the board:
How do you study for a math test?
Collect their ideas on the board. Once you have discussed these, you may also give them the half-
sheet with suggestions from us. There will probably be a lot of overlap — that is fine. You could
also give out the half-sheet in the next class, possibly revising it according to what students said.
Either way, the most important message to communicate is that students need to do math
problems. How they choose or create them can vary.
If you have encouraged students to use highlighters or any device for marking trouble spots, you
can mention this in the bulleted list. You can edit the last item on the list to suit your campus.
Remind students that they will also have their second Times Table test alongside the first exam,
and they should be working on both their quickness and accuracy.
My Review Problems for CUNY Start Exam #1
We will have some problems to give them to practice from (in the next class), but we also want to
encourage them to go back and find/create problems themselves. Give students a defined period
of time to go back, review their work in the binder, and begin to collect/create problems on the
mostly blank sheet My Review Problems for CUNY Start Exam #1. They should note the source
of each problem, including date, handout title, and problem number.
Encourage students to copy down several problems before they try and solve any. If they only
copy one or two problems, they will probably remember the solutions that appeared beside them
in the notes or on the handout. Tell them that if they create interesting practice problems, we will
include them in a pre-exam review in the following class. If you do not convey that this is a
serious assignment, many students will not take this seriously.
AlgeCross I
Some of your students may not have any experience with crossword puzzles. To orient them on
how it works (before giving it to them), you could draw the following on the board with the clues:
Across
2. Ravaged by Hurricane Katrina.
(two words)
Down
1. Clay and sand barrier protecting
low-lying areas from floods.
Discuss crossword conventions,
particularly ―(two words)‖. After you have finished this discussion, give out AlgeCross I. You
will decide if it works best to have students complete this and discuss their answers in this class, to
have them begin it in this class but discuss at the beginning of the next class, or to give it out
purely as homework.
1.
2.
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Multiplying Terms
Simplify.
1. 223 xx
2. )2(8 2x
3. )3)(5( ba 4. ))(6)(6( xxyxy
5. ))(( 2 xyyx 6. )4)(2)(3( 22 yxxxy
Rabbits Aren’t the Only Multipliers The world’s human population in 1804 was one billion. It took 123 years for the population to reach two billion, but only 34 years to reach three billion. Six billion people inhabited the Earth in 1999, and we will reach seven billion a little over 12 years after that — sometime in 2011. Source: U.S. Census Bureau
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Suggestions for How to Study for a Math Test
Studying for a math test must include doing math problems. Looking at notes is not enough.
Go back in your binder to the beginning of the course and look for places where you were
challenged by the math in class notes, vocabulary, classroom activities, or extra practice. It helps if
you have marked these for review, whether by highlighting, circling, or another method.
Copy any problems that challenged you (along with the date and page number of the handout) on a
fresh sheet of paper (but not the answers!). After doing these problems, check your work against
the solutions in your binder.
Create similar problems to ones that challenged you. Sometimes you may create problems that are
much easier or harder than what we faced in the class, but it is still worth trying to do this.
Form a study group, or work with another student in the class. You can create problems for each
other to do, do problems together, or quiz each other on vocabulary and times-table problems.
Ask your instructor if there are any extra problem sets that you can work on to help you prepare for
the exam.
Go to math tutoring for help and additional practice problems. (When you are in credit math
classes, go to the math tutoring lab in the library.)
Suggestions for How to Study for a Math Test
Studying for a math test must include doing math problems. Looking at notes is not enough.
Go back in your binder to the beginning of the course and look for places where you were
challenged by the math in class notes, vocabulary, classroom activities, or extra practice. It helps if
you have marked these for review, whether by highlighting, circling, or another method.
Copy any problems that challenged you (along with the date and page number of the handout) on a
fresh sheet of paper (but not the answers!). After doing these problems, check your work against
the solutions in your binder.
Create similar problems to ones that challenged you. Sometimes you may create problems that are
much easier or harder than what we faced in the class, but it is still worth trying to do this.
Form a study group, or work with another student in the class. You can create problems for each
other to do, do problems together, or quiz each other on vocabulary and times-table problems.
Ask your instructor if there are any extra problem sets that you can work on to help you prepare for
the exam.
Go to math tutoring for help and additional practice problems. (When you are in credit math
classes, go to the math tutoring lab in the library.)
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144
My Review Problems for CUNY Start Exam #1
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My Review Problems for CUNY Start Exam #1 (continued)
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AlgeCross I
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Clues to AlgeCross I
Across
7. We cannot ________ 4x and 2y.
8. The number of cubes in this figure (2 words).
10. 88 , for example.
11. The straight distance across a circle passing through the center.
13. 6xy, for example.
14. (4, 11), for example (2 words).
15. yx 62 , for example.
17. The 2 in five-squared.
18. The ____ of –10 and 8 is –80.
Down
1. 5 and –5, for example.
2. Eight times eight can be re-written as ________ (2 words).
3. A function has three views — the equation, the table of values, and the ________.
4. Its six faces are squares.
5. (3, 10) for the function 162 xy .
6. 5x and –100x, for example (2 words).
9. Italian scientist who disagreed with the Greeks over the speed of falling objects.
10. The distance above or below sea level.
12. The ____ of –10 and 8 is –2.
15. The process of calculating the value of 3xy when 9x and 2y .
16. Re-writing an expression using fewer terms.
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CUNY Start Core Math Curriculum, Class 12
Rectangle _______
You will need 1-inch color tiles and rulers for Rectangle _____. First ask students to measure a
single tile. Then ask students to cover the rectangle with tiles and discuss this measurement. (See
the CUNY GED lesson set Area Matters for more discussion on this.) Students should write Area
in the title once you have reached this point in your discussion. Depending on the classes the
students come from, they may or may not have had this ―covering with squares‖ experience with
area. It is a good quick review for all of them. Allow students to verbalize how the area can be
found in at least two ways — by counting the number of squares, and by multiplying the side
lengths (there are also methods that involve adding groups — 2 groups of 6 tiles, for example). In
order to emphasize the concrete representation for your students, have them record the following
below the figure before moving on to Rectangle Area with Variable Side Lengths.
Area of a figure = the number of squares required to cover the surface
Area of Rectangle CDBA = 62
= 12 square inches
Rectangle Area with Variable Side Lengths
Sketch the first rectangle and corresponding table of values
on the board. Provide the headings and lengths for the table
of values you see to the right, and ask students to calculate
the appropriate areas.
Discuss this as you did with perimeter.
Class Contents N EA F
12
Rectangle _____ and Rectangle Area with Variable Side Lengths
Combining and Multiplying Terms
Student- and Instructor-Generated Exam Practice (Extra Practice #6)
Length x Rectangle Area
10 6
25 x
Shapes used to
cover the figure.
Length of one side
of each square.
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Sketch the second rectangle on the board and its
corresponding table of values, including the headings
and the lengths 8 and 5 for f and g.
Students should do the calculations and discuss their results. There
are two likely solution methods. One method involves calculating
the area of each of the two smaller rectangles.
96128 and 4085
Some students may be unsure of what to do now. Do we add the
individual areas or multiply them? Some will want to multiply as a
reflex because multiplication is the operation that echoes in their
head when they think about area. One way to help them is to have
them think back to what the multiplication represented in the
opening figure — the number of squares that will cover the surface.
What does the 96 tell us? What does the 40 tell us? How many squares are needed to cover the
largest figure?
You will also see some students who figure out that the longer side of the large rectangle is 512
or 17. The area of the largest figure then becomes:
136817
Give your students the remaining lengths and ask
them to complete the area calculations.
Length f Length g Large Rectangle
Area
8 5
Length f Length g Large Rectangle
Area
8 5 136
9 6
2.5 4
f g
5
8
12
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Of course the most interesting discussion comes in the last
row. If students follow the ―two smaller areas‖ model, they
should have areas of 12f and fg. Combining them, then,
should result in .12 fgf Encourage your students to write
the individual areas in the figure.
For students who follow the second solution method outlined
above, the longer side has a measure g12 . The area, then,
can be calculated by multiplying the two sides of the large
rectangle:
)12( gf or fg )12(
The reasoning here is more sophisticated. If it does not come up, you do not have to bring it up
because we will get to it later. Anyone who uses this approach must use parentheses to indicate
that the multiplication involves the entire longer length. This is an absolutely correct
representation of the area of the larger figure. For now, though, we are going to avoid talking
much about this method because our work with the distributive property comes later. Salute your
students‘ work on this, but let them know that for now, we will emphasize the method that
involves combining the smaller areas. Ask them to use this method for now.
Combining and Multiplying Terms
Discuss the first problem on the board with the entire class.
Encourage them to write the area inside the rectangles but also to
clearly indicate the expressions that represent area and perimeter
below each figure.
Once your students have gained some facility multiplying terms by
expanding, reorganizing, and then compressing the numbers and
variables, you may gradually let students introduce quicker methods.
What I mean by this is the ability to look at an expression such
as )2)(4( aa and see that you can multiply the numbers first, and then
the variables. It is best to spot students taking these shortcuts themselves and having that form the
basis for the discussion.
As you are reviewing these problems with your students, occasionally ask if terms are like terms,
and if they need to be for us to complete the operations.
Student- and Instructor-Generated Exam Practice (Extra Practice #6)
If you have a chance when students are entering the class or during one of the class activities, take
a look at the problems that students have devised to practice for the exam. When you reach this
later point in the class, you can talk about what strategies students may have used to study for the
exam since the last class. If students have created some good problems, get some of them up on
the board. Finally, you can give them our practice problem set and have them get started.
fg
12f
g
f
12
a
6
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Rectangle ______________
A B
C D
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Rectangle Area with Variable Side Lengths
x
8
g12
f
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Combining and Multiplying Terms
1. For each of the following, calculate the perimeter and area of the largest figure. Clearly label each
of your measures. If you need more space, you may sketch the figures on separate notebook paper
and do the calculations there. All four-sided figures are rectangles.
a.
b.
c.
d.
a
6
b
c
9
8
x
2t
2.5
r
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e. f.
2. Simplify by multiplying and/or combining terms.
a. )3)(3)(3)(3( xxxx b. )2)(4)(3( cba
c. xx 27 d. xx 27
e. ))(3( 2xx f. 23 xx
2x
5x
b
b
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g. yxx 4112 h. )4)(11)(2( yxx
i. xxxx 93)6)(4( j. )()3( 2 xyx
k.* True or False: 43 32)5(.)4( aaa
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Extra Practice for CUNY Start Exam #1 (Extra Practice #6)
Your job is to sell headbands. For each headband that you sell at Headbands-
R-Us, you earn a commission of $1.25. If you do not sell any headbands, you
do not earn any money.
Complete the table below. Using what you know about functions, and thinking
of the columns as inputs and outputs, include the appropriate function equation.
1. Is (5, 6.5) a solution to the above function? Explain why or why not.
2. Identify a solution to the headbands function that does not appear in the table.
Function Rule:
x y
Solution Number of
Headbands Sold
Your
Commission ($)
0
1
2
3
7
15
32.50
Ben Wallace of the Chicago Bulls.
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3. Simplify if it is possible to do so. No calculators!
a. )6(4 aa b. )10()1(14
c. )8(2 d. xyx 1042
e. ))(3)(2( xxx f. 30100
g. 264 bbb h. 05.
8.
i. xx 1410 j. )14)(10( xx
k. 1032 yx l. )10)(3)(2( yx
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4. Calculate the perimeter and area of the largest figure.
Clearly label each measure. All four-sided figures are
rectangles.
5. Evaluate xy4 when 2x and 2y . 6. Evaluate 2422 xx when 5x .
7. Create an expression whose value will be 14 when 3x . Be creative!
8. On January 3 in Minneapolis, Minnesota, the low temperature at 5 a.m. was F11 , and the high
temperature at 2 p.m. was F9 . How many degrees did the temperature rise from 5 a.m. to 2 p.m.?
9. The lowest point in the continental United States is in Death Valley, California. It is 282 feet
below sea level. The highest point in New York State is Mount Marcy, at 5,344 feet above sea
level. What is the difference in elevation between these two points? Include a small drawing that
illustrates the information that has been given in this problem.
y
10
2y
x
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10. Complete the missing outputs and solutions in the table of values. Then graph the solutions.
11. Complete the missing outputs and solutions in the table of values. Then graph the solutions.
104 xy
x y solution
5
4
3
2
1
0
92 xy
x y solution
4
3
2
1
0
–1
–2
–3
–4
x
y
x
y
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CUNY Start Core Math Curriculum, Class 13
Class Contents N EA F
13
Review Extra Practice and student-devised practice
Times Table Test #2
CUNY Start Exam #1
Patterns in Functions and Their Graphs, part 1
Review Extra Practice for CUNY Start Exam #1 and student-devised practice
Times Table Test #2
CUNY Start Exam #1
Make sure to leave at least 75 minutes for this exam.
Patterns in Functions and Their Graphs, part 1
Students can begin doing these tables and graphs when they have finished the exam. Those who
do not get them started in class can do them for homework.
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Times Table Test #2
44 212 22 1111 43 86
118 1010 112 74 88 711
310 912 127 102 84 115
106 93 126 77 92 97
73 125 83 76 1212 28
55 99 75 810 72 98
109 105 94 62 63 124
107 311 52 128 1210 32
64 42 411 85 66 116
69 87 119 65 1110 33
54 1211 53 312 104 95
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CUNY Start Math Exam #1
1. Complete the missing values. (3 points total)
2. Simplify as much as possible by finding each sum, product, difference, or quotient. (1 point each)
a. 78 b. 5
0 c. 112
d. 7
14
e. )11(2 f. 37
13 10
-24
+
x
36
+
xx
+
5 1.5
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g. 27 h. )5(5 i. 15.
6.
j. )2)(3( k. )20(2 l. 0
5.1
3. Check ―Yes‖ or ―No‖ for each of the
following. (1.5 points total)
4. Calculate and simplify as much as possible
the perimeter and area of the largest figure.
All four-sided figures are rectangles.
(1 point each)
Perimeter =
Area =
Yes No
Is 62 x an example of an equation?
Is 62 x an example of two like terms?
Is 62 x an example of a function?
Is 62 x an example of an integer?
Is 62 x an example of an expression?
Is 62 x an example of a levee?
b
a
12
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5. Is )5,3( a solution to the function 12 xy ? (Answer yes or no.) (1 point)
6. Explain how you determined your answer to the last question. (1 point)
7. Complete the missing outputs and solutions in the table of values. Then graph the function
solutions. (3 points total)
8. Evaluate xx 2 when 5x . (1 point)
9. Evaluate 22xy when 2x and 3y . (1 point)
32 xy
x y solution
–3 9
–2
–1
0 3
1 1
2
3 –3
4
x
y
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10. A new gym for women called Belinda’s Biceps has
opened in Midtown. Each month the cost to a member
is calculated using the following function:
285.2 xy
where x represents the number of times a member visits
the gym in one month, and y represents the cost for that
month in dollars.
