The At Home in College Core Math Curriculum

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


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


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


3


9


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


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


4


17


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


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


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


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)


5


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


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


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


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


6


33


417 More y-Intercept In and Out of Contexts

Multiplying Binomials II

34

Multiplying Binomials III

431 Multiplying Binomials IV

Extra Practice #17

35


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


483

Factoring Trinomials II

Fraction Slopes I

Fraction Slopes II – equations and tables

Extra Practice #19

39


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


7


41


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


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


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.


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?


10

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


11

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(


12

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


13

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


14

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


15

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


16

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


17

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


18

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


19

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


20

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


21

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)


22

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


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


24

CUNY Start Core Math Curriculum, Class 1

Class Contents N EA F

1

Number Puzzle 1


Uses of Parentheses



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


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?



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


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


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.


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


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.


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)


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


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.


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.


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


32

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


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


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.


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________


35

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


36


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


37

Digital Cameras You

Sell in One Week

Your Weekly Pay

0

1

2

10

16


38



2


Signed Numbers 2: Subtraction

Extra Practice


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


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.


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


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.


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.


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.


43


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


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


45


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


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


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


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


49

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

+


50



3



More Signed Numbers 2: Subtrraction


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


51

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


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.


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


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?)


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.


54


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 =


Output =


55

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


=


=


=


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


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.


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


a. 1230 b. 1230 c. )12(30 d. )12(30

Challenge!

Simplify the following. )102()153(



4





Extra Practice


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.


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.


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.


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.


62


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.


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


63

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.


64


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.


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


66

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


67

Times Table Test Scores

Student Baseline Test 1 Test 2


68


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


69


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.


70

CUNY Start Math Textbook — Vocabulary Index

Vocabulary Word Example and/or Description Date &/or

Page #


71

Vocabulary Word Example and/or Description Date &/or

Page #


72

CUNY Start Math Textbook —Index of Topics

Math Topic Example Date &/or

Page #


73

Math Topic Example Date &/or

Page #


74

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


75

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.


76

4. Calculate any missing inputs and outputs.


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


77



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.


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.


5


Return Times Table Baseline Test and flashcard introduction



78

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


80


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!


82



6


Signed Numbers 4: Division

Extra Practice


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.


83

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.


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!)


84

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.


85

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


87

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


88


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.


89

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


90

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


91

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


92

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.


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


93

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?


94



7


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.


95

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.


96

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


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.


98

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


99

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.


100

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


101

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


102



8

2xx

Terms and Expressions discussion

Rectangle ________ and Rectangle Perimeter with Variable Side Lengths


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


103

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


104

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.


105

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.


106

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


107

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


108

Rectangle ______________

A B

C D


109

Rectangle Perimeter with Variable Side Lengths

x

8

g12

f


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


111

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


112

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


113

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.


116



9

Review Extra Practice (and making Student Number Puzzles handout)

Exponents 2


Integers



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 .

?


117

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.


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


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


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.


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


120


Fill in the missing circles for each Number Puzzle.

x

+

x

+

x

+

x

+

x

+

x

+

x

+

x

+

x

+

x

+

x

+

x

+


121

x

+

x

+

x

+

x

+

x

+

x

+

x

+

x

+

x

+

x

+

x

+

x

+


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(


Use pencil!

x y

0

1

2

3

4

y

x


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


125


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.


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)


126

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)


127

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.


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 =


128

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


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


132

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?


133

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?


134

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



High 20 42 39 28 30 34 42

Low 17 33 23 21 23 30 33



High 3 8 –8 0 2 –5 15

Low –10 –4 –19 –7 –11 –13 –2


135

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


136

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


137

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


138

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

Original material under copyright, 2008 College Transition Math at the CUNY Adult Literacy/GED Program

139



11


Remaining Group Presentations on elevation activity

Multiplying Terms



AlgeCross I


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‖.


140

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.


141


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.


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.


142

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


143

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.)


144



145

My Review Problems for CUNY Start Exam #1 (continued)


146

AlgeCross I


147

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.


148


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.


12

Rectangle _____ and Rectangle Area with Variable Side Lengths

Combining and Multiplying Terms


Length x Rectangle Area

10 6

25 x

Shapes used to

cover the figure.

Length of one side

of each square.


149

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.


Area

8 5


Area

8 5 136

9 6

2.5 4

f g

5

8

12


150

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.


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.


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


151

Rectangle ______________

A B

C D


152

Rectangle Area with Variable Side Lengths

x

8

g12

f


153


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


154

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


155

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


156


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.


157

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


158

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


159

10. 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


160



13

Review Extra Practice and student-devised practice

Times Table Test #2

CUNY Start Exam #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.


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.


161

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


162

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


163

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


164

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


165

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


166

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


167

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


168

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


169


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


170

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


171



14


Patterns in Functions and Their Graphs, part 2 and Discussion

Extra Practice


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.


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.


172

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.


173


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


174

7.

8.

