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Lesson Plan -- Graphing
Chapter Resources
- Lesson 5-12 Ordered Pairs
- Lesson 5-13 Plot Points in the Coordinate Plane
- Lesson 5-15 Graph Linear Equations
- Lesson 5-12 Ordered Pairs Answers
- Lesson 5-13 Plot Points in the Coordinate Plane Answers
- Lesson 5-15 Graph Linear Equations Answers
1
Name ———————————————————————
LESSON
5-12
7ZLMZML8IQZ[
California
Standards
Words to Remember
Gr. 3 AF 1.0: Students
select appropriate
symbols, operations,
and properties to
represent, describe,
simplify, and solve
simple number
relationships.
Ordered pair: A pair of numbers represented by x and y and written in
the form (x, y)
Input: The values of x or of the first numbers in the ordered pairs
Output: The values of y or of the second numbers in the ordered pairs
Equation in two variables: A rule relating input and output values
Gr. 5 SDAP 1.5: Know
how to write ordered
pairs correctly; for
example (x, y).
Also included: Gr. 3
AF 2.1, Gr. 3/4/5/6/7
MR 1.1
Date ————————————
Getting Started You can use a rule to describe how the output values
are related to the input values. The ordered pairs from the rule can be
displayed in an input-output table.
E@)584-
Making an Input-Output Table
Complete the input-output table for the rule y 5 x 2 1.
Substitute each input value for x in the rule. Then simplify to
find the output values y.
Input, x
Output, y
T:A <0 1;
23
21
23 2 1 5 24 21 2 1 5 22
0
5
0 2 1 5 21
52154
Make a table.
1. Complete the input-output table for the rule y 5 2x.
Input, x
22
0
1
4
Output, y
2. Complete the input-output table for the rule y 5 x 1 5.
Input, x
Output, y
48
Math Intervention
Book 5 Algebraic Thinking
23
21
2
4
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Solution
Name ———————————————————————
E@)584-
Date ————————————
Writing Ordered Pairs from a Table
Write the values in the table as ordered pairs.
3FNFNCFS
In an ordered pair,
you write the
x-values first and
the y-values second.
x
23
21
0
2
5
y
1
3
4
6
9
Solution
(23, 1), (21, 3), (0, 4), (2, 6), (5, 9)
T:A <0 1;
Write ordered pairs.
3. Write the values in the table as ordered pairs.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
-PPLGPSB
1BUUFSO
How can the
y-values be
obtained from the
corresponding
x-values?
x
215
25
0
10
y
23
21
0
2
E@)584-
Writing a Rule from an Input-Output Table
Write a rule that shows how y relates to x.
x
26
23
0
5
y
22
1
4
9
Solution
The y-values are 4 more than the corresponding x-values.
Therefore, the rule is y 5 x 1 4.
T:A <0 1;
Write a rule.
4. Write a rule that shows how y relates to x.
x
22
21
0
1
2
y
26
23
0
3
6
5. Write a rule that shows how y relates to x.
x
24
22
0
2
4
y
27
25
23
21
1
Math Intervention
Book 5 Algebraic Thinking
49
Name ———————————————————————
Date ————————————
Summarize
Making an Input-Output Table
Using a rule describing a relationship between two values x and
y, substitute input values for x to find the corresponding output
values for y.
Writing Ordered Pairs from a Table
The x-values or input values are the first number in each ordered
pair. The y-values or output values are the second.
Writing a Rule from an Input-Output Table
Look for a relationship between the x-values and the y-values.
Find the pattern that determines how each value of y can be
obtained from the corresponding value of x.
Practice
Complete the input-output table using the given rule.
1. y 5 4x
Input, x
1
2
5
7
2. The number of sailboats x is 20 less than the number of speed
boats y.
Sailboats, x
0
2
3
5
Speed boats, y
Write the values in the table as ordered pairs.
3.
x
26
22
1
4
y
28
24
21
2
____________________________________________________________________________
4.
x
23
22
21
0
y
4
5
6
7
____________________________________________________________________________
50
Math Intervention
Book 5 Algebraic Thinking
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Output, y
Name ———————————————————————
Date ————————————
Write a rule that shows how y relates to x.