Using the same function and its context (visits to the gym
and cost), describe the meaning of the numbers shown in the
table at the right. (1 point)
11. Complete any missing inputs or outputs for the following function: 63 xy (2.5 points total)
x 10 –4 1.5
y 30 –15
12. Simplify each of the following as much as possible. (1 point each)
a. xyx 1123 b. )11)(2)(3( xyx
x y
12 58
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c. xxx 825 d. 8452 22 xxxx
13. Simplify by re-writing each of the following as a single term with an exponent. (1 point each)
a. ))()(( 3 xxx b. 23)(b
14. In Portland, Oregon, the temperature one afternoon was F11 . If it dropped to a low of F8
that evening, how many degrees did the temperature drop? (1 point)
15. More than 60% of the population of the Netherlands lives below sea level. If the top of a levee in
Amsterdam, one of the biggest cities in the Netherlands, is 32 feet above sea level, and the
neighborhood below it sits at 18 feet below sea level, what is the difference between these two
elevations? Make a small diagram that illustrates this problem and your answer. (1 point)
16. Re-write the following expression with an equivalent expression that uses addition: 113
(1 point)
Challenge Problem
Determine the function that corresponds to this
table of values: x 2 5 12
y 1 –14 –49
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Scoring Guide for Exam #1
1. Each entry is worth .5 points. The maximum is 3 points.
sum = 6.5
product = 7.5
base numbers
are 9 and 4
base numbers
are 12 and –2
2a. –1
2b. 0
2c. –9
2d. –2
2e. –13
2f. –10
2g. –14
2h. 0
2i. 4 (Students who ―start at the bottom‖ — see the divison lesson — are more likely to get this than
those that use the traditional algorithm)
2j. 6
2k. 22
2l. undefined
3. Each correct response is worth .25
points. The maximum score is 1.5
points.
4. Perimeter = 2422 ba (The order of the terms is not important.)
Area = aba 12 (The order of the terms is not important.)
5. yes
Yes No
Is 62 x an example of an equation? x
Is 62 x an example of two like terms? x
Is 62 x an example of a function? x
Is 62 x an example of an integer? x
Is 62 x an example of an expression? x
Is 62 x an example of a levee? x
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6. Students must refer to the input of 3 and output of –5 ―fitting the rule‖ or ―making the equation
true‖ or must demonstrate that when 3 is the input in the function, –5 is the output.
7. Each output the student must calculate is worth .25 points. The total for the outputs is 1 point.
Each solution should be written as an ordered pair. Each is worth .125 points. The total for the
solutions is 1 point. If a student made an error calculating one or more outputs and then wrote
those outputs in the solution column, they should not be penalized again. The graph is worth 1
point and must include a line connecting at least two correct function solutions. We cannot
penalize a student who correctly draws the line through two solutions rather than plotting all 8
because the line of course represents all the solutions. Plotting all 8 points but not including a line
is worth .5 points.
8. 30
9. –36
10. Do not give any credit if a student simply restates the meaning of the variables without mentioning
the numbers — ―x represents the number of times a member visits the gym and y represents the
cost for the month in dollars.‖ You are instead looking to see if the student can appropriately
attach the numbers to their context — ―If a member visits the gym 12 times, the cost for the month
will be $58.‖ Half-credit (.5) may be awarded if the student mechanically attaches the numbers to
the variables but does not really link them together — ―12 is the visits. 58 is the cost.‖
11. Each input or output is worth .5 points. The maximum is 2.5 points.
12a. yx 214 (Alternate order is acceptable.)
12b. yx266 (It says simplify as much as possible, so it must be one term)
12c. x ( x1 is acceptable, though it is good to discuss this later)
12d. 83 2 xx (Alternate order is acceptable.)
13a. 5x
13b. 6b
14. 19 degrees
15. 50 feet (.5 points for a clear, accurate diagram and .5 points for the correct difference.)
16. )11(3 (Many students will both write the expression and simplify. That is okay.)
Challenge. 115 xy (Don‘t give any additional points for this, but write something positive if
they get it.)
Total Points = 38
x 10 –4 8 –7 1.5
y 36 –6 30 –15 10.5
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Patterns in Functions and Their Graphs, part 1
For each function, complete the missing outputs and graph the solutions. If you can draw a line or
curve to represent all solutions, please do so.
1.
2.
12 xy
x y
–4
–3
–2
–1
0
1
2
3
4
5
12 xy
x y
–3
–2
–1
0
1
2
3
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3.
4.
xy 3
x y
–4
–3
–2
–1
0
1
2
3
4
62 xy
x y
–4
–3
–2
–1
0
1
2
3
4
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CUNY Start Core Math Curriculum, Class 14
Class Contents N EA F
14
Review Patterns in Functions and Their Graphs, part 1
Patterns in Functions and Their Graphs, part 2 and Discussion
Extra Practice
Review Patterns in Functions and Their Graphs, part 1
You will be able to identify errors in the graphs from the homework almost instantly. It is difficult
to review the graphs on the board unless you put up large sheets of 1‖ graph paper or dry-erase xy-
grids and have students transfer their work. We do not recommend that you free-hand virtually
any graphs because it can be quite confusing for students. Rather than giving explicit attention to
a point that appears to be graphed in error, ask students to describe the orientation of the points
they graphed overall for a function, and whether any point(s) seem to deviate from that
orientation. When they see a point does not seem to visually fit in with the others, it can be a
reason for them to recheck their calculations. Another way to avoid telling students where they
may have gone wrong is to have them begin by comparing their graphs in pairs. Students can
usually work out any errors on their own.
Patterns in Functions and Their Graphs, part 2
Students need practice calculating outputs and graphing functions, and here there is plenty of that.
Still, our goals in this activity go beyond that. We want students to be able to spend time
observing function equations, the changes in inputs and outputs, and the shapes of the graphs of
several functions in order to determine some patterns for themselves. The teacher will help to
guide their work and also help students attach some new vocabulary to these observations.
Combined with part 1, there are lots of graphs here — 12 in all. That many are needed so that
students can have enough examples of the various kinds of linear and non-linear functions to
actually see some patterns. If your students had to do all eight of these graphs in class, it would
take up too much class time. We have to have a quicker way of giving them some graphing
practice, while arriving in a reasonable amount of time at a full set of completed tables and graphs.
Our suggestion is that you complete some of the function tables and graphs by hand in advance,
leaving 4-6 functions for the students to do themselves. Try and leave a mixture of function types
for them — at least one each that is linear, quadratic, and cubic. Also, keep in mind that #6 and #9
are considerably more challenging. (An alternate approach to those two functions is to ask your
students about the ―number part‖ of 2x and 3x , which may help them realize to do the
exponent first and then multiply by –1.) Once you have filled in the tables and graphs that you
want, photocopy the whole packet — one per student. In the class, form groups of three or four
students and have them work together to complete the handout. Be sure they are using pencils.
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In order to facilitate the Discussion, it is best if a group of students can look at all of the graphs at
once. If the photocopies are double-sided, it will be impossible for a single student to look at her
own 12 functions at once. However, we do not recommend wasting paper by copying on one side
only. You could double-side (but not staple) the copies, and then have different group members
turn different sides up so that everyone can see all 12 functions at once. Try to leave time so that
students can carefully assemble their packets (of 12 functions) later, and remember to bring a
stapler to class. Give your students plenty of time to investigate these tables, equations, and
graphs, and provide newsprint and markers for students to use to record their observations.
In the discussion that follows the group work, be sure that all of the following ideas come out:
a. Functions that have no (visible) exponent for x have solutions that form a straight line
when graphed. Ask students if they know a name for these functions. If not, tell them
they are called ―linear functions.‖ Students should label all of the linear functions.
b. Functions that include 2x are U-shaped. They might face up or they might face down. We
call these ―quadratic functions‖ — label these beside all the appropriate graphs. If you ask
students if these functions are also linear, they should be able to say no. You can also
label these functions ―non-linear‖.
c. Functions that include 3x have an unusual curved shape. You can ask your students to
describe what this looks like to them. If no one has a better idea, use ―climbing snake.‖
We call these ―cubic functions‖ — label these beside all the appropriate graphs. These
may also be labeled ―non-linear‖.
d. In linear functions, there is a constant change in the outputs when inputs are consecutive.
e. In functions that are not linear (non-linear), the change in outputs is not constant.
Your students will probably surprise you with some of their clever observations. This is an
important opportunity for our students to think like scientists — “What do I see here? What
seems to be going on? Is this always true?” When they make an observation that you want them
to explore further, have them create an additional example using the blank xy-grids that follow the
Discussion handout. We recommend you copy these grids on the back of the Discussion handout
in case you need them. Take time for the groups to report their observations to the class.
If you have ―Math Workshop‖ classes, you will follow this work with some more exploration of
function graphs on the graphing calculators.
Extra Practice #7
Note that the directions for problem #1 leave out the informal name for the shape of a cubic
function (―climbing snake‖ or other). You should either add this in by hand or electronically
before you copy and hand these to the students, or ask students to write in whatever informal name
your class will use.
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Patterns in Functions and Their Graphs, part 2
5.
6. You can think of 2x as ―the opposite of 2x .‖ This means that you should square the input first,
and then determine the opposite of that result.
xxy 42
x y
–2
–1
0
1
2
3
4
5
6
102 xy
x y
–4
–3
–2
–1
0
1
2
3
4
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7.
8.
44 xy
x y
–2
–1
0
1
2
3
4
13 xy
x y
–2
–1
0
1
2
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9. You can think of 3x as ―the opposite of 3x .‖ This means that you should cube the input first,
and then determine the opposite of that result.
10.
33 xy
x y
–2
–1
0
1
2
22 xy
x y
–3
–2
–1
0
1
2
3
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11.
12.
12 xy
x y
–3
–2
–1
0
1
2
3
xy 4
x y
–3
–2
–1
0
1
2
3
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Discussion of Patterns in Functions and Their Graphs
You should discuss the following questions in small groups formed by your
teacher. After you have had a chance to explore your ideas, write what you
have discovered on newsprint provided by your teacher.
1. Look at the equations and graphs for the 12 functions that you graphed on the previous pages.
What patterns do you notice? Is there a way to predict the shape of the graph based on the
equation?
2. Look at the tables of values, equations, and graphs. What patterns do you notice? Is there a way
to predict the shape of the graph based on the table?
3. What else do you notice?
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Extra Practice #7
1. Identify the shape of the graph of each function. For each, you should answer one of the
following — straight line, U-shape, or
a. 34 xy
b. 113 xy
c. xy
d. 162 2 xy
e. xy
f. )6(3 xy
2. Classify each function. You should answer one of the following — linear function, quadratic
function, or cubic function.
a. 34 xy
b. 113 xy
c. xy
d. 162 2 xy
e. xy
f. )6(3 xy
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3. Complete any missing information in the table of values. Then graph the solutions.
4. Complete the missing outputs and solutions in the table of values. Then graph the solutions.
42 xy
x y
–3
–2
–1
0
1
2
3
xy
x y solution
–5
–4
–3
–2
–1
0
1
2
3
4
5
x
y
x
y
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5. Congressman Babble recently gave a speech outlining his new policy
proposals. A researcher recorded the number of people listening by
counting the number who stayed in their seats and remained awake.
The following function was created to describe the number of listeners:
60015 xy
where x represents the number of minutes that Congressman Babble was speaking, and
y represents the number of listeners.
Complete the table of values.
a. What does the table of values show about the speech and the number of listeners?
b. Congressman Babble is being challenged in the next election by Judith Wrightspeak. She
wanted to bring a group of reporters to Congressman Babble‘s speech in order to embarrass
him. If the speech began at 10:00 a.m. and she wanted to bring the reporters in just as no one
was left listening, what time should they have entered? Show how you calculated your
answer.
Number of Minutes Elapsed
in Congressman Babble‟s Speech
Number of
Listeners
0
5
10
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6. Find numbers that satisfy each of the following equations.
10________ 10))((
10________ 10)(
)(
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CUNY Start Core Math Curriculum, Class 15
Class Contents N EA F
15
Review Extra Practice
Returning CUNY Start Exam #1 and Reviewing selected problems
Exploring Squares
Introduction to Square Roots
Introduction to Factoring and Factors and Finding Factors
Review Extra Practice
If you have whiteboard xy-grids, have students put the homework graphs beside the table of values
that they also put on the board. You may also have a few students put up their work on the ―–10‖
problems. Much of the rest of the Extra Practice is discussion-based.
Returning CUNY Start Exam #1 and Reviewing selected problems
You can take a number of possible approaches to returning students‘ tests, reviewing selected
problems, and having students correct their errors. We have found that if we simply return the
tests and then immediately try to have students review selected test problems together, many
students will focus primarily on looking through their own tests and figuring out what mistakes
they made. This makes it difficult to have a productive, useful class-wide discussion.
One way to prevent students from being distracted by their own tests is to review the most missed
problems before returning the tests. You could make a handout with the most missed problems
(cutting and pasting from the test) and give that out so that students have the problems in front of
them when they go over them as a class.
Another option is to return the test as soon as you have graded it (probably at or near the end of a
class to avoid the distraction during class time), not review any problems in that class session, but
instead assign students to correct their errors and turn in their corrections a class or two later. You
could use the form provided here for students to record their errors and corrections. After
collecting and reviewing students‘ corrections, you can decide which of the most missed problems
would still be worth reviewing as a class.
No matter which approach you use, you should return the exams to students within a couple of
classes after the test (possibly in student conferences or in Class 14 if you have graded them and
recorded their results by then). Students should keep the tests in their binders, possibly in a
separate section from the rest of their math materials.
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Exploring Squares
Provide color tiles for students, and encourage (or tell) them to work in pairs for the initial two-
sided handout with no title. Also have rulers available for students who may forget the
measurement of the tiles. Emphasize appropriate units in these first figures. After you have
discussed them both, (or as you discuss each one), connect the work to Counting Cubes.
Ask how we might use an exponent to express what is described in the statement. We want to use
a mathematical statement that will show what we have described in words. Once you have gotten
the equations on the board, make certain that students have written them near their statements, and
if you believe that it is important, they can label the elements in each equation.
A square with one side measuring 3 inches
has an area of 9 square inches.
The first side of the handout Exploring Squares focuses on the exponent equation and how we can
link it to squares, side length, and area. It probably makes sense to do the first problem together
because units are not apparent here. Students may assume that we are continuing with inches and
square inches. It is best that you discuss this as a group. When units are unknown, we either use
―units‖ and ―square units‖ or even ignore them entirely (especially when using variables for side
lengths or area measures). Notice that on the back of the handout, we gradually strip units from
the figures. For the items where units are given, however, it is good to discuss what these
measures mean — especially something as conceptually challenging as ―square miles‖.
The problems on the back hone in on the instance where the area of a square is given and students
must determine the length of one side. The ―squares‖ are also subtly or not so subtly changing to
look more and more like square root symbols. After students have completed these and you
discuss the side lengths, it will be interesting to see if any of them figure out what we are doing.
