5 Day Unit Plan Algebra/Grade 9 JenniferJohnston Geometer’s Sketchpad Graph Explorer Algebra I TI-83 Plus Topics in Algebra Application Transform Application
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
5 Day Unit Plan
Algebra/Grade 9
JenniferJohnston
Geometer’s Sketchpad
Graph Explorer
Algebra I
TI-83 Plus
Topics in Algebra Application
Transform Application
2
Overall Objectives
• Students will use a variety of technologies and resources to expand their knowledge
of graphing ordered pairs and equations.
• They will become efficient in working with calculators and computers for
mathematical purposes.
• The students will help each other throughout the unit by presenting their ideas,
working together as groups, and assessing one another.
NYS Standards Covered:
• Number and Numeration
• Operations
• Modeling/Multiple Representation
• Measurement
• Patterns/Functions
NCTM Standards Covered:
• Algebra
• Data Analysis and Probability
• Problem Solving
• Reasoning and Proof
• Communications
• Connections
3
Resources
• Algebra I. Simon & Schuster, 1998. Chapter 5; pgs.116-137.
• Doody, Cecelia. Hands-on Activities For your Mathematics Classroom. I2T2
Workshop. November 17, 2001
• Larson, Ron, Laurie Boswell, Timothy Kanold, and Lee Stiff. Algebra I. Evanston,
IL: McDougal Littell, Inc., 2001. Chapter 4; pgs.210-233. Chapter 5; pgs.273-299.
• Schoaff, Dr. Eileen. “Introduction to Lines”
4
Materials
Day 1:
• Geometer’s Sketchpad
• Computer Projection Device (In-Focus)
• Opening handout
• Handout for Main Activity
• File saved on Geometer’s Sketchpad for students to open
• Homework Assignment
Day 2:
• Opening handout
• Graph Explorer
• Algebra I: Equation Match
• Graph Explorer handout
• Homework Assignment
Day 3:
• TI-83 Plus
• Transform Application
• Topics in Algebra Application
• Calculator Links
• Handout for Closing Activity
• Homework Assignment
5
Day 4:
• Opening handout
• Algebra I
• Handout for Main Activity
• Closing handout
• Homework Assignment
Day 5:
• TI-83 Plus
• Opening handout
• Calculator Links
• Stop watch
• Handouts for Main Activity
• Homework Assignment
6
Overview of Graphing Linear Equations
The students will be reviewing graphing ordered pairs on the coordinate plane by
doing activities on Geometer’s Sketchpad.
They will use basic reasoning to conclude patterns between sets of ordered pairs.
Graph Explorer will help them see the graphs of the equations, while Algebra I
will force them to draw the graph on their own.
The graphing calculator will help the students make conjectures about the
connection between steepness and an equation of a line. “Topics in Algebra” will
show the students an animated lesson about the slope formula along with a game
to help them practice using the formula.
Students will use Geometer’s Sketchpad and Algebra I to graph a line by using
the intercepts. They will be using the slope-intercept form to write equations.
Also, they will use graphs of equations to find the intercepts.
The TI-83 Plus will be helpful to plot data and make conjectures about the data.
They will look at the line regression and slope-intercept form on the calculator
involving best-fit lines.
Day 1: Graphing Ordered Pairs
Day 2: Graphing Equations of Lines
Day 3: Slope of a Line
Day 4: Graphs Using Intercepts
Day 5: Scatter Plot and Best-Fit Lines
7
Day 1: Graphing Ordered Pairs
Objectives:
• Ninth grade students will use their knowledge to graph ordered pairs on the
coordinate plane.
• They will use Geometer’s Sketchpad to plot sets of ordered pairs that make up
pictures by connecting line segments.
• They students will test their peers on the knowledge they have of graphing ordered
pairs.
• They will work together in groups to make a presentation at the end of the class.
Opening Activity:
Students will be given a handout with a set of ordered pairs. They will need to
label the x- and y-axis and the point of origin on the coordinate plane. Each student is
required to plot each ordered pair, label its coordinates, and label the number of that
particular point.
Main Activity:
Students will open Geometer’s Sketchpad on their computer and use the program
to plot another set of ordered pairs. After the students have graphed their points; have
them discuss with each other the different ways they plotted the points. They should be
able to see that it can be done more than one way. Let the students connect the points to
form the picture. Two volunteers can present to the class the different ways to connect
the points on the computer that is connected to the overhead projector. Students need to
print out their work to hand in.
8
Closing Activity:
Students will be given a more difficult picture on Geometer’s Sketchpad. They
are asked to label the points with a letter and label the coordinates of each point.
Students will need to print out their work to hand in. They can discuss the different
methods they used to get the coordinates of the set of ordered pairs that makes up this
picture.
Homework:
Students will be given a worksheet with instructions on their assignment and what
is required of them for tomorrow.
9
Teachers’ Notes
Opening Activity:
Plotting ordered pairs should be a review for this class. This worksheet will tell
me if there are any students who might need more instruction for this lesson. Make time
to go over a few examples of plotting ordered pairs with (x,y) coordinates if it seems like
some students do not understand.
Main Activity:
The students should be discussing two main ways of plotting points on GSP.
First, using the arrow and clicking on the point. Second, going under graph and plot
points. In the small presentation, students should be able to show that after using “plot
points”, if nothing else is clicked on, you can go to construct; segments and the fish is
formed. You should have your overhead screen ready for this in case a student presents
in this way. The similar way to construct segments is by highlighting each point in the
order you want the segment to appear and go to construct; segment. Another way to
construct segments is to go to the tool bar on the left and choose the line segment box.
The students can connect points by clicking on the particular point and dragging the line
to the next point.
Closing Activity:
The students will need to open this file through GSP to begin. The students can
highlight each point and go to measure; coordinates. They can also make a text cell for
each ordered pair and label its coordinates.
