Lesson Plan -- Finding Slope - Blackboard

Lesson Plan -- Finding Slope

Chapter Resources

- Lesson 5-16 Slope
- Lesson 5-16 Slope Answers
- Lesson 5-18 Slope-Intercept Form
- Lesson 5-18 Slope-Intercept Form Answers

1

Name ——————————————————————— Date ————————————

LESSON ;TWXM

5-16

California Words to Remember y
Standards 4
Slope: vertical change 5 }rriusne 5 }xy22 22 xy11 Zero 3 Negative slope
Gr. 7 AF 3.3: Graph linear }horizonta}l change slope
functions, noting that 1
the vertical change
(change in y-value) 23 22 O 1 2 4 x
per unit of horizontal Slope can be positive, negative, Undefined
change (change in zero, or undefined. slope 22 Positive slope
x-value) is always the
same and know that the 24
ratio (“rise over run”)
is called the slope of
the graph.

Getting Started In this lesson you will use graphs of linear equations to
find the slopes of the lines.

Finding the Slope of a Line Using the

E@)584- Ratio }RRiusne

Find the slope of the line through (3, 3) and (6, 5).

Solution

;\MX Find the ratio of rise to y Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
run between (3, 3) and (6, 5).
3FNFNCFS }rriusne 5 }32 6 run 5 3
The slope of a line is 5
the same between ;\MX Check the slope using
any two points on (3, 3) and (0, 1). 4 rise 5 2 (6, 5)
the line.
}rriusne 5 }2232 5 }23 3 (3, 3)
/PUJDF 2 rise 5 22
A rise and a run can ANSWER The slope is }23.
be either positive or (0, 1)
negative. run 5 23

O 1 2 3 4 5 6x

22

E@) 5 8 4- Understanding the Slope of Horizontal
and Vertical Lines

Find the slope of y 5 3 and x 5 –2. y
2 y53
Solution 1

There is no vertical change between the points on the x 5 22 O 1x

line y 5 3. Therefore }rriusne 5 }ru0n 5 0. The slope of y 5 3 is 0. 23

There is no horizontal change between the points on the

line x 5 22. Therefore }rriusne 5 }ri0se. Since division by zero is not
allowed, the slope of the line x 5 22 is undefined.

Math Intervention

62 Book 5 Algebraic Thinking

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T :A <0 1; Fill in the missing information to find the slope of

the line.

/PUJDF 1. y 2. y
1 (1, 1)
The slope of any 2
horizontal line is 0. 24 23 22 O 1 2 3 4 x 1

22 24 23 22 O 1 2 3 4 x

24 22
(21, 25) 25 23

slope 5 }rise The line has no
change. Therefore
______
}rriusne 5 }run 5 .
5 }2 5
The slope is . The slope is .

Slope Formula The slope of a line containing the points (x1, y1) and
(x2, y2) can be found by using the formula slope 5 }xy22 22 yx11.

E@)584- Using the Slope Formula

Copyright © by McDougal Littell, a division of Houghton Mifflin Company. What is the slope of the line containing the points (5, 7) and (22, 4)?

3FNFNCFS Solution Slope formula
When calculating slope 5 }xy22 22 xy11
the slope from the 5 }2422275 Substitute values for y , y , x , and x .
coordinates of two 5 }2237 212 1
points, subtract
the x-values in the 5 }37 Simplify.
same order that you
subtract the Simplify.
y-values.
T :A <0 1; Find the slope of the line containing the given points.
/PUJDF
The slope of any 3. (28, 2) and (6, 9) 4. (8, 2) and (8, 23)
vertical line is slope 5 }xy22 22 xy11 slope 5 }xy22 22 xy11
undefined.
92 22
5 5

2 28 82

5}5 5}

The slope is . The slope is .

Math Intervention 63
Book 5 Algebraic Thinking

Name ——————————————————————— Date ————————————

Summarize

Finding the Slope of a Line using Rise over Run

You can find the slope of a line by counting the rise (vertical change)
and run (horizontal change) between two points on the graph. Then
write the fraction for }rriusne and simplify.
Finding the Slope of a Line Using the Slope Formula

The slope can also be found by calculating x}y22 22 xy11. A horizontal line has
0 slope. A vertical line has undefined slope. A line sloping up to the right
has a positive slope. A line sloping down to the right has a negative slope.

Practice

Tell whether the slope of the line is positive, negative, zero, or undefined.

1. y 2. y 3. y x 4. y
2
1 22 1 2 2
22 O 1 2 x 1
O 1 2x 22 O 1 2 x
22 22 O 1 22

22

Fill in the missing information to find the slope of the line.