Go back to student notes and introduce this new symbol. Students should be introduced to the
language — square root. You may want to talk about the meaning of ―root‖. Certainly, it may
help them if they think of the ―root of the square‖ if they forget the language. Below the name for
the symbol, we should be certain that students have two ways to imagine this — a number-based
concept and an area-based concept. Seek both of these out from your students. Remember to
have the students look at the number that ultimately was inside the modified square root symbol
on the handout — square area. You can show these two ways of thinking of square root as shown
below.
932
Length of
one side.
Turned into
a square.
The area of
the square.
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Square Root
Before going to the next handout, sketch one square on the board that
should be put in student notes.
Ask students to create an exponent equation and a square root
equation that both describe this square in terms of area and side
length.
Introduction to Square Roots
With your weakest students, if they are having trouble knowing how to interpret a square root
expression, encourage them to ―complete the square‖ by sketching an addition to the symbol. This
may remind them of a way to conceptualize the number inside the square root as the square‘s area.
Of course it helps if students, at this point, have grown accustomed to writing area inside of the
rectangles in our earlier drawings.
Introduction to Factoring and Factors
In student notes, ask students to record the following: 3412
What have I done to the 12 here?
You have shown that 12 can be re-written as the product of 4 and 3.
Are there other ways to write 12 as the product of two whole numbers?
Does anyone know what this process is called — this process where we re-write a number in two
parts using multiplication? It is called factoring. It is a verb — something we do.
Have students add one of these definitions in their notes:
Factoring is re-writing a quantity as two (or more) quantities multiplied together.
OR
Factoring is re-writing a quantity as the product of two (or more) quantities.
When the area of a square is 16,
what is the length of one side?
What number times
itself is 16?
16
Area =
25 square
inches
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186
The word ―quantity‖ may seem a little formal here, but it helps make the definition applicable not
just to factoring numbers, but also to factoring terms and expressions later in the semester. To
help make the definition clear, you may want to ask what the quantity we factored was and what
were the quantities being multiplied together.
After we have factored 12 as many different ways as we can ( 34 , 62 , and 121 ), we can list the
numbers that were a part of those products as the factors of 12. Factors are things — nouns. The
factors of 12 are 1, 2, 3, 4, 6, and 12. Another way to think about factors is the ways that the
original number can be divided up without having any remainders.
It is time to give your students a little practice with this. What are the factors of 27? of 25? of
17? One of your students may be able to identify numbers that only have 2 factors (1 and itself)
as prime numbers. Your students are ready for Finding Factors. They should work in groups of
2-3. Finding Factors has seven solutions, mostly multiples of 12 (48, 60, 72, 80, 84, 90, and 96).
If a group finds a solution quickly, push them to find more or even all solutions.
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187
My Exam Problems to Review
Problem (Copy the problem here.)
Mistake (What was the mistake?)
Correction (How could you correct it?)
Topic(s) (What math topic(s) did this problem involve? What section(s) in your notes can you look at for
more problems like these? Dates and page numbers of handouts can be helpful here.)
Problem
Mistake
Correction
Topic(s)
Problem
Mistake
Correction
Topic(s)
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188
Problem
Mistake
Correction
Topic(s)
Problem
Mistake
Correction
Topic(s)
Problem
Mistake
Correction
Topic(s)
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189
Use color tiles to cover this figure. Then use the figure to complete the statement below. Use
appropriate units.
A square with one side measuring ___________________
has an area of __________________________.
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190
Build a square using 16 color tiles. Then use your figure to complete the statement below. Use
appropriate units.
A ______________ with an area of ________________________
has sides that each measure _________________________.
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191
Exploring Squares
1. Complete the equation and the related description.
A square with one side measuring ________________
has an area of _____________________.
2. For each row, complete the missing measurements, equations, and statements.
Diagram Equation Description
_____
_____52
Area =
36 square
inches
Side
length = 36_____
A square with a side measuring
________________ has an area
of _______________________.
A square with a side measuring
eight miles has an area of
_______________________.
Area =
Side
length =
A square with a side measuring
________________ has an area
of _______________________.
27
Area =
Side
length =
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192
3. For each of the following, determine the missing side length. All four-sided figures are squares.
i.h.g.
f.e.d.
c.b.a.
____________ ____________
____________ ____________ ____________
= 12114416
1100 9
____________
81 sq. units
4 sq. feet ________________________
64 sq. inches
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193
Introduction to Square Roots
1. Simplify each of the following.
a. 36 b. 20100
2. Complete the missing measurement for each diagram. Then create an exponent equation and a
square root equation that relate to the diagram.
3. Evaluate 10x when 25x .
Diagram Exponent Equation Square Root Equation
Area = 8
Area = 144 ___
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194
4. Evaluate yx 2 when 49x and .50y
5. Consider the expression x8 . There is an operation in this expression, but the symbol for that
operation is not shown. This also happens in the expression )11)(6( . Mathematicians agree on the
operation even though one is not visible.
The expression 1005 is similar to the above examples — an operation is involved even though it
is not shown.
Can you figure out how to simplify 1005 ? Show your work.
6. Complete the table of values and graph the solutions for the following function: xy
x 4 16 9 36 25 1 0
y
y
x
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195
Finding Factors
Find a number that is smaller than 100 and that has more factors than 100.
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196
CUNY Start Core Math Curriculum, Class 16
Class Contents N EA F
16
__________ Fractions
Writing Fractions in Higher Terms
Large Numbers of Eligible Comm. Coll. Students Do Not Apply for FA
Writing Fractions in Lowest Terms
Extra Practice
_____________ Fractions and Writing Fractions in Higher Terms
Shade the first rectangle, ask what fraction this represents (or illustrates), record the fraction to the
right, and discuss the meaning of the numerator and denominator in the following manner.
Not all fractions refer to part-whole relationships, but we will think of them that way in this
demonstration of equivalent fractions.
Direct your students to shade the same portion of each subsequent rectangle. For each, they
should record the appropriate fraction (showing the number of shaded and total sections) to the left
of the rectangle.
For example, the fourth rectangle should appear as follows:
Review the fractions. Ask if they all represent the same portion of a whole, or the same amount.
3
1
Shaded sections
Total sections
18
6
Shaded sections
Total sections
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We have a name for fractions that look different but that represent the same portion of a whole.
Does anyone know what we call these fractions? (Equivalent fractions.) At this stage, you can
have students complete the title of the handout — Equivalent Fractions.
On the handout to the right of the figures and their individual fractions, direct your students to
record this relationship.
These fractions are equivalent because they all mean that the same portion is shaded.
18
6
12
4
6
2
3
1
These fractions indicate different numbers of sections or pieces, but represent the same amount, or
the same portion of the whole. The extra lines (or slices) do not change the amount shaded.
We will not always be able to draw rectangles in order to determine when fractions are
equivalent. We have another way to know this. Your students still should have enough room to
record the following (next to the rectangle illustrating 2/6):
6
2
23
21
3
1
We have already shown that the outer two
fractions are equivalent using the physical
model. The inner fraction is the new piece
— showing that 1/3 is hiding inside of 2/6.
Point out the horizontal line segment in the
2/6 rectangle. Ask what it did to the total
number of sections and to the number of shaded sections. (It doubled both.)
It will help to have the other two examples
there in front of students, and on their
papers.
6
2
23
21
3
1
Doubling both the number of shaded sections and total sections results in an equivalent fraction.
12
4
43
41
3
1
Quadrupling both… results in an equivalent fraction.
18
6
63
61
3
1
Multiplying both by six… results in an equivalent fraction.
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198
So why do these changes all result in equivalent fractions? The simplest way for us to help them
recognize this is to lead them to see the ―1‖ in each calculation.
To the side, ask students how to
show/shade 5/5. How much is 5/5? How
else can we write this? Do this again for
2/2, 12/12, and 100/100. Get agreement
that each of these is a whole, or one.
Then look back at the previous equations.
At this point, you should judge whether students need another visual example before moving on.
If you think that they do, you may direct your students to turn the handout over and to shade ¾ of
the first rectangle. Ask them to shade an equivalent fraction of each rectangle, write the new
fraction to the left of its rectangle, and show using an equation such as the one above how each
new fraction is equivalent to ¾.
You should probably model the beginning of each equation on the board showing
4
3
to clarify that you want each equation to start with ¾ and end with the new fraction.
When we re-write a fraction after multiplying the numerator and denominator
by the same number, we call this writing a fraction “in higher terms”.
If you think that students could use another abstract numeric example before trying the handout,
write 9/10 on the board and ask students to write one or two equivalent fractions in higher terms.
Then go to the handout.
When going over the handout, occasionally ask students which fractions are equivalent and, more
importantly, why they are equivalent. We hope that they can point out the ―1‖ involved. It is
particularly worth asking about at least one of the examples that involve negative numbers.
Watch out for the problem that asks students to multiply the numerator and denominator by 0.
When they do this, the fraction will be undefined. Instead of relying on the rule that an
expression is undefined whenever we have 0 in the denominator, it would be great to discuss 0/0.
This is a more subtle case than other instances with 0 in the denominator. According to our ―from
the bottom‖ approach to division, we would ask ourselves, ―How many groups of 0 does it take to
make 0?‖ Unlike our previous examples with 0 in the denominator, it is possible to make 0 from
groups of 0. However, that question can be answered correctly by 0, 1, 10, 100, or any number.
Are all of these answers correct for this division? Students may not be able to settle this on their
own. You may need to ask a more leading question, such as “Is there one defined answer?”
6
2
23
21
3
1
Doubling both the number of shaded sections and total sections results in an equivalent fraction.
We are really multiplying by 2/2 here. What is the value of 2/2? What happens when we multiply a number by one?
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199
Large Numbers of Eligible Community College Students Do Not Apply for Financial Aid
and Writing Fractions in Lowest Terms
The figures here are certainly alarming. ―Poorest‖ is used here to describe students whose
household income was between $0 and $9,999. The percentages that did not file for aid were
actually 29% for independent students without children and for dependent students in these
households. We are using 30% because it will help us to demonstrate why it can be useful to write
fractions in lowest terms.
How can we describe the 30% figure in another way without using the word „percent‟?
We are looking for ―30 out of 100 eligible students do not file‖. We can write this as a fraction.
100
30
Now let‟s look at this fraction compared with another one. Are these fractions equivalent? Why?
How could we translate this new (equivalent) fraction into the context of the handout?
Saying that 60 students out of 200 eligible students do not file for financial aid is the same as
saying that 30 out of 100 do not do so.
Now we are going to explore writing our fraction in “lower” terms. We will talk about why we
might want to do this a little bit later.
To do this, we need to reverse what we did when writing fractions in higher terms.
100
30
We want to re-write 30 and 100 in such a way that we reveal the “one” that brought it up that
high. Can we re-write the 30 and 100 in two parts each using multiplication so that we reveal that
“one” (or so that we reveal a common number)? What do we call the process of writing each
number in two parts using multiplication? (Factoring.)
100
30
1010
103
This is the point where we should ask about 10/10 so that students see why the fraction will be
equivalent with or without this ‗one‘. Our last step then is to permit students to eliminate the 10s
— the ―common factors‖ — because they equal 1. Multiplying 3/10 by 1 does not change it. We
can write it as 3/10.
100
30
1010
103
10
3
?
200
60
100
30
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200
Once students have guided you through this whole process of reversing what we did previously,
you will probably want to re-write it in the following order to show the new direction that we will
be going.
10
3
1010
103
100
30
So what are the equivalent fractions? 3/10, 30/100, (and 60/200).
If those are equivalent, we can use 3/10 to say something about students and financial aid. What
can we say?
If it has not come up already, you may want to have the following discussion with them:
What do you think about this?
20
6
520
56
100
30
Did I just write 30/100 in lowest terms?
As another approach here, instead of creating this second example yourself, you could ask students
to find another way to re-write 100
30 in lower terms, other than as
10
3.
Which would you prefer to use in this case — 30/100, 60/200, 3/10, or 6/20? Basically, it can give
us a clearer picture when we refer to fractions or ratios in their lowest terms. You can emphasize
that we are often asked to write fractions in lowest terms.
It is important that students recognize that these certainly are all equivalent fractions. However
this is an example of writing a fraction in ―lower‖ terms, but not in ―lowest terms‖. A student may
be able to point out that 6/20 can still be written in lower terms and explain how to do so.
―Lowest‖ terms can mean that we have found the ―one‖ made up of the largest common factors
possible. In a more practical sense, it means there is no more ―work to do‖ — there are no more
common factors that we can find and cancel. It is not always easy for students to simplify to
lowest terms in one step, because they may not be good at identifying the greatest common factor,
in part due to weaknesses in their multiplication tables. It is fine for them to simplify in more than
one step. When they have written a fraction in lower terms, encourage them to check to be sure
that it is in lowest terms — that there are no remaining common factors.
Give your students a few more fractions to write in ―lowest terms‖ in their notes, such as:
16
12
20
35
45
18
Insist that they show the intermediate step that shows how they factored the numerator and
denominator to reveal the ‗one‘. Students may be able to simplify these fractions without writing
that step (possibly by dividing on top and bottom), but this method is critical preparation for
algebra work that we will be doing over the course of the semester.
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201
With the brief practice given in the handout, students should be clear that to write a fraction in
lowest terms, they need to factor in such a way as to identify common factors and then eliminate
the common factors.
Problems involving negatives can be solved in two ways. It is helpful for you to discuss these
methods with the class. One way — and probably the trickier one — is to factor in such a way as
to identify the common factors.
2
3
22
23
4
6
Another way — likely the way that will work best for most students — is to remember that a
negative divided by a positive is negative. Then you can do the problem as if it was 4
6 and simply
drop the negative sign in at the end. The question, though, is where to place it.
Show the students these four division problems:
2
8
2
8
2
8
2
8
The first two are 4 because division involving a positive and a negative is negative. However,
the third expression equals 4, not 4 .
The fourth one is the hardest. Students will often initially think that the third and fourth
expressions are equivalent; so be sure to discuss this.
To help students think about this, write x on the board, and ask what it means. Then ask what
they think the third expression means. A useful way to think about the fourth one is ―the opposite
of 8/2‖. That again is 4 . All of these mean the same thing. Therefore,
2
8
2
8
2
8
We usually recommend to students that they bring a single negative sign to the front of any
fraction. This can help to reinforce the second approach mentioned above (dealing with the
negative symbol separately). Of course if there are negative signs in both the numerator and
denominator, this will not be appropriate. You may wish to tell students that it is conventional
(though not absolutely necessary) to avoid leaving a negative sign in the denominator.
The last two problems are included so that students will be careful when only a 1 remains in the
numerator or denominator. Ask students if 1
5 is equal to 5. Ask students if
4
1 is equal to 4.
Extra Practice #8
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202
______________________ Fractions
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203
204
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Writing Fractions in Higher Terms
1. For each of the following, write a fraction that is equivalent to 3
2, as indicated.
a.
53
52
3
2 b.
123
122
3
2
c.
x
x
3
2
3
2 d.