10
Homework:
Read over the homework assignment with the class and make sure everyone
understands. If there are no questions, explain to them that the assignment is easiest if
they look through a coloring book or children’s book. The pictures are creative and you
can add points to make it a connect the dots picture.
11
Name:________________________ Math 9
Label the x-axis and the y-axis. Label the origin and its coordinates. Plot these ordered
pairs. Label the number of the ordered pair and label its coordinates.
1. ( 1, 2 )
2. ( -1, 6 )
3. ( 0, 3 )
4. ( 4, 0 )
5. ( 1, -4 )
6. ( 4, 1 )
7. ( -2, 5 )
8. ( -5, -3 )
9. ( -6, -1 )
10. ( 6, -5 )
(Handout for Opening Activity)
-10 10
10
-10
12
Name: KEY__________________ Math 9
Label the x-axis and the y-axis. Label the origin and its coordinates. Plot these ordered
pairs. Label the number of the ordered pair and label its coordinates.
11. ( 1, 2 )
12. ( -1, 6 )
13. ( 0, 3 )
14. ( 4, 0 )
15. ( 1, -4 )
16. ( 4, 1 )
17. ( -2, 5 )
18. ( -5, -3 )
19. ( -6, -1 )
20. ( 6, -5 )
(Handout for Opening Activity)
10
10
1 (1, 2)
2 (-1, 6)
3 (0, 3)
4 (4, 0)
5 (1, -4)
6 (4, 1)
7 (-2, 5)
8 (-5, -3)
9 (-6, -1)
10 (6, -5)
X-axis
Y-axis
Origin (0, 0)
13
GSP Activity Math 9
Make sure you put your name on your GSP worksheet with a text cell. Make text cells
on your page to answer the questions below.
I. Plot the following set of ordered pairs.
1. ( 8, 1 ) 2. ( 7, 2 )
3. ( 4, 4 ) 4. ( 1, 5 )
5. ( -2, 4 ) 6. ( -5, 1 )
7. ( -6, 2 ) 8. ( -8, 4 )
9. ( -8, -5 ) 10. ( -6, -3 )
11. ( -5, -2 ) 12. ( -2, -3 )
13. ( 2, -4 ) 14. ( 5, -3 )
15. ( 7, -1 )
II. What type of command or method did you use to plot your points?
III. Are there any alternative ways to plot this set of ordered pairs? Discuss with
your neighbor the method they used to plot their set of points.
IV. What picture is made when you connect the points with the line segments?
V. Which method did you use to connect the points with line segments?
(Handout for Main Activity)
14
-10 -5 5 10
10
5
-5
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
Student's Sample Worksheet
II. I went to Graph; Plot Points...I then typed in the ordered pairs and clicked on plot. This plotted the set of ordered pairs.III. You can individually click on the point with the arrow. IV. A fishV. After I plotted the points, I went directly to Construct; Segment. Since I went in the order of 1 through 15 when I plotted my ordered pairs, I could use only one command to get my entire picture.
15
GSP Activity Math 9
Label each point with a capital letter. Label the coordinates of each point. There are 17
ordered pairs.
(Open this file through GSP for Closing)
-10 10
10
-10
16
GSP Activity Math 9
Label each point with a capital letter. Label the coordinates of each point. There are 17
ordered pairs.
(Open this file through GSP for Closing)
10
-10
-10 10
S
R
Q
PO
NM
L
KJ
IHG
F
E
D
C
S: (2.0, 7.0)
R: (3.0, 4.0)
Q: (4.0, 0.0)
P: (7.0, -4.0)
O: (4.0, -4.0)
N: (3.0, -6.0)
L: (2.0, -4.0)
M: (1.0, -6.0)K: (-1.0, -6.0)
J: (-3.0, -6.0)
I: (-2.0, -4.0)
H: (-4.0, -4.0)G: (-7.0, -4.0)
F: (-4.0, 0.0)
E: (-3.0, 4.0)
D: (-2.0, 7.0)
C: (0.0, 10.0)
17
Graphing Ordered Pairs: Homework Assignment
Your assignment for tonight is to create a picture of your choice. It must consist
of a set of at least 10 ordered pairs with connecting line segments, like we did in class
today. Be as creative as you would like. On one sheet of graph paper, draw your picture
with its points, but do not label the coordinates of the points. On a separate sheet of
graph paper, state the set of ordered pairs of your picture. The assignment is due at the
beginning of class tomorrow. You will give the first sheet to one student and the second
to another student. It is their job to find the set of ordered pairs for your picture for the
first sheet. For the second sheet, they are to plot the ordered pairs and connect the dots,
like we did in class. They will do your assignment for homework and give it back to you
the next day. It is your job to then correct the assignment and turn both papers in to the
teacher.
(Homework Handout)
18
Day 2: Graphing Equations of Lines
Objectives:
• Ninth grade students will use a table of values to determine an equation of a line.
• They will use Graph Explorer to graph sets of ordered pairs and lines.
• Students will practice graphing linear equations on Algebra I by using a game.
Opening Activity:
Students are given a table of values with four ordered pairs. It is their job to
determine two more ordered pairs that follow the same pattern.
X Y2 6
1.5 4.53 94 12
Students are asked to state how they concluded that the two points follow the same
pattern as the other four. State in words the pattern of these ordered pairs. Write an
algebraic expression for these ordered pairs.
Main Activity:
On Graph Explorer, students are going to determine whether or not this set of
ordered pairs satisfies the algebraic statements they have come up with. Students will
enter their ordered pairs into a table. They can plot these points and enter their equation
into y1. If all of these ordered pairs lie on this line, they are correct. If they are wrong,
they need to examine their ordered pairs and line before moving on. Make a print out of
the correct graph.
19
Students will do the same activity for the worksheet given to them. They will find
the pattern and come up with two new ordered pairs. The students will use Graph
Explorer to plot their data and sketch a graph.