5.
x
21
0
1
2
y
4
0
24
28
_________________________________________________________________________________
6.
x
21
0
1
2
y
22
21
0
1
_________________________________________________________________________________
7.
x
23
0
6
12
y
21
0
2
4
_________________________________________________________________________________
8. Explain how to identify the input values in the ordered pairs
(22, 21), (7, 6), and (10, 9).
_________________________________________________________________________________
_________________________________________________________________________________
9. The distance d that Maria travels in h hours can be described by the
rule d 5 50h. Write three ordered pairs that make the rule true.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
_________________________________________________________________________________
,1,A7=/ -<1<'
10. Fill in the missing words. In the ordered pair (x, y), the value
of ________ is the input and the value of ________ is the output.
11. Describe a process. How would you write a rule that shows
how y relates to x ?
x
22
21
0
1
y
28
24
0
4
__________________________________________________________________________
__________________________________________________________________________
12. Explain your reasoning. Can the relationship between the
number of weeks that have passed and the number of years that have
passed be described by a rule relating two variables w and y ? Explain.
__________________________________________________________________________
__________________________________________________________________________
Math Intervention
Book 5 Algebraic Thinking
51
Name ———————————————————————
LESSON
5-13
Date ————————————
8TW\8WQV\[QV\PM
+WWZLQVI\M8TIVM
California
Standards
Words to Remember
Gr. 4 MG 2.0: Students
use two-dimensional
coordinate grids to
represent points and
graph lines and simple
figures.
y
4
x-axis: The horizontal axis
3
y-axis
2
y-axis: The vertical axis
Origin: The point (0, 0), where the
horizontal and vertical axes intersect
Gr. 5 AF 1.4: Identify and
graph ordered pairs in
the four quadrants of
the coordinate plane.
1
x-axis
24 23 (0, 0)
origin
1 2
3 4 x
22
23
24
Getting Started In Lesson 5-12 you learned how to write ordered pairs
from an input-output table. In this lesson you will graph ordered pairs in a
coordinate plane.
Identifying Points in a Coordinate Plane
Name the two points on the graph with the coordinates
(24, 0) and (1, 23).
y
B
24 23 22
A point with
y-coordinate 0 lies
on the x-axis. A point
with x-coordinate 0
lies on the y-axis.
3
2
A
1
C
D
3FNFNCFS
4
O
G
1 2
3
4 x
22
F
23
24 E
Solution
Starting at the origin, if you move left
4 units, then up 0 units, you get to C.
Point C has coordinates (24, 0).
Starting at the origin, if you move right
1 unit, then down 3 units, you get to F.
Point F has coordinates (1, 23).
52
Math Intervention
Book 5 Algebraic Thinking
y
4
3
2
1
C (24, 0)
24 23 22
O
22
23
24
2
3
4 x
F (1, 23)
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
E@)584-
Name ———————————————————————
T:A <0 1;
Date ————————————
Use the graph in Example 1 to name the point
with the given coordinates.
1. (4, 0)
If you move __________ 4 units, then up
units you get to point
__________. Point __________ has coordinates (4, 0).
2. (23, 22)
units, then _________________ 2 units you get to
If you move left
point __________. Point __________ has coordinates (23, 22).
E@)584-
Graphing Points in a Coordinate Plane
Graph the points A(25, 2), B(4, 1), and C(0, 23).
Solution
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
3FNFNCFS
When you graph
a point, always
start at the origin.
Move right or up for
positive coordinates.
Move left or down
for negative
coordinates. First
move along the
horizontal axis. Then
move up or down.
To graph A move left 5 units.
Then move up 2 units.
y
4
3
2
A
B
1
25 24 23 22
O
1
2
3 x
22
C
23
24
To graph B move right 4 units.
Then move up 1 unit.
To graph C do not move right or left.
Just move down 3 units.
T:A <0 1;
3. Graph the points X(22, 24), Y(21, 3), and Z(2, 0).
To graph X move left
units.
Then move ______________ 4 units.
y
To graph Y move ______________ 1 unit.
O
x
Then move up
units.
To graph Z just move ______________
units. Do not move ______________
or ______________.