5.13
5.12
3
2
e.
)4(3
)4(2
3
2 f.
2
2
3
2
3
2
a
a
2. Write the following fractions in higher terms, as indicated.
a.
65
61
5
1 b.
53
52
3
2
y
x
y
x
c.
yy
y
y
44 d.
)2(8
)2(3
8
3
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3. Begin with the fraction 5
2. For each of the problems below, find an equivalent fraction by
multiplying the numerator and denominator by whatever is indicated. Show your calculations.
a. by 5: b. by –3:
5
2
5
2
c. by x: d. by 3x:
5
2
5
2
e. by a number or term of your choice: f. by 0:
5
2
5
2
4. For each of the following, determine the missing numbers that go into writing the fraction in
higher terms, and complete the fraction in higher terms.
a. 217
2
7
2
b.
16
5
4
5
4
c. 36
11
6 d.
204
x
e. 222 x
y
x f.
aba 6
10
2
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Large Numbers of Eligible Community College Students
Do Not Apply for Financial Aid
In October 2008, a federal advisory committee released a report
titled “Apply to Succeed: Ensuring Community College Students
Benefit from Need-Based Financial Aid.”
The study found that nearly 30% of the poorest community
college students who are dependents or independent without
children did not apply for financial aid. These students almost
certainly would have received significant support from the
government if they had requested it.
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Writing Fractions in Lowest Terms
Re-write each fraction in lowest terms. In each case, show the
stage where the numerator and denominator are factored to
reveal a common factor.
1.
43
42
12
8
2. 12
20
3. 6
15 4.
9
35
5.
4
6 6.
22
20
7. 3
15 8.
28
7
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Extra Practice #8
1. Complete missing values for this unusual function.
2. Write the factors of 20 that are not also factors of 12.
3. Write the following fractions in lowest terms. In each case, show the stage where the numerator
and denominator are factored to reveal a common factor.
a. 4
10 b.
15
20
c. 42
24 d.
14
18
y the number of factors of the number x
x y
4
9
16
25
50
8
9
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4. Write three fractions that are equivalent to 11
3. One of them must include one or more variables.
5. Simplify.
a. 25.
5.1 b.
0
8 c. 42 )(t
6. Simplify each of the following.
a. 121 b. 48
c. )3)(2( xx d. )3(2 xx
7. Evaluate 112 x when 4x .
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8. Evaluate 11x when 16x .
9. Create a linear function that has a solution at (–2, 6). Include a calculation that demonstrates why
this is a solution to your function.
10. Create a non-linear function that has a solution at (–2, 6). Include a calculation that shows why
this is a solution to your function.
11. Observe the figures below.
a. On the following page, sketch the next two figures.
Figure 3Figure 2Figure 1
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b. Record the information regarding the figures in the table of values, where x represents the
figure number and y represents the number of little squares in that figure.
x
y
c. Challenge — can you determine the function equation using this table of values?
Class 16 Challenge
Calculate the perimeter of a rectangle whose area is 212x . There is more than one correct answer.
Check yours with your College Transition Math teacher.
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CUNY Start Core Math Curriculum, Class 17
Class Contents N EA F
17
Review Extra Practice
Dividing Terms
…for all 0x .
Expression Puzzles
Review Extra Practice
Dividing Terms
Most of the hard work has been done already. Students need to apply what they have learned
about writing fractions in lowest terms to expressions — if they factor the numerator and the
denominator, they can then eliminate common factors. Encourage students to write the expansion
before they do any elimination. Give students the following examples in their notes, asking them
to help you simplify them.
b
b
7
5
7
5
7
5
7
5
b
b
b
b
x
x
14
6
7
3
27
23
14
6
x
x
x
x
When students tell you to cancel factors, ask why we can cancel them. They can hopefully
explain in some way that when we cancel a b from the top and a b from the bottom, we are
essentially canceling a 1, which does not change the result. The expression remains equivalent to
the original expression.
Give your students a couple of minutes to try the following problems, and then have them guide
you through them in the same way. Be sure that they show the expansion of the
variables/exponents.
3
2
35
10
xy
yx
29
12
x
x
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Students should move to the handout. Watch out for the problem 26
)3)(2(
x
xx. Students will often
―cancel‖ everything and decide that the result is 0. You will probably need to question them to
help them remember that the elimination occurs because we are multiplying by 1. In this case,
though, all we have are the 1s.
…for all 0x .
We need to familiarize students with the language ―for all 0x ‖, ―when 0x ―, ―for all non-
zero x‖, and similar wording. Sooner or later, students will be confronted with it. If we do not
talk about this vocabulary, students will often think that they need to do something with the
information, confusing ―simplifying‖ expressions with ―evaluating‖ expressions. Not knowing
what to do, they will sometimes substitute 0 for x even though that is exactly what the direction
says you must not do. Give the handout to the students and ask them what they think it means. It
can help for us to compare the problem with one such as the following:
Evaluate 102 x when 5x . or maybe even better: Evaluate 20
3 2x when 2x .
Review the task in this example. The problem gives the value of x and is asking for the value of
the expression. The problem from the handout does not ask for the value of the expression. It
only asks students to simplify it. Another way to think about it is to say that we will write the
fraction in ―lowest terms.‖ The expressions will still be equivalent.
Rather than giving students values for the variables, the problem rules one value out — 0. Seek a
student who will be able to interpret this from the symbol.
Why do you think it says for all 0x ?
Here is a good hint that does not give too much away — I wonder why it says x cannot be zero but
it does not mention y. Hmmm…
In the end, it is important for students to realize there is nothing ―to do‖ with this information. It
is only provided because the denominator cannot be 0, or else the expression will be undefined.
Expression Puzzles
In the last puzzle in the second row and the first puzzle in the last row, the bases are not like terms.
Some students conclude that they cannot add them and may leave the sum blank (or write
something like ―cannot combine‖, ―N/A‖, or even ―undefined‖). Lead a discussion of this. In the
end, we hope that students will realize that we can indeed add 3a and 4b. The sum is 3a +4b. We
just cannot combine them.
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Dividing Terms
1. A group of 200 students were questioned, and 25 reported
that they were left-handed.
What is the fraction of left-handers in this group? Can you
express this fraction in lowest terms?
Try to describe the meaning of this particular lowest-terms
fraction in this context as if you were speaking to a friend
who does not understand or care about the math involved.
2. According to the Kingsborough Community College website, 55% of the students enrolled at
KCC in 2008 were enrolled full-time, and 45% were part-time.
Express each of these percents as a fraction and then as a fraction in lowest terms.
3. According to the most recent statistics from the MTA, 80% of the subway cars on the Q line were
found to have clean seats and floors.
Express this percent as a fraction and then as a fraction in lowest terms.
Rich Lefties? In 2006, researchers at Lafayette College and Johns Hopkins University* in a study found that left-handed men are 15 percent richer than right-handed men for those who attended college, and 26 percent richer if they graduated. The wage difference is still unexplainable and does not appear to apply to women. *Sinster and Rich: The Evidence That Lefties Earn More by Joel Waldfogel. Appeared in Slate on August 16, 2006
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4. Consider the figure at the right. What fraction of the small
triangular sections have a point inside them? Express this
fraction in lowest terms.
Simplify as much as possible.
5. zx
xy2
3 6.
a
ab
20
5
7. 2
3
8
4
x
x 8.
yzx
xyz212
6
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9. 26
)3)(2(
x
xx 10.
4
100xyz
11. st
sr
8
3 22
12. d
bc
12
5 2
13. xyz
zyx
8
4 23 14.
22
2
4
)2)(5(
yx
yxx
*15. 22
22
18
)2()3(
yx
xyx
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…for all 0x .
Simplify x
yx
6
24 2
for all 0x .
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Expression Puzzles
-6b
+
x
36x 2y 2
20xy
x
++
x
5xy 2
xy
b
11y5.5a
4a
x
+ +
x
4b3a
x
+
24y 2
8x
+
x
6x
12x 2
-4x
x
++
x
3x x
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CUNY Start Core Math Curriculum, Class 18
Class Contents N EA F
18
Perfect and Non-Perfect Squares
More Squares and Square Roots
Introduction to Polynomials
Adding Polynomials
Extra Practice
Perfect and Non-Perfect Squares
Give out this handout right away, and allow time for students to try and determine the
characteristics that lead to numbers being classified as perfect squares or not perfect squares.
In the discussion, we think that it is important to think about perfect squares in two ways. If a
number has a whole-number square root, it is a perfect square. Another way to say this (and this is
closer to our area-based discussions earlier) is to have a student imagine the number as the area of
a square. If the length of one side of this square is a whole number, then the original number is a
perfect square. We have discussed integers with students, and you could use the word ―integer‖
rather than whole number in these instances. The examples of 6.25 and 2.25 are in the ―Not
Perfect Squares‖ box because they are 2.5 and 1.5 squared respectively. You might be able to lead
some of your stronger students to see that and conclude that perfect squares have to be integers.
You may wish to write some of the following formulations on the board:
36 is a perfect square because 636 , and 6 is an integer.
(As a corollary, 36 is a perfect square because 3662 , and 6 is an integer.)
36 is a perfect square because when a square has an area of 36, the length of one side will be an
integer — 6.
In student notes, we should look more closely at non-perfect squares. Have
your students sketch the following:
What can you tell me about this square?
Clearly, the area of the square is 40. Determining the length of one side is
the tricky part. There is no whole number that we can multiply by itself to make 40.
What can you tell me about the length of one side?
If no student raises it, you can ask them more direct questions to keep the discussion going.
Area = 40
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Is the length of one side 5? Why or why not?
Students might think that a ―non-perfect square‖ means that the shape is not perfectly a square. In
that case, students might think that the lengths in this figure are 5 and 8. To illustrate a square
root, the shape still must be a square, regardless of whether the area is a perfect square or not.
Is the length of one side 6? Why or why not?
The goal is for all students to be able to determine that the length of one side is between 6 and 7.
A higher level of understanding would be determining if the length is closer to 6 or closer to 7. A
length of 6 would result in an area of 36, while a length of 7 would result in an area of 49. Since
40 is closer to 36, the area shows us that the length is a bit closer to 6 than to 7.
Depending on how concretely your students may be thinking, a student could raise the objection
that it is not possible to have an area of 40. They could be thinking about the color tiles and
realize that there is no way to have 40 of them formed into the shape of a square. In this instance,
we can add that 40 squares, including some that may have to be cut into pieces, can be assembled
into a square.
Now you need to move to the square root symbol.
Is 40 a perfect square? Why or why not?
What can you tell me about the following: 40
Some students might conclude that 40 is between 5 and 8, because 4085 . It actually is
between 5 and 8, but for the square root of a non-perfect square, we want our students to estimate
the value down more precisely — to between two consecutive integers. We still want them to be
able to identify perfect squares exactly.
More Squares and Square Roots
It can be useful to have some graphing calculators handy. When discussing students‘ work on
More Squares and Square Roots, you might distribute calculators so that they can confirm the
square root of the non-perfect squares. Encourage your weaker students to sketch a square when
they encounter a square root, and then try to determine the length of one side. Trial and error here
is fine.
Pick a possible length of the side. Okay, would that give us an area of 70?
No — is it too big or too small? Okay, then what is your next guess?
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Introduction to Polynomials
Students should title their notes ―Polynomials‖. You might ask them the meaning of the prefix
poly-. Think of polygon, polygamy… You can translate polynomials as ―many terms‖. Write:
Expression
52 x
What do you see?
Look here for ―expression‖ and ―two terms‖. Some may also give you ―polynomial‖ because it is
the topic of the discussion.
Yes, we can call it a polynomial, and we have another name that emphasizes that there are two
terms. Does anyone know it? After a student gives ―binomial‖, remind others of the prefix in
―bicycle‖ which indicates two wheels. (Also bilingual, bipartisan, bisexual, biped, bifocals…)
Expression Alternate names
52 x polynomial or binomial
Add the following expression:
Expression Alternate names
52 x polynomial or binomial
342 xx
How do you think we can name this second expression? When a student gives you ―trinomial‖,
remind the student of other words like ―tricycle‖, ―triangle‖, triathlon, tripod, trilogy…
Expression Alternate name/s
52 x polynomial or binomial
342 xx polynomial or trinomial
Consider the following expression: 5)7(2 xx . We would not name this expression a
trinomial. Why do you think not?
Allow students to talk this out. Yes, there are three terms here, but looking back at the other
examples, someone should suggest that we are able to combine terms in this last example. When
we name expressions binomials and trinomials, we are typically talking about expressions with
two or three unlike terms, or with two or three terms in their most simplified forms.
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Write this expression on the board: y
x
5
2
This expression is not a binomial. Why do you think it is not a binomial? What is different about
this expression?
The difference is the operation. A binomial is really an expression with two unlike terms being
added or subtracted, and similarly with trinomials and polynomials.
Add the following expression to the table:
How do you think we can name this expression?
122 23 xxx is a polynomial. Some students may offer ideas such as quadrinomial or
tetranomial — this is good thinking, but the convention is to refer to expressions with 4 or more
(unlike) terms simply as polynomials.
Expression Alternate name/s
52 x polynomial or binomial
342 xx polynomial or trinomial
152 23 xxx polynomial
What do you notice about the order of the terms in these polynomials? It is conventional to order
the terms of a polynomial so that the variable exponents are in descending order.
If you like, you may also include monomial in this vocabulary review. It is a tricky idea because
―polynomial‖ implies more than one term, but a monomial is also considered a polynomial.
Adding Polynomials
Continue with student notes. Give them a minute or two to decide whether they believe the
equation is true or false.
31026)310()26(
In discussing student ideas, be sure that there is understanding about the appropriate order of
operations on both sides. On the left, the parentheses tell us that we need to do any possible
calculations inside before going outside of the parentheses. On the right, we proceed left to right.
Expression Alternate name/s
52 x polynomial or binomial
342 xx polynomial or trinomial
152 23 xxx
?
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99
369
3104134
31026)310()26(
Once students are comfortable that both sides are equal, it is time to talk about exactly what is
going on here. Some possible questions for your students include:
What is the difference between the two sides?
Is the presence or absence of parentheses the only difference?
Thinking about the two groups in parentheses, what are we doing with them? (Adding them)
When we are adding numbers grouped by parentheses, does it change anything to remove the
parentheses and then follow that new order of operations?
Are you convinced this is always true?
It would be best if they/you suggest a new example to test that includes subtraction inside one or
both groups of parentheses. Remember, though, addition must be used between the parentheses.
1)4(610)14()610(
After two or three numeric examples, it is time for an example that includes variables.
)74()13( xx
What do you see? (An expression…two binomials…two polynomials…4 terms…)
After getting to ―two binomials‖, What are we doing with these binomials? (Adding them.)
What does the order of operations say that we should do first in order to simplify this expression?
(It says to do any calculations inside of parentheses before going outside.)
Okay, we have )13( x . Can we combine these? Why not? What about )74( x ?