Closing Activity:
Students will use Algebra I to play a game that involves the same type of
reasoning. To start the game from Algebra I, they can use the following commands:
Open Algebra I; File; Open; Equation Match. The students are required to match their
equation to the target equation by moving the graph with the mouse arrow. They do not
know y = mx + b form for an equation of a line. They can make conjectures about the
form of the equation after they play the game a few times. They should be able to
conclude, for example, the slope is 3 for this graph and it crosses the y-axis at –1. This
10-10
-10
10
Go!
m b
y = 3x-1m b
y = -x+4
New
Target EquationYour Equation
Help
10-10
-10
10
Go!
m b
y = 3x-1m b
y = 3x-1
New
Target EquationYour Equation
Help
20
will help them understand the slope-intercept form much better when we work with the
equation later in the unit. They will do 3 different trials and make a printout of each of
the 3 screens.
Homework:
Students will be doing a worksheet that involves substituting ordered pairs into an
equation to verify the ordered pairs lie on that particular line. This process is the opposite
from what they did in class.
21
Teachers’ Notes
Opening Activity:
If the students are having a hard time determining the relationship, have them tell
you the pattern in words. They might understand the equation if it is said in a complete
sentence. The setup of the table of values should be a review.
Main Activity:
Make sure each student makes a print out of the correct version of this problem
before doing the worksheet. Have them leave their printouts on their desk. You want to
be able to walk around and see that their printouts are correct so you know they
understand the lesson.
Closing Activity:
These are a bit more difficult so offer help to the students who are struggling.
Conduct a class discussion on the conjectures that you want the students to realize. If
they are having trouble getting motivated to think about this, encourage them with
questions that have hints. Do not give them the answers. You should let them think on
their own.
Homework:
Make sure they know that there is no difference between y = (3x + 2)/4 and
4y – 3x = 2. They should see that, by algebra, these are equivalent.
22
Graphing Equations of Lines Math 9
This is a table of values, or ordered pairs. There is a pattern and two ordered pairs are
missing. Can you fill in the blanks with two ordered pairs.?
X Y
2 6
1.5 4.5
3 9
4 12
1. How did you conclude that these were the missing ordered pairs?
2. State in a complete sentence the pattern you found in this table of values.
3. Write an algebraic expression for the pattern of this table of values.
4. Before we move on, write down your ideas on how we could check to see if your
expression is the correct equation of the line made up of these values.
(Handout for Opening Activity)
23
Sample Student Worksheet
Graphing Equations of Lines Math 9
This is a table of values, or ordered pairs. There is a pattern and two ordered pairs are
missing. Can you fill in the blanks with two ordered pairs.?
X Y
2 6
1.5 4.5
3 9
4 12
1 3
-5 -15
5. How did you conclude that these were the missing ordered pairs?
The right side is three times the left side for each pair.
6. State in a complete sentence the pattern you found in this table of values.
The pattern is that y is 3 times bigger than x.
7. Write an algebraic expression for the pattern of this table of values.
Y = 3*x
8. Before we move on, write down your ideas on how we could check to see if your
expression is the correct equation of the line made up of these values.
Graph the ordered pairs and draw a line. If all of the points are on the line, they are
correct.
(Handout for Opening Activity)
24
Graph Explorer Activity Math 9
Name:____________________
Add the 2 missing ordered pairs. Write an algebraic expression for each of the tables.
Plot the points on Graph Explorer. Enter your equation into y1. Make a sketch of the
graph with the ordered pairs on the
coordinate plane to the right.
1.
X Y1 12 2-1 -14 4
_____________________
2.
X Y1 35 74 60 2
_____________________
15
10
5
-5
-10
-15
-10 10
15
10
5
-5
-10
-15
-10 10
25
3.
X Y1 3
2.5 64 90 1
_____________________
(Handout for Main Activity)
15
10
5
-5
-10
-15
-10 10
26
Graph Explorer Activity Math 9
Name:Sample Student Worksheet_
Add the 2 missing ordered pairs. Write an algebraic expression for each of the tables.
Plot the points on Graph Explorer. Enter your equation into y1. Make a sketch of the
graph with the ordered pairs on the coordinate plane to the right.
1.
X Y1 12 2-1 -14 45 50 0
_____y = x____________
2.
X Y1 35 74 60 2-1 1-3 -1
________y = x + 2______
10
8
6
4
2
-2
-4
-6
-8
-10
-10 -5 5 10
-10 -5 5 10
10
8
6
4
2
-2
-4
-6
-8
-10
27
3.
X Y
1 32.5 64 90 1-1 -13 7
____y = 2x + 1_________
(Handout for Main Activity)
-10 -5 5 10
10
8
6
4
2
-2
-4
-6
-8
-10
28
Homework Handout Math 9
Name:______________________
Verify each ordered pair works for the given equation by substituting the values for x and
y. Sketch the graph of the equation and plot the points. Are each of these ordered pairs
solutions for these equations?
1. y = 2 – 3x
( 0, 2 ) ( 3, -7 )
( 1, -5 ) ( -1, 5 )
( -2, 8 ) ( -3, -3 )
2. y = -x/3 – 2
( 3, -3 ) ( 2, -6 )
( -3, 4 ) ( 0, -2 )
( -6, 0 ) ( 9, -4 )
15
10
5
-5
-10
-15
-10 10
15
10
5
-5
-10
-15
-10 10
29
3. 2y = x – 3
( 0, 3 ) ( 5, 1 )
( -3, -3 ) ( 1, -1 )
( 2, 0 ) ( 6, 1 )
4. 4y – 3x = 2
( 1, 5/4 ) ( -1, -1/4 )
( 0, 1/2 ) ( 2/3, 1 )
( -1, 0 ) ( -1/3, 1 )
5. 2x + 5y = 1
( 2, 1 ) ( 0, 5 )
( 0, 1/5 ) ( 5/2, -4/5 )
( 1/2, 0 ) ( -1/2, 2/5 )
(Homework Handout)
15
10
5
-5
-10
-15
-10 10
15
10
5
-5
-10
-15
-10 10
15
10
5
-5
-10
-15
-10 10
30
Homework Handout Math 9
Name:KEY___________________
Verify each ordered pair works for the given equation by substituting the values for x and
y. Sketch the graph of the equation and plot the points. Are each of these ordered pairs
solutions for these equations?