5. y 6. y Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
(26, 6) 6
1 4
(26, 0) 2
22 O 1 2 x 28 24
Ox
22

______ The line has no
slope 5 run ____________ change.
So }rriusne 5 }rise
1
5 Division by is
undefined. Therefore,
The slope is . the slope is ____________ .

Find the slope of the line.

7. y 8. y 9. y
(21.5, 0) 1
4 8
22 O 1 2x (1, 7)
2
(21, 22) 1 6
O 1 2 3x 4

(0, 3)

24 2 4 x

Math Intervention

64 Book 5 Algebraic Thinking

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Find the slope of the line containing the given points.

10. (0, 5) and ( 21, 4) 11. (3, 6) and (8, 6)

slope 5 }xy22 22 xy11 slope 5 }xy22 22 yx11

42 26
5 5

20 32

5} 5}

5 5
The slope is . The slope is .
12. (2, 5) and (2, 21) 13. (211, 29) and (28, 23)

14. (212, 22) and (29, 13) 15. (22, 26) and (8, 212)

1 2 1 216. 2}31, }12 and 2}12, }32 1 2 1 217. 2}25, 2}43 and }53, }41

Copyright © by McDougal Littell, a division of Houghton Mifflin Company. ,1,A7=/-<1<'

18. Fill in the missing words. The slope of a line can be found
by writing the fraction ____________ /____________.

19. Find the error. Lihue found the slope of the line containing
(5, 6) and (3, 9) to be 2}32. What was his mistake?
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________

20. Explain your reasoning. A line contains (2, 4) and (23, 5).

You found the slope using }2532242. Your friend found the slope
using }2 422(253). Who is correct? Explain.

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

Math Intervention 65
Book 5 Algebraic Thinking

Answer Key

Lesson 5-16, pp. 62–65

Try this:

1. run; 6; 3; 3

2. vertical; 0; 0; 0
3. 2; 6; 7; 222; 2} ​272  ;​  2} ​272   ​
4. 23; 8; }​2 01  ;​ undefined
Practice:

1. zero

2. positive

3. undefined

4. negative
5. rise; 2; 1; 2

6. horizontal; 0; 0; undefined

7. 24

8. 0

9. 4

10. 5; –1; –1; –1; 1; 1
11. 6; 8; 0; –5; 0; 0
12. undefined
13. 2
14. 5
15. 2} ​53 ​
16. 21
17. 1
18. rise; run
19. Sample answer: He put the difference of the x-values in the

numerator and the difference of the y-values in the denominator.
He should have put the difference of the y-values in the numerator
and the difference of the x-values in the denominator.

20. Sample answer: You are both correct. The order you subtract the
coordinates of the points does not matter as long as you are consistent
and subtract the coordinates in the numerator and denominator in the same order.

Name ——————————————————————— Date ————————————

LESSON ;TWXM1V\MZKMX\.WZU

5-18

California Words to Remember
Standards y-intercept: The y-value of the point where a line crosses the y-axis
Slope-intercept form: The equation of a line in the form y 5 mx 1 b,
Gr. 7 AF 3.0: Students where m is the slope of the line and b is the y-intercept
graph and interpret
linear and some Getting Started In Lesson 5-15 you learned how to graph a linear
nonlinear functions. equation using a table of values. In this lesson you will learn another way to
graph a linear equation.
Gr. 7 AF 3.3: Graph
linear functions, noting
that the vertical change
(change in y-value)
per unit of horizontal
change (change in
x-value) is always the
same and know that the
ratio (“rise over run”)
is called the slope of
the graph.

E@)584- Finding the Slope and y-Intercept

Remember Find the slope and y-intercept of the line y 5 3x 1 5.

The slope m is the Solution
coefficient of x. Compare the equation to y 5 mx 1 b.

y 5 mx 1 b m is the slope; b is the y-intercept. Copyright © by McDougal Littell, a division of Houghton Mifflin Company.

y 5 3x 1 5 In this equation, m 5 3 and b 5 5.

ANSWER The slope m is 3. The y-intercept b is 5.

T :A <0 1; Fill in the missing information to solve the problem.

Remember 1. Find the slope and y-intercept of the line y 5 }34x 2 2.
Compare y 5 }43x 2 2 to y 5 mx 1 .
The equation
y 5 }34x 2 2 can be y 5 mx 1 m is the slope; is the y-intercept.
rewritten as
y 5 }34 x 1 (22). y 5 x 2 2 In the equation y 5 }34x – 2, m 5 , and b 5 .
The slope m is . The y-intercept b is .