Math Intervention
Book 5 Algebraic Thinking
53
Name ———————————————————————
Date ————————————
Summarize
Identifying Points in a Coordinate Plane
(1) To name the coordinates of a point on a coordinate graph, first
find the x-coordinate by counting how many units the point is
to the left or right of the origin.
(2) Then find the y-coordinate by counting how many units the
point is above or below the origin.
(3) Write the coordinates as an ordered pair with the x-coordinate
first and the y-coordinate second.
Plotting a Point in a Coordinate Plane
(1) To plot a point, start at the origin and move right or left the
number of units corresponding to the x-coordinate. Move right
if the x-coordinate is positive, move left if it is negative.
(2) Then move up or down the number of units corresponding to
the y-coordinate. Move up if the y-coordinate is positive, move
down if it is negative.
(3)
Make a dot at this location.
8ZIK\QKM
In Exercises 1–7, write the coordinates of the points.
1. Point A is 1 unit right and 6 units down from the origin, so the
coordinates of A are ___________ .
C
B
G
2. Point B is 4 units right and 1 unit up from the origin, so the
coordinates of B are ___________ .
3. Point C is
D
units right and 3 units ___________ from the origin,
so the coordinates of C are ___________ .
4. Point D is
units ___________ and
units ___________ from the
origin, so the coordinates of D are ___________.
5. Point E is
units ___________ from the origin, so the coordinates
of D are ___________ .
6. The coordinates of F are ___________ .
7. The coordinates of G are ___________ .
54
Math Intervention
Book 5 Algebraic Thinking
28 26 24
O
24
26
28
2 4
E
A
6
8 x
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
y
8
F 6
4
2
Name ———————————————————————
Date ————————————
Plot the point.
y
8. M(26, 2)
9. N(0, 1)
10. P(23, 24)
11. Q(8, 21)
8
6
4
2
28 26 24
12. R(5, 6)
13. S(22, 0)
14. O(0, 0)
15. U(0, 27)
16. V(21, 8)
17. W(2, 26)
18. X(0, 22)
19. Y(3, 1)
O
2 4
6
8 x
24
26
28
20. Describe the coordinates of a point that lies on either the x-axis or
the y-axis.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
,1,A7=/ -<1<'
21. Fill in the missing words. The horizontal axis in a coordinate
plane is called the ____________ and the vertical axis is called the
____________.
22. Describe a process. How would you explain to your friend
how to graph the point (6, 21)?
__________________________________________________________________________
__________________________________________________________________________
23. Explain your reasoning. Fred says the coordinates
of point M are (4, 22). Vicki says the coordinates
are (22, 4). Who is correct? Explain.
M
y
4
3
2
1
24 23 22
O
1 2
3
4 x
22
23
24
__________________________________________________________________________
__________________________________________________________________________
Math Intervention
Book 5 Algebraic Thinking
55
Name ———————————————————————
LESSON
5-15
/ZIXP4QVMIZ-Y]I\QWV[
California
Standards
Words to Remember
Gr. 4 AF 1.5: Understand
that an equation
such as y 5 3x 1 5
is a prescription for
determining a second
number when a first
number is given.
Gr. 5 AF 1.0: Students
use variables in simple
expressions, compute
the value of the
expression for specific
values of the variable,
and plot and interpret
the results.
Also included: Gr. 4
Date ————————————
y
Graph of a linear equation: All the points
in the coordinate plane that are solutions of
the equation
2
1
23
The graph is a horizontal, vertical, or diagonal line.
O
2
3 x
22
23
Getting Started In Lesson 5-13 you learned how to plot points in a
coordinate plane. In this lesson you will learn how you can plot points to
graph a line.
MG 2.0, Gr. 4 MG 2.1
E@)584-
Graphing Linear Equations Using a Table
of Values
Solution
;\MX Complete the table.
3FNFNCFS
x
Find y by substituting
each value of x in the
equation y 5 x 1 3.
y
–2
y5x13
y 5 –2 1 3
51
(x, y )
(–2, 1)
–1
y5x13
y 5 –1 1 3
52
(–1, 2)
0
1
y5x13
y5013
53
y5x13
y5113
54
(0, 3)
(1, 4)
;\MX Plot the points.
y
4
;\MX Draw a line through the points.