1. y = 2 – 3x
( 0, 2 ) * ( 3, -7 ) *
( 1, -5 ) ( -1, 5 ) *
( -2, 8 ) * ( -3, -3 )
2. y = -x/3 – 2
( 3, -3 ) * ( 2, -6 )
( -3, 4 ) ( 0, -2 ) *
( -6, 0 ) * ( 9, -4 )
-10 -5 5 10
10
8
6
4
2
-2
-4
-6
-8
-10
-10 -5 5 10
10
8
6
4
2
-2
-4
-6
-8
-10
31
3. 2y = x – 3
( 0, 3 ) ( 5, 1 ) *
( -3, -3 ) * ( 1, -1 ) *
( 2, 0 ) ( 6, 1 )
4. 4y – 3x = 2
( 1, 5/4 ) * ( -1, -1/4 ) *
( 0, 1/2 ) * ( 2/3, 1 ) *
( -1, 0 ) ( -1/3, 1 )
5. 2x + 5y = 1
( 2, 1 ) ( 0, 5 )
( 0, 1/5 ) * ( 5/2, -4/5 ) *
( 1/2, 0 ) * ( -1/2, 2/5 ) *
(Homework Handout)
-10 -5 5 10
10
8
6
4
2
-2
-4
-6
-8
-10
-10 -5 5 10
10
8
6
4
2
-2
-4
-6
-8
-10
-10 -5 5 10
10
8
6
4
2
-2
-4
-6
-8
-10
32
Day 3: Slope of a Line
Objectives:
• Ninth grade students will use the TI-83 Plus to review the steepness of a line as
rise/run.
• They will look at an overview on the Topics In Algebra Application.
• Students will use the application to play two different games on their own and with a
partner.
Opening Activity:
Students will look at “Transform”; an application on their TI-83 Plus. They will
use this to evaluate the steepness of a line and set of lines. They will enter Ax + 2 into
y1.
We will look at a slide show that starts with A = 0 and ends with A = 10.
We can see by the example below there is a big difference in A = 1 and A = 5. The first
graph is less steep than the second graph.
33
The students will see the same graphs on their calculators for 11 different values of A. If
they would like to stop the slide show at any time, they can press enter. We will have a
class discussion about what happens to the graphs as A gets closer to 10. We will talk
about what “A” represents in this equation of a line.
Students will choose another equation example and we will observe the slide
show again. We can discuss the similarities and differences to the first slide show. They
are encouraged to use negative numbers for the value of “A” in order to see an important
change in the graph. There should be another class discussion about the representation of
“A”. The students should make the conclusion that the constant in front of x represents
the steepness of a line. They can be told that this steepness is a line’s slope.
Main Activity:
Students will use their TI-83 Plus applications to look at an overview of slope as
rise/run. They will play a game in “Topics in Algebra, Chapter 3”.
In order to look at the overview and play the game, they have to use the following
commands: APPS; ALG1CH3 (ENTER); ENTER; 3:LINEAR FUNCTIONS (ENTER);
1:SLOPE WITH GRID (ENTER); 1:OVERVIEW (ENTER). They can use the arrows to
go through the slide show. After they see the slide show, they can press 2nd: QUIT. This
will take them back to the menu screen. They can choose 3:ACTIVITIES and this will
get them started on the game.
34
Students will look at the same game and are asked to think about the picture being
displayed on the coordinate plane. The starting point would represent something and the
ending point would represent something similar. They are to consider what type of
algebraic expression can be used to find the slope of a line that connects these points.
They can record any ideas they have on this expression and save them for later in the
activity.
Students will continue evaluating slope of a line by looking at another overview
in “Topics in Algebra, Chapter 3”. The students will see the formula for slope is (change
in y)/(change in x) = (y2 - y1)/(x2-x1). They can compare their algebraic expression with
the one given by the calculator overview. This calculator overview can be found in a
similar way. For the first overview, the students chose 1:SLOPE WITH GRID. For this
overview, we want them to choose 2:SLOPE USING COORD. All of the other steps are
the same.
Closing Activity:
Students will link their calculators with a partner and play the game that involves
the slope formula.
35
The game can be found under the following commands: Apps (ENTER);
ALG1CH3 (ENTER); ENTER; LINEAR FUNCTIONS (ENTER); 2:SLOPE USING
COORD (ENTER); 3:ACTIVITIES (ENTER); > (ARROW); 2:SCREEN CROSS
(ENTER); NEXT (F4); NEXT (F4); PLAY (F4). The object of the game is to get the
correct slopes for each line segment before your partner. The first person to get from the
left side of the screen to the right side is the winner. They should play against two
different people.
Homework:
Finish the questions about your observations that you made about the game you
played with a partner. Do the worksheet involving the slope formula.
36
Teachers’ Notes
Opening Activity:
Remind the students that a line has steepness, we call this slope. The variable we
are using to express slope is a constant.
So although we used, A, they should know we can use any letter to represent the slope.
We can conclude from the slide show that as our constant gets larger, our line gets
steeper.
Main Activity:
Make sure the students review what rise/run is. They should see what the
relationship is to slope.
The problems on the game should be fairly simple with the help of scrap paper.
Let them discuss the algebraic expression amongst each other. Write down any
expressions that they feel are correct. After looking at the next overview, have the class
discuss what is wrong, if anything, with the expressions they gave.
Closing Activity:
They may need scrap paper to do their arithmetic.