Wait a minute — if the order of operations says that we have to do operations inside parentheses
before doing the operations outside of parentheses, and yet we cannot combine terms in )13( x
or )74( x , then are we stuck? It seems like we can do nothing…
This is a real opportunity for us to help our students understand why removing parentheses is even
necessary. In most presentations of adding and subtracting polynomials, it is simply stated as a
fact that students must ―clear‖ parentheses (in the subtraction case being careful about signs) as a
pre-cursor to combining terms.
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If only we could work with these terms when there were no parentheses around, we would be able
to combine like terms. If only there was a way to re-write this expression in such a way that
removes the parentheses (but that does not affect the value)…
Encourage students to re-write the expressions without parentheses before they move on to
combining terms.
The final problem on Adding Polynomials is a great counterpoint to the previous problem from
Dividing Terms:
16
)3)(2(2
x
xx
The Adding Polynomials problem involves combining terms, and so the terms are not eliminated
to produce a series of 1s, but are eliminated producing a series of 0s. The value of the sum is 0.
0)354()354( 22 xxxx
Extra Practice #9
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Perfect Squares Not Perfect Squares
6.25
27
2.25
1. If you believe you know what makes a number a perfect square, add a few more examples of
perfect squares to the box of perfect squares above.
2. What appears to be true about all perfect squares?
36
49
1
121
81
25
9
10
2480
109
2
45
52.5
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More Squares and Square Roots
1. Circle the numbers that are perfect squares.
2. Simplify each square root expression if you can. If the number inside a square root is not a perfect
square, indicate the two consecutive integers that the square root will lie between.
a. 49 b. 70
c. 32 d. 2016
e. 812 f. 140
120
144
121
44
72
9
18
35
64
100
90
25
50
36
42 10
8
1
2
3
4
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3. Complete the missing measurement for each diagram. Then create an exponent equation and a
square root equation that relate to the diagram.
4. Complete the table of values for the following function: 8 xy
x 100 64 9 36 25
y –8 1
Diagram Exponent Equation Square Root Equation
Area = 5
Area = 81 ___
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Adding Polynomials
Add the following polynomials:
1. )12()54( xx
2. )223()752( 22 xxxx
3. )242()15( 2 xxx 4. )10()13()52( xxx
5. )242()44( 22 xxxx 6. )12()12( 22 xxxx
7. )354()354( 22 xxxx
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Extra Practice #9
1. Johana works in the crime lab at the New York City
Police Department. A woman‘s shoe was found at a
crime scene. Johana measures it and then uses the
following function to estimate the height of the
suspect:
5619.1 xy
where x represents a woman‘s shoe size and y
represents the estimated height in inches of the
woman.
a. Using a calculator (it does not need to be a graphing calculator), determine an ordered pair
that is a solution to this function.
b. Interpret your solution in the context of this problem, using one or more complete sentences.
Use the information given in the problem when describing the values of your input and output.
2. Simplify.
a. y
yx210 when .0y b.
36
24
a
abcwhen .0a
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3. Which of the following is equivalent to the expression x20
5 when 0x ?
4. Simplify x
yx
21
14 2
when 0x .
5. Calculate the area and perimeter of the largest figure.
All four-sided figures are rectangles.
Perimeter =
Area =
6. Simplify.
a. )4)()(2( 2yxx b. )144()142( xx
a. x4 b. 1
4x c.
x4
1 d.
x4
1 e. undefined
7
y
x
5
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c. 105 xx d. 124
e. )116()14( 22 xxxx f. 6)2(
g. 64 h. )16(16
7. Complete the table of values and graph the function solutions.
22 xy
x y solution
–4
–3
–2
–1
0
1
2
3
4
x
y y
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8. Simplify.
a. )20(5 b. xxx 1032
c. ww 324 d. 4
15.6
9. The expression )142()3( xx is equivalent to which of the following?
a. 112 2 x b. 112 2 x c. 113 x d. 113 x e. 233x
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CUNY Start Core Math Curriculum, Class 19
Class Contents N EA F
19
Review Extra Practice
Rate of Change
Identifying and Interpreting Rate of Change
Review Extra Practice
Rate of Change
Before mentioning the word ―slope‖, we will use the language ―rate of change.‖ It is both correct
and a more friendly way to introduce the concept. ―Slope‖ will appear soon. If a student raises it
now (this is rare), you can compliment them on making the connection, but we will not talk about
slope right now.
We are going to be working with integer and decimal slopes for several sessions before
introducing the more difficult fraction slopes. Try and avoid those fraction complications for now.
Students should complete the tables of values on the first side of Rate of Change. They should be
able to classify these as linear functions based on the function equation. Could they have done it
only by looking at the inputs and outputs? How?
What do you notice that is similar or different about the
functions and their corresponding tables of values?
Push for accurate, precise statements. Incorporate vocabulary that was introduced earlier such as
―consecutive‖.
Here are some possible student responses:
They have the same inputs.
The outputs are different.
Both say 2x in the rules/equations.
The inputs grow by 1 (or, are consecutive).
The outputs are odd in the first function and
even in the second.
The outputs increase by 2. (Mark these
differences on the board and have your students
do the same.)
They are both linear functions
12 xy
x y
0 1
1 3
2 5
3 7
4 9
5 11
6 13
2
2
2
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The outputs on the right are 3 larger than the ones on the left. (A great observation! Can
anyone figure out why this is the case?)
Ultimately, the feature that we want to focus on initially is the difference in the outputs. This
constant change (when the inputs are increasing by one) is what can tell us that they are linear
functions. It is important that we give a formal name to this difference. Ask students to write the
following statement below the function tables, and tell them this is an absolutely critical concept.
The “rate of change” of a function is the constant change in the outputs
when the inputs are increasing by 1.
Ask students about the rate of change of each of the two functions in this handout. Once they
answer this, you can write below each function:
Rate of change = 2.
It is important to note that the single number that we use for the rate of change gives us
information about how both the inputs and outputs are changing. Write the following important
sentence on the board (possibly just beginning it and eliciting the rest from students):
As the inputs increase by 1, the outputs increase by 2.
Emphasize that this is what a rate of change of 2 really means.
Move students on to the additional examples. Tell them that for each function, they should first
complete the table, then identify the rate of change, and finally write the sentence that explains the
rate of change in terms of inputs and outputs.
In all of these cases, we want students to be able to identify the rate of change as a single number,
but also describe it in terms of how the inputs and outputs are changing.
For our reappearing Best Buy function, this is an opportunity not only to explain the rate of change
in terms of inputs and outputs, but also to describe that measure using the context of the problem.
As the number of cameras sold increases by 1, the weekly pay increases by $18.
This is a statement that students could make during the first session of the course, but now they
can see it as a more formal math measurement.
Once you have done the introductory handout, you may want to put one non-linear function table
on the board (such as 32 2 xy ) and ask students to complete a table of values (giving them
inputs that are increasing by one) and then determine the rate of change. This will be helpful so
that you can emphasize that we only discuss rate of change for linear functions. They should have
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a strong enough grasp of how linear and non-linear function tables appear to say why this is the
case. It is not absolutely necessary to do this before giving out the next handout. If you do not
have time to do this in this class session, come back and do this another time.
Ask your students to look at the examples that they just completed to see if they notice whether
the rate of change seems to appear anywhere else, aside from the change in the outputs. Someone
should notice that in each case, the rate of change also appears directly in
front of the x term in the function.
Go back to the first page to discuss why the 2 in this function rule is related
to the change in the outputs. This is challenging because in the
rule, it appears as a part of a product while in the table, the 2
appears as a difference.
Have your students verbalize how they originally completed the
outputs in the very first table.
“How did you calculate the output when the input was 3?”
Remind them that multiplication can be thought of as a number
of groups. When the input is 3, there are three groups of two.
When the input is 4, there are four groups of two. Each time
the input increases by one, this adds an additional group of two
— the difference in the outputs, then, should be 2. Once you
have had this discussion, students should label the rate of
change in the function rules on their handout. It is important
that they are able to identify the rate of change either from the
table or from the rule.
Identifying and Interpreting Rate of Change
Before giving students this handout, you should do one more problem with them in their notes.
A linear function has a rate of change of 6 and a solution at (10,65).
What does the rate of change tell us about the function equation?
We know that the function must begin with the following:
...6xy
Having a solution helps us know that when 10 is the function input, 65 is the output. You will
have a few students that can describe how to complete this function and test it to be sure it works.
12 xy
x y
0 1
1 3
2 5
3 7
4 9
5 11
6 13
Rate of change = 2
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Rate of Change
12 xy
x y
0
1
2
3
4
5
6
42 xy
x y
0
1
2
3
4
5
6
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Rate of Change
Rate of change = Rate of change =
As the inputs As the inputs
Rate of change = Rate of change =
As the inputs As the inputs
5.24 xy
x y
2
3
4
5
16 xy
x y
1
2
3
4
425. xy
x y
4
5
6
7
105 xy
x y
0
1
2
3
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Do you remember Best Buy Commissions? This function shows the
amount of your weekly pay (y) in dollars, based on the number of
cameras you sell (x).
What is the rate of change for this function?
Describe the meaning of this rate of change using the context.
Do you remember the reckless driving function? This function shows
the number of hours of ―good driving‖ classes you attend (x) after
getting a ticket, and the ultimate fine that you must pay in dollars (y):
What is the rate of change for this function?
Describe the meaning of this rate of change using the context.
15018 xy
x y
0
1
2
3
4
36040 xy
x y
0
1
2
3
4
5
6
7
8
9
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Identifying and Interpreting Rate of Change
1. Identify the rate of change in each of the following functions.
a. b.
c. d.
2. Identify the rate of change in the following function, and interpret the rate of change using
―inputs‖ and ―outputs‖.
14 xy
x y
0 42
1 49
2 56
3 63
4 70
x y
0 –8
1 –3
2 2
3 7
4 12
x y
0 13
1 5
2 –3
3 –11
4 –19
x y
–5 1.5
–4 2.25
–3 3
–2 3.75
–1 4.50
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3. A scientist tested a medicine in order to determine how effective it is in
producing antibodies. The following is a function that represents the
number of antibodies in a sample of blood from a patient:
10020 da
In this function, d represents the number of days that passed in the
experiment, and a represents the number of antibodies in a sample of
the patient‘s blood.
a. Complete the table of values for this function.
b. How many days did it take for the patient‘s blood sample to have 360 antibodies?
c. What is the rate of change for this function? How can it be interpreted using the context of
the problem? In other words, can you describe the rate of change in terms of ―days‖ and the
―number of antibodies‖?
Data for Medicine A
d a
0
1
2
3
4
5
Antibodies are y-shaped molecules of the immune system.
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4. Create a linear function that has a rate of change of 7. After you have written the function, write
an explanation of why your function has a rate of change of 7. If it helps, include a table of
values.
5. Create a linear function that has a rate of change of 5 and a solution at (4, 32).
6. Create a linear function that has a rate of change of –3 and a solution at (5, 2).
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CUNY Start Core Math Curriculum, Class 20
Class Contents N EA F
20
A Moment for Mental Math
Three Scenarios and the Distributive Property
Uses of the Distributive Property
Extra Practice
A Moment for Mental Math
If your students will go for it, make a big deal about trying to do these calculations in their heads
as they might in a store. You might instruct them to put their pencils down or on the floor to
emphasize this point.
Most of your students will use the distributive property in these problems without realizing it.
Allow students the opportunity to discuss their mental math solutions to both problems. Once you
have discussed their solutions, your students will need notepaper in order to record a more formal
treatment of this strategy.
Write the following equation on the board, ask students to consider if it is true, and then discuss.
)10.00.3(8)10.3(8
You can connect this separation of 3.10 into dollars and cents to the students‘ own solution
methods. This is a formalization of what they did. As you discuss the critical next step, also
connect it to students‘ own work on the problem.
80.24$
80.24
10.800.38)10.00.3(8)10.3(8
See if students can relate this to our work with multiplication as ―groups‖. See if someone can
articulate that eight groups of $3.10 would be the same as 8 groups of $3.00 and 8 groups of .10.
You might try setting up the next problem, and then see if students can follow the same steps to
formalize the separation process in the same way. The $12 base cost of the pizza might make
things a little more confusing, but see how they handle it.
)05.1(612
After going through those problems, give them two more to see if they can show the multiplication
―in parts‖ without your help. Ask students questions about the ―groups‖ involved so that students
are not merely following the procedures from the preceding problems.
)12.1(4 )102(35
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Three Scenarios and The Distributive Property
Our goal is to put three distributive property scenarios on the board at the same time so that
students can see all of them at once. Discuss each in turn, but try to keep all of them in front of
the students. You will need to think of this as you plan your work on the chalkboard/whiteboard.
Scenario #1
Is there a way to do the following multiplication “in parts” so that the arithmetic is simple?
)03.1(15
As you discuss student ideas on this problem, see if your students can describe it in the language
of groups — Fifteen groups of $1.03 is the same as fifteen groups of $1 and fifteen groups of .03.
Ultimately, the board work should explicitly show the multiplication in parts:
45.15
45.15
03.1500.115
)03.1(15)03.1(15
Scenario #2
Write the following instructions on the board:
1) Describe the multiplication using the language of “groups”.
2) Re-write the expression using addition.
3) Simplify the expression.
Start by just writing the first example and asking student(s) to take you through these three steps.
53
Once you go over this first example together, write the following expressions and ask students to
try them on their own.
)6(4 (Think of money.)
)2(5 x
The third example will challenge students because they may not see )2( x as a single quantity.
We are multiplying two things in the expression — 5 and )2( x . After this clarification, they
will hopefully be able to say ―five groups of )2( x ‖ and write the work that you see below. You
may let students know that professors may say ―five times the quantity 2x ‖ to indicate the
grouping in parentheses.
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If students do not show the step where parentheses are removed, ask how they simplified. If they
do show it, ask why parentheses can be removed.
105
22222
)2()2()2()2()2()2(5
x
xxxxx
xxxxxx
Scenario #3
Sketch two versions of the same rectangle on the board:
Working with the rectangle on the right, have your students determine the areas of the two
interior rectangles. When they are done, calculate the total area.
Once you have done this, tell your students that there is another way of thinking about the area of
the largest figure. They may come up with it on their own, but make sure it is discussed clearly.
This is the point where you should ask students the length of the horizontal side of the entire
rectangle on the left. Some will say 2x while others will say x2 . If they cannot work it out
themselves, show a numerical example for them to see that the operation must be addition.
How long is the total segment? How did you do this?
Now, ask students to think more generally about the area of rectangles. How do we calculate the
area of a rectangle? So how can we do this here if one side is 2x ?
In the end, your drawings should look like this:
)2(8 xAreaTotal 168 xAreaTotal
2x
8
2x
8
612
2x
8
2x
8
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Are these two rectangles the same? (Yes.)
How do their areas compare? (They must be the same.)
Therefore:
168)2(8 xx
At this point, we must synthesize these scenarios.
Scenario 1: 03.1500.115)03.00.1(15
Scenario 2: 105)2(5 xx
Scenario 3: 168)2(8 xx
What do you notice?