44 xy

x y

–2

–1

0

1

2

3

4

13 xy

x y

–2

–1

0

1

2


175

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


176

11.

12.

12 xy

x y

–3

–2

–1

0

1

2

3

xy 4

x y

–3

–2

–1

0

1

2

3


177

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?


178


179

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


180

3. Complete any missing information 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


181

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


182

6. Find numbers that satisfy each of the following equations.

10________ 10))((

10________ 10)(

)(


183



15



Exploring Squares


Introduction to Factoring and Factors and Finding Factors


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.


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.


184

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.


185

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.


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


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.


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)


188

Problem

Mistake

Correction

Topic(s)

Problem

Mistake

Correction

Topic(s)

Problem

Mistake

Correction

Topic(s)


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 __________________________.


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 _________________________.


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 _______________________.


eight miles has an area of

_______________________.

Area =

Side

length =


________________ has an area

of _______________________.

27

Area =

Side

length =


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


193


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 ___


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


195

Finding Factors

Find a number that is smaller than 100 and that has more factors than 100.


196



16

__________ Fractions


Large Numbers of Eligible Comm. Coll. Students Do Not Apply for FA


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


197

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.


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?


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


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.


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


202

______________________ Fractions


203

204



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

205


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

206


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.

207



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

208


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

209


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


a. 121 b. 48

c. )3)(2( xx d. )3(2 xx

7. Evaluate 112 x when 4x .

210


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

211


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.

212




17


Dividing Terms

…for all 0x .

Expression Puzzles


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

213


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.

214


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

http://en.wikipedia.org/wiki/Lafayette_College

http://en.wikipedia.org/wiki/Johns_Hopkins_University

http://en.wikipedia.org/wiki/Johns_Hopkins_University

http://en.wikipedia.org/wiki/Slate_%28magazine%29

http://en.wikipedia.org/wiki/August_16

http://en.wikipedia.org/wiki/2006

215


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

216


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

217


…for all 0x .

Simplify x

yx

6

24 2

for all 0x .

218


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

219




18


More Squares and Square Roots


Adding Polynomials

Extra Practice


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

220


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.


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?

221



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


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


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.

222


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.




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.




152 23 xxx

?

223


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.

224


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

225


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

226



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

227


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 ___

228


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

229


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

230


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

231


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

232


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

233




19


Rate of Change



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

234


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

235


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.


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

236


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

237


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

238


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

239



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

240


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.


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.

241


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).

242




20


Three Scenarios and the Distributive Property


Extra Practice


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

243


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.

244


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

245


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

246



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?

247



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

248


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

249


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

250


g. 9______)(9 x h. 6012___)___( xx

i. 3510___)2___( xx j. caba 18___3___)8(___

k.* 156___)4___( xx l.* 5015___)6___( xx

251


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

252


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

253


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

254


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

255




21


Pencil Packs

The Distributive Property — Addition and Subtraction

Identifying and Interpreting Rate of Change from graphs


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.

256


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.

257


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.

258


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

259


e. )62(4 x f. )68(3 2 xx

g. )8(2 xx h. )83(10 a

i. )( 22 xxx

260


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‖.

261


Identifying Rate of Change from a Graph – More Practice

1. Identify the rate of change for each function.

262


263


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.

264


6. Identify the rate of change for each function.

265




22



Rate of Change = _____________

Extra Practice


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(

266


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.

267


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.


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(

268


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(

269


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


6)3(124 xx Because we are now adding polynomials, we

can remove parentheses.

18 x Combine terms.

270



Subtract.

1. )103()124( xx

2. )112()67( xx

3. )26()103( xx



Remove parentheses.

Combine like terms.



Remove parentheses.

Combine like terms.



Remove parentheses.

Combine like terms.

271


4. )97()15( xx

5. )1212()128( xx

6. )95()16( xx

272



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


Remove parentheses.

Combine like terms.




Remove parentheses.

Combine like terms.




Remove parentheses.

Combine like terms.

273


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

274


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

275


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

276


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)

277


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!

278


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

279


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

+

280


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

281




23


Cube roots



AlgeCross II


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?”

282


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.


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.

283


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.


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.

284


Cube Roots


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

285



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

__

__

286


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

287



288


My Review Problems for CUNY Start Exam #2 (continued)

289


AlgeCross II

290


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

291




24


Finding Linear Function Rules



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

292


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

293


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.


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

294



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

____ ____

295


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

296



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

297


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

298


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).

299



(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‖.

300


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

301


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

302


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.

303




25

Review Student-Devised Problems and Extra Practice for Exam #2


Times Table Test #3

CUNY Start Exam #2


Review Student-Devised Problems and Extra Practice for CUNY Start Exam #2


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.

304


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.


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

305



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

306


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

307


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

+

308


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?

309


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)

310


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

311


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

312


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

313


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

314


Scoring Guide for Exam #2


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


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

315


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

316



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( .

The At Home in College Core Math Curriculum

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