We canuse anyconstant.
37
TI-83 Plus Math 9
Name:__________________________
Definition: A set of points that are located on the same line are called COLLINEAR
POINTS.
1. Is any set of three points collinear? Give some examples to justify your answer.
2. What type of line has a negative slope? Make a sketch of the line.
3. What does the line that has undefined slope look like?
After playing the slope game with at least two different partners, write a short paragraph
( 4 to 5 complete sentences ) about anything that interested you while playing that game.
Are there any other observations that can be made?
(Handout for Closing)
10-10
-10
10
10-10
-10
10
38
TI-83 Plus Math 9
Name: KEY__
Definition: A set of points that are located on the same line are called COLLINEAR
POINTS.
4. Is any set of three points collinear? Give some examples to justify your answer.
No
Ex:
5. What type of line has a negative slope? Make a sketch of the line.
6. What does the line that has undefined slope look like?
After
playing the slope game with at least two different partners, write a short paragraph
( 4 to 5 complete sentences ) about anything that interested you while playing that game.
Are there any other observations that can be made?
(Handout for Closing)
10-10
-10
10
10-10
-10
10
39
Slope Homework Math 9
Name:________________________
Find the slope of the line that contains each pair of points.
1. ( 5, 7 ) and ( 3, 4 ) 2. ( 7, 6 ) and ( 4, 3 )
3. ( -1, 3 ) and ( 2, -2 ) 4. ( 0, -2 ) and ( 4, 3 )
5. ( 6, 0 ) and ( -3, 5 ) 6. ( -4, -1 ) and ( 1, 3 )
7. ( 1, 3 ) and ( -4, -5 ) 8. ( -5, -6 ) and ( -1, -5 )
Find the slope of each line from its graph.
1. (the scale goes by 1) 2.
(Homework Handout)
15
10
5
-5
-10
-15
-10 10
15
10
5
-5
-10
-15
-10 10
40
Slope Homework Math 9
Name:KEY______________
Find the slope of the line that contains each pair of points.
1. ( 5, 7 ) and ( 3, 4 ) m = 3/2 2. ( 7, 6 ) and ( 4, 3 ) m = 1
3. ( -1, 3 ) and ( 2, -2 )m = -5/3 4. ( 0, -2 ) and ( 4, 3 ) m = 5/4
5. ( 6, 0 ) and ( -3, 5 ) m = -5/9 6. ( -4, -1 ) and ( 1, 3 ) m = 4/5
7. ( 1, 3 ) and ( -4, -5 ) m = 8/5 8. ( -5, -6 ) and ( -1, -5 ) m = 1/4
Find the slope of each line from its graph.
1. (the scale goes by 1) 2.
m = 0 m = 1
(Homework Handout)
15
10
5
-5
-10
-15
-10 10
15
10
5
-5
-10
-15
-10 10
41
Day 4: Graphs Using Intercepts
Objectives:
• Ninth grade students will be using their problem solving skills as groups to find the
x-intercepts and y-intercepts of equations.
• They will use Algebra I to make tables of values, graph the equations, and check their
solutions.
Opening Activity:
Students are given some basic vocabulary by looking at a graph on the computer
overhead. They are told the definitions of x- and y-intercepts and are able to look at a
picture on GSP. They will discuss the values, or coordinates, of these points. Students
will eventually come to the conclusion that the x-intercept has coordinates of (x, 0), and
the y-intercept has coordinates of (0, y). They will be given an equation and are asked to
find the x- and y-intercepts. They can discuss, in groups of 3 or 4, how they can find the
intercepts given the equation and realization they just made about the coordinates.
Main Activity:
Students will use the information from the previous activity to graph this equation
using Algebra I. On their handout, they can plot their two points and draw the line that
goes through each of the points. On Algebra I, they can set up a table of values by
choosing the command from the left side of the screen. They should enter their two
ordered pairs, or two intercepts, into their table of values on Algebra I. They will make a
graph box and connect the two boxes by clicking on the double arrow from the table and
dragging it to the double arrow on the graph box. Students can click on the table and
press enter to get their graph. They should answer the questions on the handout. They
42
can graph the equations from the worksheet by using similar steps, which are the
following: determine the slope; find the x-intercept and the y-intercept; enter the values
into the table; graph the equation; and make a sketch of the graph from the computer onto
your paper. They should make a y= box for each graph to check their answer and make
sure it is the correct equation with graph. The opening handout has the students write
down the x- and y-intercepts for the beginning equation, so they do not need to rewrite
them on the main handout.
Closing Activity:
Students will be given a variety of graphs and they are asked to find the
x-intercepts and y-intercepts by looking at the graph.
Homework:
Students will be doing a handout that has a variety of questions. The questions
will be like the activities we did today in class. One part of the assignment asks the
students to write an equation of a line given its graph. The students will need to
determine the x- and y-intercepts. They will need to find the slope between these two
points by using the slope formula. They will then write the equation of this line by using
the formula: y = mx + b. Students know that m can stand for the slope of the line. They
are told that the y-coordinate of the y-intercept can be substituted for b. The x and y are
part of the equation of the line, which they should remember from writing and graphing
equations the other day.
43
Teachers’ Notes
Opening Activity:
The overhead screen should show an axis with grid. Draw any line on the grid
that crosses the x- and y-axis. The y-intercept is the value of y when x = 0. The
x-intercept is the value of x when y = 0. Explain the definition and point to each of these
points. The students should be able to write their own definition of these two terms.
When they work in groups, they should see that 0 can be substituted in for y then the
same for x. For example, 2x + 3y = 6. Let x = 0, we have 2(0) + 3y = 6. This gives us
3y = 6 and we can solve for y to get its intercept. The y-intercept is 2. The opposite can
be done for the x-intercept.
Main Activity:
The students should be able to follow the handout and answer the questions
without any problems.