Normally the order of operations tells us that we must execute operations inside parentheses
before going outside of them. We have operations inside of parentheses in all three scenarios. In
#2 and #3, though, we have unlike terms and cannot combine them. It seems that we would not
be able to do the multiplication (in the same way that we appeared to be stuck when adding
polynomials).
What these scenarios show us is that we can do the multiplication when we carefully
follow one important rule — can you tell me what the rule appears to be?
Once your students have verbalized that the multiplication has to apply to both terms inside
parentheses, reinforce this by reminding them that parentheses (as discussed in class #1) can
indicate that numbers or variables are grouped together. The multiplication is of the entire group,
not just the first term.
The name for this important idea is the Distributive Property. Think of ―distributing‖ chocolates
(or color tiles or whatever) to students. The idea is that I am giving out chocolates to each student.
―Distributing‖ in algebra can mean that we will apply a calculation to all terms that are grouped
within parentheses.
Illustrate this ―distribution‖ visually in these expressions, connecting the number in front of
parentheses to each term inside parentheses.
Students should now be ready for the handout.
Extra Practice #10
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A Moment for Mental Math
Attempt the following problems only by using mental math. That means without a pencil and paper, a
calculator, or a cell phone.
1. In preparation for a seminar, a seminar leader buys new binders for 8
participants. Binders cost $3.10 each. What is the total cost of the binders
before any taxes are added? (Remember — think about how to do this in
your head!)
2. Pizza Amore charges $12 for a medium pizza, plus $1.05 for each
topping.
You decide to order a medium pizza with 6 toppings. How much does the
pizza cost before any taxes are added?
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Uses of the Distributive Property
1. For each of the following, re-write the second number
―in parts‖ and use the Distributive Property to simplify
the arithmetic.
a. )03.2(5 b. 10.69
c. 1215 d. )108(5
2. Consider the largest figure at the right. As we did in Scenario #3 in
class, write the total area of the largest figure using two different
expressions. All four-sided figures are rectangles.
20
10
a
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3. Consider the largest figure at the right. Write its total area using two
different expressions. All four-sided figures are rectangles.
4. Go back and review ―Scenario #2‖ from your class notes. Using this method of writing
multiplication as repeated addition (think about ―groups‖), write the following using addition and
then simplify. Show all work.
a. )6(2 x
b. )(4 yx
c. )10(3 2 x
d. )65(2 2 xx
5
x
x
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5. Multiply.
a. )4(10 x b. )5.2(6 x
c. )1(5 2 x d. )3( xx
e. )11(2 xx f. )5(2 x
g. )( yxx h. )43(5 rqp
6. Determine the missing variables or numbers.
a. 84)2___( xx b. 217___)(7 xx
c. ___5)11(___5 y d. 182___)(___2 x
e. 488___)___( xx f. 484___)(___4 x
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g. 9______)(9 x h. 6012___)___( xx
i. 3510___)2___( xx j. caba 18___3___)8(___
k.* 156___)4___( xx l.* 5015___)6___( xx
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Extra Practice #10
1. Write the area of the largest figure using two different
expressions. All four-sided figures are rectangles.
2. Multiply.
a. )8(3 x b. )8(10 x
c. )2( aa d. ))3((12 x
3. Show how the Distributive Property can simplify the arithmetic in the following problems.
a. )15.50(3 b. )05.3(12
20
12
x
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4. Write the following fractions in lowest terms.
a. 16
6 b.
36
80
5. Identify the rate of change for each of the following functions.
a. b.
c. 5.155.7 xy d. xy
6. Create a linear function that has a rate of change of 3 and a solution at (6, 12).
7. Create a linear function that has a rate of change of –2 and a solution at (–5, 16).
x y
0 –21
1 –18
2 –15
3 –12
4 –9
x y
0 .25
1 .75
2 1.25
3 1.75
4 2.25
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8. Simplify.
a. xz
yx
12
20 2 when x and z are non-zero. b. )112()13( xx
9. Before reading any further, look back at the handout
Identifying and Interpreting Rate of Change from the
previous class session. Find the function that examined
how well a medicine produced antibodies in a patient.
Imagine that the same scientist tested a second
medicine (Medicine B) and recorded the data in the
table at the right after following a patient for several
days.
a. How many days would it take for the patient‘s
blood sample to have 480 antibodies? Show how
you calculated your answer.
b. What is the rate of change for this function? How can it be interpreted using the context of the
problem?
Data for Medicine B
Days Passed Antibodies in the
Sample of Blood
0 60
1 90
2 120
3 150
4 180
5 210
6 240
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c. If you were ill and you had both Medicine B and Medicine A (from the problem you did in
class) to choose from, which would you prefer? Why?
10. Simplify each square root expression. If the number inside a square root is not a perfect square,
indicate the two consecutive integers that the square root will lie between.
a. 81 b. 35
c. 925 d. 10035
11. Simplify as much as possible.
a. 25 )(n b. 25 nn c. 8
0
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CUNY Start Core Math Curriculum, Class 21
Class Contents N EA F
21
Review Extra Practice
Pencil Packs
The Distributive Property — Addition and Subtraction
Identifying and Interpreting Rate of Change from graphs
Review Extra Practice
For problem #5d, some students may identify the rate of change simply by thinking of the
―hidden‖ –1, the number part of the term –x. It is worth having students make a table of values to
check this. Try to see if a student can suggest this. For #10c, it may help for students to think
about drawing the square represented by this expression. What would be inside the square?
Pencil Packs and The Distributive Property — Addition and Subtraction
Solicit students‘ informal solution methods to this problem. Probably the easiest way for students
to solve this problem using mental math is to use the Distributive Property with subtraction —
even if they do not identify it as such. However, after the work that we have previously done,
some students will probably use the Distributive Property with addition here.
As a part of the discussion that follows the problem that students solved mentally, we will need to
divide the pencil pack price into two parts. At some point, get the following equation on the board
and ask students if it is true before discussing how the distributive property can be used.
)06.00.3(5)94.2(5
Once you have confirmed the above equation with your students, connect the informal solutions to
the use of the distributive property below.
70.14$
30.15
06.500.35)06.00.3(5)94.2(5
If some students did the mental math other ways, you could also have them help you show how
they split the 2.94. Two likely possibilities are:
)94.00.2(5)94.2(5 and )04.90.00.2(5)94.2(5
Ask students which method they think makes the multiplication easiest. You do not need to push
them to choose the method using subtraction. Some students prefer to avoid subtraction.
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It is worth mentioning or eliciting that subtraction can always be re-written as addition, and so the
application of the distributive property with addition has to work the same with subtraction.
Try a few examples that include variables, such as )52(6 x or )59(5 yx , and then proceed to
the handout.
Identifying and Interpreting Rate of Change from Graphs
Create a large version of the graph of xy 2 (use 1‖ grid paper or whiteboard grid) and bring all
your students around to look at it together.
When thinking about rate of change, students can do at least two things — they could create the
table of values from the solutions and then determine the rate of change from that table, or they
could identify the change in the inputs and outputs directly from the graph.
It will be most useful to discuss the table of values method first. This will be understandable and
connect most directly to the method discussed earlier. Students must be careful, though, to
organize the solutions so that the inputs are increasing by one. There could be some discussion
here about a solution that would be a good ―starting point‖ as well as the need for subsequent
solutions to have consecutive inputs.
Once you have done the ―table‖ method, put that off to the side. It is time to talk about identifying
the rate of change directly from the graph. This is tricky.
Start the conversation by asking a student to identify one solution on the graph. Hopefully they
will choose one not at the extreme right of the graph. Write the solution as an ordered pair right
beside the solution or off to the side of the graph.
What input and output values are indicated in this solution?
A student can be asked to restate what rate of change is — the change in the outputs when the
inputs increase by 1. Starting at this initial solution:
What is the next input after the one shown in this solution?
Where is the solution with this input?
It may help to discuss more generally where the inputs are measured on the xy-grid, and the
direction in which they increase. Because they increase to the right, we can find the rate of change
by following the graph of a function to the right.
Remember that it is okay that students stick to the ―table‖ method, though some students may need
reminders on how to organize their tables so that the inputs are increasing by one.
Students can return to their desks, and there they should add a table of values and description of
the rate of change for the graph of xy 2 . Additional practice is included.
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Pencil Packs
A pack of pencils is on sale at Office Pro for $2.94. You
want to purchase 5 packs. Can you use mental math to
calculate the total cost before taxes? ―Mental math‖ means
trying to do this without writing anything, and without
consulting a calculator. Be prepared to explain your
solution method.
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The Distributive Property: Addition and Subtraction
1. For each of the following, re-write the second number ―in parts‖ and use the Distributive Property
to simplify the arithmetic. Use addition or subtraction, depending on which operation makes the
most sense — are you calculations easier if you round the second number up or down?
a. 97.15 b. 05.69
c. )07.3)(11( d. )85.24)(4(
2. Multiply.
a. )8(3 x b. )5.1(8 x
c. )6( xx d. )(10 yx
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e. )62(4 x f. )68(3 2 xx
g. )8(2 xx h. )83(10 a
i. )( 22 xxx
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Identifying and Interpreting Rate of Change from a Graph
What is the rate of change for this function? ________
Describe the rate of change using ―inputs‖ and ―outputs‖.
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Identifying Rate of Change from a Graph – More Practice
1. Identify the rate of change for each function.
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This graph shows the relationship
between the distance traveled and
the fare in a New York City
taxicab.*
2. What does the input measure in
this function?
3. What does the output measure
in this function?
4. What is the rate of change for
this function?
5. Describe the meaning of the
rate of change for this function
using the context.
NYC Taxi Fares
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
0 1 2 3 4 5 6 7 8 9 10 11 12
Miles Traveled
Ca
b F
are
* This function makes it appear that taxi fares only rely on distance traveled. In fact, taxi fares in New York City also rise when a cab is not in
motion for 60 seconds or is traveling below 12 miles per hour. These factors are not included here, and so the function is a bit over-simplified.
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6. Identify the rate of change for each function.
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CUNY Start Core Math Curriculum, Class 22
Class Contents N EA F
22
Subtracting Polynomials, part 1
Subtracting Polynomials, part 2
Rate of Change = _____________
Extra Practice
Subtracting Polynomials, part 1
The following should be put in student notes. Give them a minute or two to decide whether they
believe the equation is true or false.
In discussing student ideas, be sure there is understanding about the appropriate order of
operations on both sides. On the left, the parentheses tell us that we need to do any calculations
inside before going outside of the parentheses. On the right, we proceed from left to right. Once
your students have guided you through the calculations, highlight the inequality of the two sides.
137
3107
3414714
3486)34()86(
Once students are comfortable that the equation is false, it is time to talk about exactly why. Some
possible questions for your students:
Looking at the first row, what is the difference between the two sides?
Is the presence or absence of parentheses the only difference?
When we are subtracting numbers grouped by parentheses, does it change anything to remove the
parentheses and then follow the ordinary order of operations?
How does this compare with a similar situation that we looked at?
Look for students‘ ideas about why the two sides are unequal. Ultimately, it will help to remind
students that parentheses can be used to ‗group‘ items (one of our first College Transition Math
conversations). It will also help to show students a subtraction example such as the following:
106
In this expression, what are we subtracting?
? 3486)34()86(
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For the expression )34()86( , what are we subtracting?
The goal is to agree that we are subtracting the group )34( , or the quantity )34( .
Starting with 106 , how can we rewrite this expression using addition?
)10(6106
And so, do you think it is possible to rewrite this expression using addition? How?
)34()86(
Once you have been reminded by your students that we are subtracting the whole group or
quantity, we can rewrite the expression in the following way:
))3(4()86()34()86(
We have now changed the group so that we are adding the opposites. We have not changed the
operation inside parentheses.
Your students should now simplify both sides separately (following the order of operations) to see
if the two sides are equal. It will help them in particular to see the next step because it will make
the subtraction-to-addition conversion clear.
77
)7(14714
))3(4()86()34()86(
What we have, now, is a way to rewrite polynomial subtraction so that it becomes polynomial
addition. We must now start looking at algebraic polynomial examples.
...
)10()(72
))10(()72()10()72(
etc
xx
xxxx
Question students about how to do each step, and why it is done. They should add these
comments in their notes in the following manner.
We could test this by evaluating the original and simplified expression for selected value(s) of x.
))10(()72()10()72( xxxx Re-writing the subtraction (of the group) as
adding the opposites.
)10()(72 xx Now that we are adding binomials, we can
remove parentheses, and it will give us the same
result. Now we are able to combine like terms.
)3( x Combining like terms.
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A student may ask at this stage if the final expression can be written as 3x . This most likely
will be raised by a student who knows and separately has been thinking about the more traditional
way of doing this sort of problem. When a student asks this question, put the two expressions
beside each other. It is another opportunity for students to connect to the subtraction/addition
relationship that we raised early in the course. Use the question as a chance to compare
)3(x and 3x , but not to highlight an alternative (in this case, the traditional) method at this
time. It will be far too confusing.
This might be the most challenging topic of all in the course, and we are sticking to our belief that
we should have defensible reasons for what we do with students (and not ask them to flip signs
without knowing why).
Start by giving students Part 1, which only includes straightforward examples where addition is
present in both binomials. Do problem #1 together, and then let them do the rest on their own or
in pairs. Have students show their simplification on the board before moving on. To ensure that
they follow the method that we have outlined, we include the description of each step for the first
few problems, and we suggest that you show this on the board as well.
Subtracting Polynomials, part 2
Before giving out Part 2, have students go back to their notes.
How is this expression different from the ones that we just worked on?
We are looking for students to notice that subtraction appears not only between the groups, but
also within the second group.
We are going to review two ways of simplifying this expression.
Let‟s start by using the normal order of operations method. Show me how we can do that.
)103()124( xx Re-write the subtraction (of the group) as
addition.
Remove parentheses.
Combine like terms.
)27()39(
)27()39(
Following the order
of operations.
7
512)27()39(
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Now, we are going to investigate an alternative method similar to what we did earlier. We want
to remove parentheses before doing any calculations. Can we remove the parentheses right away
here? (No, we need addition, but there is subtraction between groups and inside parentheses.)
So we want to re-write the subtraction as addition in order to be able to remove parentheses and
combine terms. Which subtraction do you think we should re-write as addition first? (Students
may think of ―doing what‘s inside parentheses first‖, which reminds us of the order of operations.)
This second method may appear to you and the students as a very inefficient way to do the
problem, but it is important for them to see that it produces the same result. This will give some
justification for what we are about to do with algebraic expressions.
Let‟s look at a new expression:
)63()124( xx
Can we follow the order of operations and simplify what is inside
parentheses before doing anything else? Why or why not?
After discussing why we cannot follow the regular order of operations (the terms inside
parentheses cannot be combined), we will use our alternative method — re-writing subtraction
inside and then between parentheses as addition so that we can eliminate the parentheses and
simplify.
Work through these steps with your students before going to the problems on the handout.