2
1
24
22
O
1 2
3
4 x
22
23
24
;\MX Interpret the graph. Every ordered pair that makes y 5 x 1 3 true is
a solution of the equation and lies on its graph. Since 13 5 10 1 3,
(10, 3) is a solution of y 5 x 1 3 and lies on its graph.
58
Math Intervention
Book 5 Algebraic Thinking
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Use the table of values to graph y 5 x 1 3. Interpret the graph by explaining
how you know that the point (10, 13) also lies on the graph of the line.
Name ———————————————————————
T:A <0 1;
Date ————————————
Use the equation y 5 2x 1 1.
1. Fill in the missing information to graph y 5 2x 1 1.
;\MX Complete the table.
x
–2
0
y 5 2x 1 1
y 5 2x 1 1
11
y52•
y
(
,
)
,
), and (
,
y5
5
;\MX Plot the points (
(
y5
1
y52•
5
(x, y)
1
5
(
,
)
(
),
,
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
E@)584-
)
1 2
3
y
).
3
2. The point (–8, –15) also lies on the graph of
52+
,
4
2
1
Then draw a line through the points.
y 5 2x 1 1 because
1
•
1
24 23 22
O
4 x
22
23
24
.
Graphing Horizontal and Vertical Lines
Use three ordered pairs to determine whether the graphs of y 5 – 4 and
x 5 3 are horizontal or vertical.
Solution
Choose 3 values for x.
y is always 24.
Choose 3 values for y.
x is always 3.
y 5 24
x53
(–2, –4)
(0, –4)
(1, –4)
(3, –4)
(3, 0)
(3, 3)
ANSWER The graph of y 5 24 is a horizontal line.
The graph of x 5 3 is a vertical line.
T:A <0 1;
Use three ordered pairs to determine whether the
graph is horizontal or vertical.
3. x 5 22 __________
4. y 5 1 __________
Math Intervention
Book 5 Algebraic Thinking
59
Name ———————————————————————
Date ————————————
Summarize
Graphing a Linear Equation
To graph a linear equation, make a table of values and plot the
points. Graph at least three points to determine the line.
Graphing a Vertical or Horizontal Line
For a vertical line, x will have the same value for all values of y. For a
horizontal line, y will have the same value for all values of x.
Vertical line
Horizontal line
Diagonal line
x52
x 5 –0.5
y 5 –10
y 5 12
y5x–5
y 5 –4x 1 1
8ZIK\QKM
Complete the table of values to draw the graph.
1. y 5 x 2 1
y
–2
y
y5x–1
y5
21
–1
(
)
22
4. y 5 4x 1 2
0
60
2
y
3
4
x
y
x
O
y
x
O
)
3
5
1
Math Intervention
Book 5 Algebraic Thinking
,
5. y 5 2} x 2 1
y
y
(
x
x
O
21
)
3. y 5 2x 2 5
2
y
x
,
y
0
2
5
(
2. y 5 x 1 7
x
y5
y5
5
,
x
O
y5x–1
y5
2
5
(x, y)
0
25
0
y
5
O
x
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
x
Name ———————————————————————
Date ————————————
Complete the table of values to draw the graph.
6. x 1 y 5 4
x
y
21
0
7.
3x 1 y 5 6
x
1
x
O
y
8. y 5 1.5
1.5
1.5
x
x
O
22
x
1.5
3
9. x 5 22
y
O
2
y
x
y
1
y
y
22
22
O
y
x
Tell whether the graph of the equation is a vertical, horizontal, or
diagonal line. Explain your reasoning.
10. y 5 3
_________________________________________________________________________________
11. x 5 4
_________________________________________________________________________________
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
12. y 5 2x 2 6
_________________________________________________________________________________
,1,A7=/ -<1<'
13. Fill in the missing words. The graph of a linear equation is a
___________ , ___________ , or ___________ line.
1
2
14. Describe a process. How would you graph y 5 } x 1 3?
__________________________________________________________________________
__________________________________________________________________________
15. Explain your reasoning. Your friend said that the graph of
x 5 2 is a horizontal line. Is she correct?