Closing Activity:
The students should have no problem doing this worksheet. The graphs are very
easy to understand. All they have to do is determine ordered pairs.
Homework:
Let the students know that they can still look up the slope formula in their
calculator applications if they forget the formula. If any of the students are having
difficulty understanding the assignment, they should work together to discuss their
problems. Also, they can stay after to go over an example or ask questions. If there is
time after the closing activity, it would be beneficial to do an example of one of these
problems. I think it will help them greatly if they try it on their own first. The main
44
objective of this activity is to let the students think for themselves about writing the
equation of a line.
45
Intercepts Opening Math 9
Name:_________________________
1. Define, in your own words, the following terms:
• X-intercept -
______________________________________________________
• Y-intercept -
______________________________________________________
2. Find the x-intercept and the y-intercept of the graph of the equation 2x + 3y = 6.
What should you do first to find the x-intercept?_________________________
X-intercept:_______
What should you do next to find the y-intercept?_________________________
Y-intercept:_______
(Opening Handout)
46
Intercepts Opening Math 9
Name:KEY_____________
1. Define, in your own words, the following terms:
• X-intercept – the point where a line crosses the x-axis
• Y-intercept – the point where a line crosses the y-axis
3. Find the x-intercept and the y-intercept of the graph of the equation 2x + 3y = 6.
What should you do first to find the x-intercept? Let y = 0
X-intercept: 3
What should you do next to find the y-intercept? Let x = 0
Y-intercept: 2
(Opening Handout)
47
Algebra I Math 9
Name:_________________________
Answer the questions and then graph the following equations. For questions # 1, use the
x- and y-intercept from the opening handout to find the slope since the equations are the
same.
1. 2x + 3y = 6
a. What is the slope of this
line?_______
b. Sketch the graph of the
equation.
c. Using Algebra I, enter your
coordinates
into the table of values. Press enter to get the graph of your equation. To check if
this is the right graph for the equation, make a y = box by choosing it from the tool
bar on the left of the screen. Connect the y= box with your graph box. Press enter on
your table to get the equation and the graph.
d. Does your sketch match the picture on the computer?_______________________
2. 3.5x + 7y = 14
a. X-intercept:______
b. Y-intercept:______
c. Find the slope:_______
d. Make a sketch of the graph after you have done the graph on Algebra I.
15
10
5
-5
-10
-15
-10 10
48
3. y = 4x + 8
a. X-intercept:_______
b. Y-intercept:_______
c. Find the slope:_______
d. Make a sketch of the graph.
4. x – 2y = 6
a. X-intercept:_______
b. Y-intercept:_______
c. Find the slope:_______
d. Make a sketch of the graph.
(Main Handout)
15
10
5
-5
-10
-15
-10 10
15
10
5
-5
-10
-15
-10 10
15
10
5
-5
-10
-15
-10 10
49
Algebra I Math 9
Name:KEY___________________
Answer the questions and then graph the following equations. For questions # 1, use the
x- and y-intercept from the opening handout to find the slope since the equations are the
same.
5. 2x + 3y = 6
b. What is the slope of this
line?
m = -2/3
b. Sketch the graph of the
equation.
c. Using Algebra I, enter your
coordinates into the table of
values. Press enter to get the
graph of your equation. To check if this is the right graph for the equation, make a y
= box by choosing it from the tool bar on the left of the screen. Connect the y= box
with your graph box. Press enter on your table to get the equation and the graph.
d. Does your sketch match the picture on the computer?_______________________
6. 3.5x + 7y = 14
e. X-intercept: 4
f. Y-intercept: 2
g. Find the slope: m = -1/2
10
5
-5
-10
-10 10
10
5
-5
-10
-10 10
50
h. Make a sketch of the graph after you have done the graph on Algebra I.
7. y = 4x + 8
e. X-intercept: -2
f. Y-intercept: 8
g. Find the slope: m = 4
h. Make a sketch of the graph.
8. x – 2y = 6
e. X-intercept: 6
f. Y-intercept: -3
g. Find the slope: 1/2
h. Make a sketch of the graph.
(Main Handout)
10
5
-5
-10
-10 10
10
5
-5
-10
-10 10
51
Intercepts Closing Math 9
Name:_______________________
Use the graph to find the x-intercept and the y-intercept of the line.
1. Window: (-7.5, 7.5) x (-5, 5) X-intercept:_____ Y-intercept:_____
2. Window: (-7.5, 7.5) x (-5, 5) X-intercept:_____ Y-intercept:_____
3. Window: (-9.5, 5.5) x (-5.5, 4.5) X-intercept:_____ Y-intercept:_____
4. Window: (-40, 20) x (-20, 50) X-intercept:_____ Y-intercept:_____
Scale is by 10
52
5. Window: (-100, 100) x (-100, 100) X-intercept:_____ Y-intercept:_____
Scale is by 25
6. Window: (-10.5, 10.5) x (-7, 7) X-intercept:_____ Y-intercept:_____
(Closing Handout)
53
Intercepts Closing Math 9
Name:KEY
Use the graph to find the x-intercept and the y-intercept of the line.
1. Window: (-7.5, 7.5) x (-5, 5) X-intercept: 2 Y-intercept: 3
2. Window: (-7.5, 7.5) x (-5, 5) X-intercept: -2 Y-intercept: 4
3. Window: (-9.5, 5.5) x (-5.5, 4.5) X-intercept: -4 Y-intercept: -1
4. Window: (-40, 20) x (-20, 50) X-intercept: -10 Y-intercept: 40
Scale is by 10
54
5. Window: (-100, 100) x (-100, 100) X-intercept: 25 Y-intercept: 50
Scale is by 25
6. Window: (-10.5, 10.5) x (-7, 7) X-intercept: -3 Y-intercept: 6
(Closing Handout)
55
Homework Handout Math 9
Name:______________________
Find the x-intercept of the graph.