)27()39(
Following the order
of operations.
7
512)27()39(
In order to remove
parentheses, we re-write
subtraction as addition.
7
2)7(12
2)7(39
)27()39(
)2(7()39()27()39(
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Rate of Change = _________________
The purpose of this handout is to give students additional practice using and interpreting rate of
change in all three views of a function, but also to rename it as slope. You may be surprised to see
how few of your students will have made this connection before now. Give them an opportunity
to rename rate of change before you simply tell them. Once it has been identified, put ―slope‖ in
the title of the handout, and in all the blank places on the front page of the handout. You can
simply say that from now on we will usually refer to rate of change as slope. They mean the same
thing.
There are quite a few places where students need to be careful in this handout. The second, third,
and fourth problems include a function with a slope of 0 (not 10), a non-linear function which
cannot have a constant slope (can you determine the next input and output?), and a tricky slope
because the function inputs that are shown are not consecutive. Several students will only focus
on the outputs and say that the slope is 4. Refocus them on the definition of rate of change/slope.
What do they think the outputs are for the missing (skipped) inputs? If some students are
skeptical, you could have them help you graph most of the points on a whiteboard x/y-grid, and
then ask what they think about the missing/skipped inputs and about the slope.
For the graph in this handout and for subsequent graphs that students encounter, continue to
encourage students to make a table of values to decide on the slope if they have trouble reading
rate of change directly from the graph.
In problem #9, students can again use a table to help them identify additional solutions if they are
not as strong working directly on the graph. For #10, students can find the slope if they rearrange
the solutions in a new table so that the inputs increase by one. This is challenging. You may need
to discuss whether the new table still shows the same function (and the same solutions) as the
original table if the solutions are listed in a different order.
Extra Practice
))6(3()124()63()124( xxxx Re-writing interior subtraction as addition.
)63()124( xx We are subtracting the second binomial, and
we are going to re-write this subtraction as
adding the opposites.
6)3(124 xx Because we are now adding polynomials, we
can remove parentheses.
18 x Combine terms.
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Subtracting Polynomials, part 1
Subtract.
1. )103()124( xx
2. )112()67( xx
3. )26()103( xx
)103()124( xx Re-write the subtraction (of the group) as
adding the opposites.
Remove parentheses.
Combine like terms.
)112()67( xx Re-write the subtraction (of the group) as
adding the opposites.
Remove parentheses.
Combine like terms.
)26()103( xx Re-write the subtraction (of the group) as
adding the opposites.
Remove parentheses.
Combine like terms.
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4. )97()15( xx
5. )1212()128( xx
6. )95()16( xx
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Subtracting Polynomials, part 2
Subtract.
1. )63()85( xx
2. )67()14( xx
3. )26()108( xx
)63()85( xx Re-write interior subtraction as addition.
Re-write the subtraction (of the group) as
adding the opposites.
Remove parentheses.
Combine like terms.
)67()14( xx Re-write interior subtraction as addition.
Re-write the subtraction (of the group) as
adding the opposites.
Remove parentheses.
Combine like terms.
)26()108( xx Re-write interior subtraction as addition.
Re-write the subtraction (of the group) as
adding the opposites.
Remove parentheses.
Combine like terms.
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Simplify as much as possible by adding or subtracting.
4. )52()64( xx 5. )52()64( xx
6. )2092()92( 22 xxxx 7. )2092()92( 22 xxxx
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Rate of Change = _______________
Determine the ___________ for each of the following functions. Be careful!
1. 2.
___________ = ___________ =
3. 4.
___________ = ___________ =
5. 65.1 xy ___________ =
Interpret this ___________ using ―inputs‖ and ―outputs‖.
x y
3 16
4 20
5 24
6 28
7 32
x y
5 10
6 10
7 10
8 10
9 10
x y
1 1
2 2
3 4
4 7
5 11
x y
2 –2
4 2
6 6
8 10
10 14
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6.
slope =
7. A function has a slope of –4, and one solution is given in the table. Identify the missing outputs.
8. A function has a slope of 2.5, and one solution is given in the table. Identify the missing outputs.
x y
3 12
4
5
6
7
x y
–2 –10
–1
0
1
2
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9. A function has a slope of –3 and a
solution that is shown at (1, 5).
Identify at least four more solutions to
this function, and use them to graph
the function.
10. Determine the slope for the function
that has the following table:
Challenge
A function has a slope of –4 and one solution is given in the table. Identify the missing outputs.
x y
3 27
0 12
2 22
–1 7
1 17
x y
6 –10
4
2
0
–2
Answer to Rate of Change=Slope Challenge:
The solutions are (4, -2), (2, 6), (0, 14), and (-2, 22)
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Extra Practice #11
1. You need to rent a car for
the weekend. You locate the
following advertisement for
a local rental agency.
a. Using the information
from the advertisement,
complete the following
table.
b. What is the slope of this function? How can it be interpreted using the problem context?
2. Create three functions — one linear function, one quadratic function, and one cubic function —
that all have a solution at (–5, 32).
Miles Driven 0 1 2 3 4
Total Rental Charge ($)
Brooklyn’s Best Car Rentals
Special Weekend Price!
One mid-sized car rental for a flat rate of $96, plus $0.12 per mile.
Check out our hot
new models!
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3. Multiply.
a. )11(3 x b. )42(5. 2 x
4. Consider the following function: 52 xy .
a. Classify the function as linear, quadratic,
cubic, or none of these.
b. Create a table of values and graph the
function. See if you can determine all of
the integer solutions that will fit on the
graph that you are given.
5. Simplify ab
ba
2
6 2 for all non-zero a and b.
6. Simplify.
a. 82 b. )8(2 c. )8(2
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7. Edy and Mario are math teachers working
with a group of students. They are trying to
prepare for a lesson on decimals. Some of
their students are struggling with
the following calculation:
98.37
How could Edy and Mario use the
distributive property as a way of helping the
students to do the calculation without a
calculator, and without using the traditional
method for multiplying decimal numbers?
Show and explain how they could use the distributive property to do the calculation.
8.
-9
-36
+
x
+
x
-2x
14xy
3x -3x
x
+
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9. Calculate the area of rectangle A, the area of rectangle
B, and the area of the largest rectangle.
Then, using a fraction, compare the area of rectangle
A with the area of the largest rectangle. In other
words, the area of rectangle A is what fraction of the
area of the largest rectangle? Express this fraction in
lowest terms.
10. Simplify each square root expression. If a number inside the square root is not a perfect square,
indicate the two consecutive integers that the square root will lie between.
a. )15(100 b. 23
c. 645 d. 2)49(
11. Add or subtract, as indicated.
a. )116()84( xx b. )116()84( xx
2x
2x
x
BA
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CUNY Start Core Math Curriculum, Class 23
Class Contents N EA F
23
Review Extra Practice
Cube roots
Rectangle Area Revisited
Students Begin Exam Study
AlgeCross II
Review Extra Practice
There is quite a lot here, but the remaining tasks in this class are not overwhelmingly time-
consuming. Do not feel pressured to rush through student work here.
Cube Roots
Students should record the following in their notes:
100
Review the meaning of this notation — that we are looking for the length of one side of a square
whose area is 100, or we are looking for the number that makes 100 when multiplied by itself.
Now add the following on the board and in student notes:
100 is the same as 2 100 .
When no number is visible in the upper-left corner of the square root symbol, it means
there is an invisible 2 there. Does that remind you of any other convention we have discussed?
A student might talk about the expression x, in which the coefficient of 1 is assumed, or where
there is an ―invisible‖ exponent of 1.
In the case of 2 100 , what do you think the 2 is about?
A number of things could be suggested — perhaps the friendliest version is that “a number
multiplied by itself a total of 2 times should make 100.”
Other possibilities — “We are asking ourselves what number squared makes 100.”
“What number to the second power equals 100?”
“Two copies of what number [multiplied together] make 100?”
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In addition to these kinds of suggestions, we encourage you to elicit the exponent equation that is
related to the square root expression.
101002 because 100102
Now it is time for some new notation:
What do you think is going on here: 3 8 ?
Look for a student-friendly way of defining this with the class, and get it in student notes. “A
number multiplied by itself a total of 3 times should make 8.”
Other versions — ―What number cubed makes 8?”
―What number to the third power equals 8?”
“Three copies of what number [multiplied together] make 8?”
The more formal version:
283 because 823
Students might be able to divine the name for this root without us giving it if we remind them of
square root but say that we cannot use the same name here. The connection to ―cubed‖ in the
exponent equation would help to make this possible. Have them say the problems in the handout
out loud to practice saying ―cube root of…‖.
We are not taking the time here to try and connect this to the physical conception of a cube. If we
did, we would be able to talk about ―the root of the cube‖ as we did with the square root.
Rectangle Area Revisited
Students should sketch the following diagram in their notes.
As we have done before, we will write two expressions that
represent the area of the largest figure.
168)2(8 xx
As students give you these expressions, review
how they arrived at them. Use these
explanations to label the parts on the board and
in student notes. See right.
Typically, we have asked students to determine
the area of a figure after giving them the
lengths of the sides. We are going to have
them go in reverse in this activity — starting
with the total area and writing the lengths of
the sides whose product would give that area.
8
x 2
168)2(8 xx
Length of
one side.
Multiplication.
Length of
other side.
Total area.
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That is probably enough of an introduction before giving your students the handout. When you
are circulating and looking at student work, question students to see how they can be sure that
their equations are correct. Because they also have an understanding of the distributive property,
they should be able to multiply side lengths and return to the total area.
When you have discussed solutions for these problems, take a moment to focus on the final
problem. Students should add it to their notes:
)52(3156 xx
What are we starting with on the left side of the equation? The total area. What is another name
for this two-termed polynomial? Binomial.
What are we ending with on the right side of the equation? The lengths of the sides. What
operation are we using between them? Multiplication.
Take a look at this: 248 . Is this in any way similar to what we have done above?
We have written the 8 in two parts using multiplication — that is factoring. We are doing the
same thing with our binomial that represents the total area. We are rewriting it in two parts (two
lengths) using multiplication. This is also factoring — factoring a binomial.
Students Begin Exam Study
You may want to ask students what topics we have studied so far. Compile a list on the board.
This can help students realize the breadth of topics that they need to study. Their exam review
problems should cover a wide variety of these topics.
Students should then begin looking back to record and create problems for their exam review.
Give them the handout copied back-to-back for this purpose so they are not writing problems in a
disorganized fashion. You may choose to ask them to do some of this work on their own or in
pairs or other groups if that is how they might actually study.
AlgeCross II
The trickiest one here is 8 down — polynomial. It is fine for this to be done in class, or if there is
no time, it can be given as homework.
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Cube Roots
1. Simplify each of the following.
a. 3 27 b. 3 6464
2. Complete each equation and then create the related equation.
3. Evaluate 103 x when 1x .
4. Simplify.
a. 3 2710 b. 3 8
5. Complete the table of values for the following function: 43 xy
x 0 1 –8 27
y 6 3
Cube root equation Exponent equation
3 125
310
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Rectangle Area Revisited
1. Determine the total area for the largest figure. All four-
sided figures are rectangles. After determining the total
area, complete the equation.
)10(40 x ______________
Determine the missing side lengths. After doing so, complete the corresponding equation.
2. 3.
___)___(2412 xx ___)(___62 aaa
4. 5.
___)(___6366 b ___)5___(3220 xx
__
__
2412x
x
a 2 a6a
____
10x
40
6
__ __
36 6b
5x
20x 32
__
__
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6. In each case, the total area of a rectangle is given. Determine the lengths of the sides.
a. ___)(___4208 b b. ___)___(502 xx
c. )6___(___305 a d. ___)___(9010 bb
e. ___)(___4444 x f. ___)(___3156 x
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My Review Problems for CUNY Start Exam #2
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My Review Problems for CUNY Start Exam #2 (continued)
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AlgeCross II
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Clues to AlgeCross II
Across
5. The process of writing a number in parts using multiplication.
9. The straight line formed by the graph of a linear function represents all of its _____________.
11. xy is a _________ function.
12. The ________ of 8 is 2. (two words)
13. 3 in the function 1003 xy . (three words)
Down
1. 3xy is a _____________ function.
2. 122 xx , for example.
3. 62 xy is a _____________ function.
4. In the expression )4()113( 2 xxx we are subtracting the __________.
5. 1, 2, 4, 5, 10, and 20 are the _______ of 20.
6. In the expression xy
x22 , x must be ________ or the expression will be undefined.
7. When working with the expression )03.2(6 if we multiply first before adding, that is an example
of using the __________ property.
8. 32 x , for example.
10. It has a value of –4 in this function.
x 0 1 2 3
y 2 –2 –6 –10
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CUNY Start Core Math Curriculum, Class 24
Class Contents N EA F
24
Factoring Binomials I
Finding Linear Function Rules
Extra Practice for CUNY Start Exam #2 (Extra Practice #12)
Factoring Binomials I
Draw this figure on the board and the accompanying equation that needs completion.
___)(___63018 x
After your students have helped you to fill in the missing items, discuss again why this process is
called factoring a binomial. We are rewriting the binomial as two parts or quantities (side lengths
in this case) multiplied together. Another way to think about it is that we are starting with the total
area, and we rewrite it to show the product of the sides.
Some people also think about factoring a binomial as a process of ―finding things in common.‖
I can say the following:
“I had lunch with Paul. I had lunch with Zerin. I had lunch with Roxy.”
What do those sentences have in common? So I can say the following:
“I had lunch with Paul, Zerin, and Roxy.”
I identified what the three sentences had in common, and I applied it to the group.
The expression 3018 x has two terms. What do they have in common? (6.) To see that, it can
help for us to factor each term, looking for a common factor. When else do we factor and look for
common factors? Yes, this is much like what we do when we are dividing terms and looking for
factors to eliminate.
56363018 xx
________
306 18x
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We have factored both of the terms in this binomial. If we look carefully, we can see that 6 is a
factor of both terms. Now to factor the whole binomial, what can we do with the 6? (If necessary:
How can we show that the 6 needs to be multiplied to give us both terms?)
Does this remind you of anything? (What if we look at the steps in reverse?)
We used the distributive property “in reverse” here.
How can we check to make sure that we have done this correctly?
At some point in this discussion, be sure to discuss the difference between dividing terms and
factoring binomials too. We have seen students confuse the two types of problems based on the
similarity that we look for common factors in both cases. Students have occasionally begun to
cross out common factors when factoring binomials.
When we are asked to factor in algebra classrooms, the typical assumption is that we will try to
find the greatest common factor and ―pull it out‖ rather than a smaller common factor. Instead of
telling students this, you can lead into this discussion by asking students to try to factor the
following binomial:
4024 x
Give them a little time to work on it, and discuss students‘ work.