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
Math Intervention
Book 5 Algebraic Thinking
61
Answer Key
Lesson 5-12, pp. 48–51
Try this:
1. 24, 0, 2, 8
2. 2, 4, 7, 9
3. (215, 23), (25, 21), (0, 0), (10, 2)
4. y 5 3x
5. y 5 x – 3
Practice:
1. 4, 8, 20, 28
2. 20, 22, 23, 25
3. (26, 28), (22, 24), (1, 21), (4, 2)
4. (23, 4), (22, 5), (21, 6), (0, 7)
5. y 5 24x
6. y 5 x 2 1
x
7. y 5 }
​ 3 ​ 8. Sample answer: The input values are the first values in each ordered pair.
The input values are 22, 7, and 10.
9. Sample answer: (1, 50), (2, 100), (3, 150)
10. x; y
11. Sample answer: Look for a relationship between the x-values and the
y-values. In the table each y-value is 4 times the corresponding x-value, so the rule is y 5 4x.
12. yes; 52 weeks in 1 year, so w 5 52y
Answer Key
Lesson 5-13, pp. 52–55
Try this:
1. right, 0, G, G
2. 3, down, D, D
3.
4
;
y
Y 3
2
1
1 2
O
24 23 22
Z
3
4 x
22
23
X
24
(continued )
2; down; left; 3; right; 2; up; down
Practice:
1. (1, 26)
2. (4, 1)
3. 3; up; (3, 3)
4. 5; left; 2; down; (25, 22)
5. 4; down; (0, 24)
6. (23, 5)
7. (23, 0)
y
8–19.
V
R
6
4
2 N
M
S
Y
O O2 4
28 26 24
P
X
6
Q
x
24
W
26
28 U
20. Sample answer: A point that lies on the
x-axis has y-coordinate 0. A point that lies on the y-axis has x-coordinate 0.
21. x-axis; y-axis
22. Sample answer: Start at the origin and move 6 units right and then 1 unit down.
23. Vicki is correct. Sample answer: To get to M you move left 2 units and then up 4 units.
Answer Key
Lesson 5-15, pp. 58–61
Try this: 1. (continued on next page also)
x
y
(x, y )
–2
y 5 2x 1 1
0
y 5 2x 1 1
1
y 5 2x 1 1
y 5 2 p 22 1 1
y 5 2 p 0 1 1
y 5 2 p 1 1 1
5 23
(22, 23)
5 1
(0, 1)
5 3
(1, 3)
–2; 23; 0; 1; 1; 3;
4
3
y
2
1
24 23 22
O
1 2
3
4 x
22
23
24
2. 215; –8; 1
3. vertical
4. horizontal
Practice: 1.
x
y
–2
y5x21
–1
y5x21
0
y5x21
y 5 22 2 1
y 5 21 2 1
y5021
5 23
(x, y ) (22, 23)
2
5 22
(21, 22)
y
1
22
O
2. 5, 7, 9;
1 2 x
10
O
210
y
5 10 x
5 21
(0, 21)
Answer Key
3. 21, 1, 3;
10
5
y
5 10 x
O
210
4. 22, 2, 6;
y
4
2
2 4 x
24 22
5. 2, 21, 24;
4
2
y
2 4 x
24 22
24
6. 5, 4, 3;
4
2
y
2 4
O
24
x
24
7. 3, 0, 23;
2
1
22
O
y
1
x
22
8. Sample answer: 1, 2, 3;
2
y
1
22
1 2 x
O
22
9. Sample answer: 1, 2, 3;
2
y
1
O
1 2 x
22
10. The y-values are the same for each x-value, so the line is horizontal.
Answer Key
11. The x-values are the same for each y-value, so the line is vertical.
12. The x-values are different for each different y-value, so the line is diagonal.
13. horizontal; vertical; diagonal
14. Sample answer: Make a table of values choosing at least 3 numbers for x.
To make the arithmetic easier, use even numbers for x.
Substitute them in the equation for x to
find the y-values. Then graph the ordered
pairs and draw a line through them.
15. no; Sample answer: For every ordered pair on the graph of x 5 2, the value of x is 2.
This means the graph goes through the points (2, 21), (2, 0), and (2, 1), for example.
This describes a vertical line 2 units to the right of the y-axis.