1. x + 3y = 5
2. 2x + 2y = -10
3. 3x + 4y = 12
4. 5x – y = 45
5. –x + 3y = 27
6. –12x – 20y = 60
Find the y-intercept of the graph.
7. y = -2x + 5
8. y = 3x – 4
9. y = 8x + 27
10. 4x – 5y = -35
11. 6x – 9y = 72
12. 3x + 12y = -84
Choose two of the equations from 1 – 6. Make sketches of the two that you have chosen.
15
10
5
-5
-10
-15
-10 10
15
10
5
-5
-10
-15
-10 10
56
Choose two of the equations from 7 – 12. Make sketches of each that you have chosen.
Follow each step in order to do the following problems.
Example 1
Step 1: Given these two intercepts, find the slope.
(0, 5) and (10, 0)
m = _______
Step 2: Substitute m into y = mx + b
Step 3: Substitute the y-intercept in for b. This is the equation of the line.
Problems:
1. (0, 9) and (3, 0) 4. (0, .01) and (.5, 0)
2. (4, 0) and (0, 8) 5. (-35, 0) and (0, 105)
3. (-1/2, 0) and (0, 3/4) 6. (.7, 0) and (0, -2.1) (Homework Handout)
15
10
5
-5
-10
-15
-10 10
15
10
5
-5
-10
-15
-10 10
57
Homework Handout Math 9
Name:KEY
Find the x-intercept of the graph.
13. x + 3y = 5 5
14. 2x + 2y = -10 -5
15. 3x + 4y = 12 4
16. 5x – y = 45 9
17. –x + 3y = 27 -27
18. –12x – 20y = 60 -5
Find the y-intercept of the graph.
19. y = -2x + 5 5
20. y = 3x – 4 -4
21. y = 8x + 27 27
22. 4x – 5y = -35 7
23. 6x – 9y = 72 -8
24. 3x + 12y = -84 -7
58
Choose two of the equations from 1 – 6. Make sketches of the two that you have chosen.
x + 3y = 5 -x + 3y = 27
Choose two of the equations from 7 – 12. Make sketches of e ach that you have chosen.
y = -2x + 5 y = 3x - 4
10
5
-5
-10
-10 10
r x =5-x
3
10
5
-5
-10
-10 10
s x = 27+x
3
10
5
-5
-10
-10 10
u x = -2 x+5
10
5
-5
-10
-10 10
v x = 3 x-4
59
Follow each step in order to do the following problems.
Example 1
Step 1: Given these two intercepts, find the slope.
(0, 5) and (10, 0)
m = -1/2
Step 2: Substitute m into y = mx + b y = -1/2x + b
Step 3: Substitute the y-intercept in for b. This is the equation of the line. y = -1/2x + 5
Problems:
4. (0, 9) and (3, 0) y = -3x + 9 4. (0, .01) and (.5, 0) y = .02x + .01
5. (4, 0) and (0, 8) y = -2x + 8 5. (-35, 0) and (0, 105) y = 3x + 105
3. (-1/2, 0) and (0, 3/4) 6. (.7, 0) and (0, -2.1) y = 3x – 2.1
y = 3/2x + 3/4
(Homework Handout)
60
Day 5: Scatter Plot and Best Fit Lines
Objectives:
• Ninth grade students will be using their TI-83 Plus to do an activity that involves a
scatter plot.
• They will be constructing the best-fit line of this scatter plot after doing a hands-on
activity.
Opening Activity:
Students will be doing a quick review of the previous night’s homework. The
review will be a handout that asks them to write equations in slope-intercept
form (y = mx + b).
Main Activity:
Students will be doing an activity with their TI-83 Plus using stat plot. For 20
seconds, they will put the first letter of their last name in one-inch squares of grid paper.
They will be doing this with their dominant hand, or the hand they write with. They will
count how many times they wrote the letter and record that number, along with their
name, on the spreadsheet. For 20 seconds more, students will use the opposite hand and
do the same activity. They should count and record their data.
Name of Student Number of Letters -Dominant
Number of Letters –Non-Dominant
1)2)3)4)5)6)7)8)9)10)
X Y
61
The next step in this activity is to graph the data. On a separate piece of graph
paper, make a first quadrant graph of the data by plotting the points from the table. The
dominant hand numbers go into the x values; they represent the x-axis. The non-
dominant hand represents the y-axis. The students will link their calculators with the
calculator at the front of the classroom. The front calculator has two lists stored. The
first list, DOM, and the second list, NON. The students will be receiving both of these
lists. The students will need to use the following commands: 2nd Link; Arrow over once
to Receive; Press enter. The front calculator has to follow these commands in order to
send the two lists: 2nd Link; 4:List…; Arrow down to DOM; Press enter (there should be
a little square next to the chosen list); Arrow up or down to NON; Press enter; Arrow
over once to Transmit; Press enter. By choosing both of these lists, the students should
receive both of them into their calculators as long as the little squares appeared next to
both lists.
The students will be using these lists to construct a scatter plot of the data. They
will use the following commands: 2nd y =; 1:Enter; Press enter when the cursor is
blinking to ON; Arrow down to TYPE; Press enter on the graph that has a bunch of dots;
Arrow down to Xlist; 2nd Stat; Under NAMES, arrow down to DOM; Press enter (DOM
should appear in the Xlist spot); Arrow down to Ylist; 2nd Stat; Under NAMES, arrow
down to NON; Press enter (this should appear in the Ylist); Arrow down to Mark; Press
62
enter on the box; ZOOM; 9:ZOOMSTAT. The scatter plot of the classes’ data should
appear on their screen.
Closing Activity:
Students will be using this data to find a best-fitting line. They will use linear
regression to find the best-fitting line. They will need to follow these steps: 2nd Stat;
CALC; 4:LinReg(ax+b); 2nd LIST; Arrow down to DOM and press enter; comma; 2nd
LIST; Arrow down to NON and press enter; comma; VARS; Y-VARS; 1:FUNCTION;
1:Y1; Enter. The students should get a screen that gives them values for a, b, r^2, and r.