It would be best if students came up with different ways to do this. If the class does not come up
with two or three different ways to factor 4024 x on their own, put the following on the board:
)2012(24024 xx )106(44024 xx )53(84024 xx
Make sure that students agree that these are all correct ways to factor 4024 x . All three are
examples of how to factor 4024 x in that each re-writes the expression in two parts using
multiplication. You might ask which one of these math professors typically prefer. We can say
that the last example is the ―complete factorization‖ of the expression 4024 x (and not the other
two factorizations) because 8 — the largest common factor — has been pulled out front.
Ask if students think that there is a way to look at a factorization and tell if the largest common
factor has been pulled out front. Ideally someone will realize that we can look at the quantity
inside parentheses and see if the terms there still have any common factors.
6 multiplied by both 3x and 5, shown
with parentheses.
)53(6
56363018
x
xx
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Once you have completed that example, include one with a subtraction symbol:
Factor: 1015 x
We will generally restrict our examples in this class to binomials where there are only numeric
common factors. The last problem has a variable common factor. It is nice to throw it in there to
see how students react to it, but we will save a fuller discussion of it for a later class.
Finding Linear Function Rules
Students should put the following table in their notes.
Can you tell what kind of function this is? Does it have a slope?
They should be able to identify this function as a linear function with
a slope of 5.
Students have already had a little practice determining function rules
using a slope along with a single solution. Have a student explain
their method for determining the function equation. Hopefully, they
will say that a slope of 5 indicates that the function equation must include the following:
...5xy
Once students have this much of the equation, they can test an input and figure out what
adjustment will be needed so that it matches the table of values. In this case, testing an input of 0,
they will see that they need to add 4 to result in an output of 4.
Note problems #8 and #14. It is perfectly appropriate for students to offer function rules written
as 50 xy and 120 xy , as well as the more simplified versions. This is a good discussion,
and students should not get the idea that one is better than the other. One is more simplified, yes,
but a function written as 50 xy can be helpful because it can more clearly show what the rate
of change is.
Problem #10 is a trick because it is not a linear function. They will not be able to use slope to
determine the rule, but there still is a function rule. It is more challenging to find it, and so they
should consider it a challenge problem. If you want to indicate this from the start, you could add
an asterisk for the problem, but this probably tips students off too much that something different
could be going on.
Notice that the language (but not the meaning) changes in the final problems. This is to reflect the
different ways that examiners can phrase these questions. Highlight the different wordings when
you review student work.
Extra Practice #12
x y
0 4
1 9
2 14
3 19
4 24
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Factoring Binomials I
1. Determine the missing lengths and complete the equation.
___)(___103020 x
Factor.
2. 224 x 3. 273 a
4. 812 b 5. 162 a
6. 2010 b 7. )24(6 x
20x10 30
____ ____
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Write the complete factorization of each expression.
8. 217 x 9. 88 x
10. 4212 x 11. 324 x
12. )50(20 x 13. xx 62
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Finding Linear Function Rules
Determine the equation for each function.
1. 2.
3. 4.
x y
0 4
1 9
2 14
3 19
4 24
x y
1 24
2 26
3 28
4 30
5 32
x y
–2 2
–1 2.5
0 3
1 3.5
2 4
x y
0 –5
1 –2
2 1
3 4
4 7
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5. 6.
7. 8.
9. 10.
x y
0 4
1 2
2 0
3 –2
4 –4
x y
2 100
3 90
4 80
5 70
6 60
x y
4 10
5 –2
6 –14
7 –26
8 –38
x y
10 5
11 5
12 5
13 5
14 5
x y
0 –10
1 –4
2 2
3 8
4 14
x y
0 1
1 2
2 5
3 10
4 17
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11. Determine the equation of the function that has a slope of 6 and a solution at (2, 20).
12. Determine the equation of the function that has a slope of –5 and a solution at (3, –10).
13. Determine the equation of the line that has a slope of 4 and a solution at (–2, –13).
14. Determine the equation of the line that has a slope of 0 and that passes through the point (5, 12).
15. Determine the equation of the line that has a slope of .75 and that passes through the
point (4, 4.50).
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Extra Practice for CUNY Start Exam #2 (Extra Practice #12)
(This is only a selection of problems and does not review all exam topics.)
1. Create a quadratic function that has a solution at (–3, 13). Demonstrate that your function has this
solution.
2. For the following function, create a table
of values and graph the function, including
all the integer solutions that fit on the grid.
xy 2
3. Consider the function: 63 xy . Interpret the slope of this function using ―inputs‖ and
―outputs‖.
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4. Simplify as much as possible.
a. x
x
12
4 2
when .0x b. )14()52( xx
c. )14()52( xx d. ))(3)(2( 2xxx
5. Multiply. No calculators!
a. )11(6 x b. )84(2 y c. )95.3(4
6. Determine the missing lengths and complete the
equation.
___)2___(4016 xx
7. A function has a slope of 3, and one solution is given in the table. Identify the missing outputs.
x 5 6 7 8 9
y –50
__
__
4016x
2x
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For problems #8 and #9, identify the slope of each function.
8. 9.
Slope = ______
Slope = ______
10. Simplify.
a. 25 b. 3 64 c. 4)2(
11.
x y
0 –2
1 –8
2 –14
3 –20
4 –26
11y
+
x
6ab
3a
x
+
-2y
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12. Tenzin is preparing an explanation of the
process she used to simplify the following
expression:
)109()14( xx
Help her to prepare for her explanation by
writing each step in the simplification. Label
each step with a number. Then write a
description of each numbered step below. If
you need additional room, attach a separate
piece of paper.
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CUNY Start Core Math Curriculum, Class 25
Class Contents N EA F
25
Review Student-Devised Problems and Extra Practice for Exam #2
Introduction to Function Notation
Times Table Test #3
CUNY Start Exam #2
Using Function Notation
Review Student-Devised Problems and Extra Practice for CUNY Start Exam #2
Introduction to Function Notation
Begin with student notes (not the handout). Write the first table and function (using f(x) notation)
on the board exactly as it appears on the handout. If you ask your students what they think they
are looking at, they will probably say that it looks like a function.
What is different and what do you think it means?
The goal is for them to verbalize that f(x) appears to be in the place of the output. Here we have a
new use of parentheses. It is not a symbol for multiplication. f(x) is the output, and the x only
appears there as way of indicating that x is the input.
What could be confusing about this use of parentheses?
Some students believe that the parentheses indicate multiplication here, but they do not. It is good
to get this out in the open.
Also be explicit about how we verbalize this function and have students record it in their notes.
A math professor would say “f of x is equal to 5x+3.”
This function is a “function of x”. What do you think that means? (That x is the input.)
Give students the first page of the handout so that students can complete the tables of values and
re-write the bottom two functions using x/y notation.
When going over this, ask students to read the functions to reinforce how we verbalize these
functions.
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One of the reasons why we use function notation is that it clarifies things when we work with
several functions at once.
“When I say — „Look at g(x)‟, do you know where to look? If all of them were written as ...y ,
it would be more confusing.”
“Does anybody know how your graphing calculator distinguishes between different functions?”
( ...1 Y , ...2 Y , ...3 Y , etc.)
After this discussion, go back to students‘ notes and have students record the three functions from
the front of the handout again:
Write this in your notes: )10(f
Which function do you think this refers to? Why?
What does the 10 represent? (If necessary: What is the 10 replacing?)
What do you think that means?
Remember in the above conversation — the x in f(x) tells you that x is
the input for that function.
The meaning of this notation is likely something that they either
remember or do not remember. They can suspect what it means, but
they cannot really ―discover‖ its meaning on their own because it is a
convention. Once you have established that it tells us to input 10 in f(x),
ask them how the same problem could be shown using a table. A student
could be asked to come to the board and show this.
Give students one or two more examples to do in their notes, such as finding )6(g . They should
now be ready to do Using Function Notation after the test or as homework.
Times Table Test #3
CUNY Start Exam #2
Add arrows to the line in problem #14, ideally before making copies.
Using Function Notation
This is for students who finish the exam early to work on in class. It is homework for other
students.
35)( xxf
x f(x)
10
35)( xxf 53)( xxg 32)( xxh
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Introduction to Function Notation
x f(x)
0
1
2
10
.5
43
15.5
x g(x)
0
1
2
10
.5
43
5.5
This function rule written
in x/y notation is:
x h(x)
0
1
2
10
.5
43
This function rule written
in x/y notation is:
35)( xxf
This function rule is the same as the
following:
y = 5x+3
53)( xxg 32)( xxh
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Times Table Test #3
85 1211 78 123 52 98
96 109 119 65 1110 33
26 42 411 54 66 107
611 105 104 128 1210 32
53 311 94 64 36 124
55 99 75 810 72 59
73 125 83 118 1212 310
106 712 126 77 92 97
28 912 93 210 84 115
1010 44 112 74 88 117
76 212 22 1111 43 68
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CUNY Start Math Exam #2
1. Complete the missing quantities. (1 point for each puzzle)
2. Which of the following is NOT a factor of 35? Circle one answer. (1 point)
3. For each of the following, complete the missing value in order to write the fraction in higher
terms. (1 point each)
a. 217
4 b.
222 c
d
c
4. Re-write 9
24 in lowest terms. (1 point)
a. 1 b. 3 c. 5 d. 7 e. 35
+
x
9y2x2x -5x
x
+
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5. Check ―Yes‖ or ―No‖ for each of the following. (1.5 points total)
6. Simplify as much as possible. (1 point each)
a. )23()132( 22 xxxx b. )116()12( xx
c. 32
3
6
12
yx
xy when 0x and 0y . d. )53()79( xx
Yes No
Is 132 xx an example of an equation?
Is 132 xx an example of a binomial?
Is 132 xx an example of a trinomial?
Is 132 xx an example of a polynomial?
Is 132 xx an example of an expression?
Is 132 xx an example of a function?
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7. Multiply. (1 point each)
a. )02.1(4 b. )95.4(8
c. )8(6 x d. )4(2 aa
8. For each function, classify the function as linear, quadratic, or cubic, and then describe its shape as
U-shaped, a ―climbing snake‖, or a straight line. (4 points total)
Function Is this a linear, quadratic,
or cubic function?
Is the graph of this function U-shaped,
a ―climbing snake‖, or a straight line?
24xy
1103 xy
3xy
10y
9. Evaluate )10(x when 64x . (1 point)
10. The value of the expression 44 lies between which two consecutive integers? (1 point)
11. Simplify. 3 27 (1 point)
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12. Consider the following table of values.
The slope of this function is _______. (1 point)
13. Consider the following function: 17 xy . What is the slope of this function? (1 point)
14. What is the slope of this function? (1 point)
15. A linear function has a slope of 5. Interpret this slope using ―inputs‖ and ―outputs‖. (1 point)
As the inputs ___________________________, the outputs ________________________.
x y
0 14
1 11
2 8
3 5
4 2
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16. Graph the following function. An appropriate
graph will include all integer solutions that can
be graphed on this xy-grid. (1 point)
24 xy
17. Graph the following function. An
appropriate graph will include all integer
solutions that can be graphed on this xy-grid.
(1 point)
13 xy
18. Fill in the missing side lengths. All four-sided figures
are rectangles, and the terms inside the rectangles are
area measurements. (1 point total)
______
4 16x 36
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19. What can you say about the characteristics of a linear function that has a slope of 0? Write a
careful explanation of your answer. You may provide a mathematical example if you wish to help
you illustrate your points, but you must also use sentences as a part of a written description.
(1 point)
20. Which of the following is equivalent to 288 x ? Circle one answer. (1 point)
21. Determine the equation of the line that has a slope of 4 and that passes through the point (–2, 12).
(1 point)
a. )72(4 x b. )3(8 x c. )20(8 x d. 36 e. x36
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22. Determine the function equation for the function with the following table of values. (1 point)
Challenge:
Determine the function equation for the function with the following table of values.
x y
4 18
5 23
6 28
7 33
8 38
x y
1 –2
2 4
3 14
4 28
5 46
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Scoring Guide for Exam #2
1. Each entry is worth .5 points. The maximum is 2 points.
product = 210x
sum = x3
product = xy18
sum = yx 92 (order does not matter)
2. b
3a. 12 3b. cd
4. 3
8
5. Each correct response is worth .5
points. The maximum is 1.5 points.
6a. 13 2 x or equivalent expressions such as )1(3 2 x or 231 x
6b. 104 x or equivalent expressions
6c. x
2 or equivalent expressions
6d. 126 x or equivalent expressions
7a. 4.08 7b. 39.6
7c. 486 x 7d. aa 82 2
8. Each entry is worth .5 points. The maximum is 4 points.
Yes No
Is 132 xx an example of an equation? x
Is 132 xx an example of a binomial? x
Is 132 xx an example of a trinomial? x
Is 132 xx an example of a polynomial? x
Is 132 xx an example of an expression? x
Is 132 xx an example of a function? x
Function Is this a linear, quadratic,
or cubic function?
Is the graph of this function U-shaped,
a ―climbing snake‖, or a straight line?
24xy Quadratic U-shaped
1103 xy Linear Straight line
3xy Cubic Climbing snake
10y Linear Straight line
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9. –2
10. 6 and 7
11. 3
12. –3
13. 7
14. The slope is –4. Arrows should be added to this line in advance.
15. …increase by 1…increase by 5 (one half point for each part)
16. A full point is awarded if all 6 integer solutions appear along with a line and arrows indicating that
it continues. Half credit is awarded if all 6 integer solutions appear, but no line is drawn. Zero
credit otherwise. (Kevin gives .25 credit for 3 to 5 correct solutions.)
17. A full point is awarded if all 5 integer solutions appear along with a curve and arrows indicating
that it continues. Half credit is awarded if all 5 integer solutions appear but no curve is drawn.
Zero credit otherwise. (Kevin gives .25 credit for 3 to 4 correct solutions, plus an additional .25
points for drawing a decent ―climbing snake‖ that includes those 3 to 4 correct solutions.)
18. 4x and 9.
19. To receive full credit, a student must essentially state that the outputs remain constant as the inputs
change (or increase by 1). The student only earns half credit for writing ―The outputs don‘t
change‖ without referring to the inputs. (Some of us are also willing to give half credit if students
write sentences in the right general ballpark AND provide good example(s), even if the sentences
are not 100% precise in their language.)
20. a
21. 204 xy
22. 25 xy
Challenge: 42 2 xy (Do not award any points for this.)
Total Points = 33.5
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Using Function Notation
1. Consider the following functions:
12)( xxr xxs 5)( xxxt 2)( 1010
)( x
xu
a. Find )50(r . b. Find )50(s .
c. Find )50(t . d. Find )50(u .
2. Create a function b(x) that has a solution (4 ,7). Try to find a function that is different from
students near you in the class.
3. Consider the function 104)( xxe . Identify two function solutions (written as ordered pairs).
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4. Consider the function 1)( 2 xxxd . Identify an ordered pair that is not a solution to this
function. Explain why the ordered pair is not a solution.
5. Consider the following functions:
52)( xxf 324)( 2 xxxg 17)( xxh
a. Calculate )6(f .
b. What is the slope of the function )(xh ?
c. Calculate )3(g .
6. Create a non-linear function )(tf whose graph passes through the point )12,3( .