If the students look at y =, they will see that they have an equation of a line in y = mx+b
form. They can use GRAPH to graph the line along with their data.
There will be a handout for this activity that the students will do together in groups of
two or three. The students should notice that “a” represents the slope and “b” represents
the y-intercept. The value for “r” tells us how well this line fits our set of data. In other
words, how well is our line in accordance with our data? The closer the absolute value of
r is to 1, the better the line fits the data.
63
Homework:
Students are given a homework assignment that involves the same activity.
However, the experiment they have to simulate involves a little twist. Instead of using
dominant and non-dominant, the people doing the activity would use their right and left
hands; it doesn’t matter which hand they usually write with.
64
Teachers’ Notes
Opening Activity:
Since there wasn’t much time spent on slope-intercept form yesterday, you should
go over more examples at the beginning of today’s class. It is not a good idea to go over
every homework problem from yesterday. If a student didn’t do their homework, then we
would just be giving them the answer. New example questions will help them learn the
concept. Also, if they did have a problem or did not do their homework, they can go
back and try it again.
Main Activity:
This is a great way to teach the students how to plot data using their calculators.
Also, the activity is fun so they will want to try it.
Closing Activity:
This will help us review slope-intercept form in a different context. It will
reinforce what each of the variables represents. When the students do the linear
regression, the diagnostics on their calculator must be on. They can go to 2nd
CATALOG; Press D; Arrow down to Diagnostics On; ENTER; ENTER.
Homework:
The homework is designed to show the students that an experiment can be done
right, but it may look like there was an error. If the experiment had been done by just
saying right- and left-handed, the students may have assumed that the whole class is
right-handed. I think it is easier for them to see the differences in the way the activities
are worded.
65
Slope-Intercept Form Math 9
Name:_____________________
Write an equation in slope-intercept form of the line that passes through the points.
1. (-3, 0) and (0, -1)
2. (-1, 0) and (0, -3)
3. (4, 2) and (-2, -4) (*HINT: After finding the slope, substitute it for m and use
one of your points for your x and y. Use algebra to solve for b.)
4. (-6, -5) and (1, 4)
5. (2, 3) and (4, 3)
6. (5, -10) and (12, -7)
(Opening Handout)
66
Slope-Intercept Form Math 9
Name:KEY
Write an equation in slope-intercept form of the line that passes through the points.
7. (-3, 0) and (0, -1) y = -1/3x - 1
8. (-1, 0) and (0, -3) y = -3x - 3
9. (4, 2) and (-2, -4) y = x - 2
(*HINT: After finding the slope, substitute it for m and use one of your points for
your x and y. Use algebra to solve for b.)
10. (-6, -5) and (1, 4) y = 9/7 x + 19/7
11. (2, 3) and (4, 3) y = 3
12. (5, -10) and (12, -7) y = 3/7 x – 85/7
(Opening Handout)
67
X Y
Name of Student Number of Letters- Dominant
Number of Letters - Non-Dominant
1)2)3)4)5)6)7)8)9)10)11)12)13)14)15)16)17)18)19)20)21)22)23)24)25)26)27)28)
(Main Activity)
68
First Quadrant Graph Paper
70
60
50
40
30
20
10
-10
20 40 60
69
Homework Assignment Math 9
Name:______________________
We did the activity using dominant and non-dominant hands for writing.
However, some people are left-handed and others are right-handed. For example, John is
writes with his left hand. He writes the first letter of his last name on the worksheet for
20 seconds with his right hand. Then, he changes hands and writes the letter in the boxes
with his left hand. What do you think would happen to the data if 29 students were right-
handed and 1 student was left-handed? What would the scatter plot look like? Your job
is to give an example of 5 students who are participating in the activity. Out of these five
students, four are right-handed and one is left-handed. Fill in the table below with five
students’ names. Fill out the table as if the experiment were conducted amongst these
five students, remembering that one student is left-handed. Try to make your data as real
as possible, like five of your friends did the experiment.
Name ofStudent
Number of Letters - Righthand
Number of Letters - Left hand
1)2)3)4)5)Draw a first quadrant graph of the data by plotting the points above.
70
Draw the y = x line on your first quadrant graph. What is the meaning of this line?
What do the points above the y = x line represent?
What do the points below the y = x line represent?
What is the significance of points near the y = x line?
(Homework Handout)
71
Homework Assignment Math 9
Name: Student Sample Worksheet
We did the activity using dominant and non-dominant hands for writing.
However, some people are left-handed and others are right-handed. For example, John is
writes with his left hand. He writes the first letter of his last name on the worksheet for
20 seconds with his right hand. Then, he changes hands and writes the letter in the boxes
with his left hand. What do you think would happen to the data if 29 students were right-
handed and 1 student was left-handed? What would the scatter plot look like? Your job
is to give an example of 5 students who are participating in the activity. Out of these five
students, four are right-handed and one is left-handed. Fill in the table below with five
students’ names. Fill out the table as if the experiment were conducted amongst these
five students, remembering that one student is left-handed. Try to make your data as real
as possible, like five of your friends did the experiment.
Name ofStudent
Number of Letters - Righthand
Number of Letters - Left hand
1)Karen 15 322)John 29 243)Megan 25 114)Shawn 35 295)Kristen 50 25
72
Draw a first quadrant graph of the data by plotting the points above.
Draw the y = x line on your first quadrant graph. What is the meaning of this line?
This is the line that represents people who are right and left handed.
What do the points above the y = x line represent?
The people who write more letters with their left hand.
What do the points below the y = x line represent?
The people who write more letters with their right hand.
What is the significance of points near the y = x line?
These people can write well with both hands . (Homework Handout)
100
80
60
40
20
50 100
w x = x
Related Documents