Math Intervention

70 Book 5 Algebraic Thinking

Name ——————————————————————— Date ————————————

E@)584- Graphing a Line in Slope-Intercept Form

Graph the line y 5 4x 1 3. y
7 (1, 7)
Solution
6 run 5 1
;\MX Identify the slope and y-intercept. rise 5 4 5
Compare y 5 4x 1 3 to y 5 mx 1 b.
This shows you that the slope m 4
is 4 and the y-intercept b is 3. (0, 3)

;\MX Plot the y-intercept at (0, 3). 3

2

;\MX Use the slope to graph a second point. 1
slope 5 }rriusne 5 }14
From (0, 3) move 4 units up and 24 23 22 O 1 2 3 4 x
1 unit right to (1, 7 ).
Remember Graph this point. y
7
Write the slope as the 6
fraction }14 to identify 5
the rise and the run. 4
3
;\MX Draw the graph of the
line between the two points.

Don’t Forget You will need to move down from 24 23 22 O 1 2 3 4 x
the y-intercept if the slope is negative.

Copyright © by McDougal Littell, a division of Houghton Mifflin Company. T :A <0 1; Fill in the missing information.

2. Follow these steps to graph y 5 2}21x 2 1. . y
;\MX Identify the slope and 4

3
2

Rewrite y 5 2}21x 2 1 as y 5 2}21x 2 1. 1
Compare y 5 2}21x 24 23 22 O 1 2 3 4 x
to y 5 mx 1 b.
22

This shows you that the slope m is 23
and the y-intercept is . 24

;\MX Plot the y-intercept at ( , ).

;\MX Use the slope to graph a second point.

slope 5 _____ 5 }2 .
}run

From b move unit down and units right.

This takes you to the point ( , ). Graph this point.

;\MX Draw the graph of y 5 2}12x 2 1 between ( , ) and the
y-intercept.
Math Intervention
Book 5 Algebraic Thinking 71

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Summarize

Finding the Slope and y-Intercept

When an equation is in the form y 5 mx 1 b, the value of m is the
slope and the value of b is the y-intercept.

Graphing a Line Using the Slope and y-Intercept

To graph a line in slope-intercept form, first plot the y-intercept
on the y-axis. Then use the rise and the run to move from the
y-intercept to a second point on the line. Graph this point and draw
a line through that point and the y-intercept.

Practice

Find the slope and y-intercept of the line.

1. y 5 8x 1 2
Compare y 5 8x 1 2 to y = mx 1 .

y 5 mx 1 m is the slope; is the y-intercept.
y 5 x 2. In the equation y 5 8x 1 2, m 5 and b 5 .

The slope m is . The y-intercept b is .

2. y 5 }15x 2 3 __________________ 3. y 5 27x 2 4 ___________

4. y 5 2}23x 1 }21 _________________ 5. y 5 x 2 10___________ Copyright © by McDougal Littell, a division of Houghton Mifflin Company.

6. Explain how you would use the slope and y-intercept to graph
y 5 24x 1 3.

The y-intercept is , so graph ( , ).

The slope 5 }r_i_s_e 5 }1 , so from the y-intercept, move units
___________ then 3 units ___________ .

Draw a line through this point and ( , ).

Graph the line using the slope and y-intercept.

7. y 5 3x 1 1 y 8. y 5 }12x 2 2 y
4 4

3 3
2 2

1 1

24 23 22 O 1 2 3 4 x 24 23 22 O 1 2 3 4 x

22 22
23 23
24 24

Math Intervention

72 Book 5 Algebraic Thinking

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Graph the line using the slope and y-intercept.

9. y 5 2x 2 6 y 10. y 5 4x – 6 y
8 8
6 6
4 4
2 2

28 26 24 O 2 4 6 8 x 28 26 24 O 2 4 6 8 x

24 24
26 26
28 28

11. y 5 x y 12. y 5 x 1 3 y
13. y 5 }53x 1 2 8 14. y 5 2}23x 1 4 4
6 3
Copyright © by McDougal Littell, a division of Houghton Mifflin Company. 4 2
2 1

28 26 24 O 2 4 6 8 x 24 23 22 O 1 2 3 4 x

24 22
26 23
28 24

y y
4 4
3 3
2 2
1 1

24 23 22 O 1 2 3 4 x 24 23 22 O 1 2 3 4 x

22 22
23 23
24 24

,1,A7=/-<1<'

15. Fill in the missing words. In the equation y 5 2x 1 6, 2 is
the ____________ and 6 is the ____________. This equation is in
_______________________form.

16. Explain your reasoning. Do you think it would be easier to
graph the line y 5 }32x 1 1.4 using a table of values or the slope
and y-intercept? Explain.

__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________

Math Intervention 73
Book 5 Algebraic Thinking

Answer Key

Lesson 5-18, pp. 70–73

Try this:

1. b; b; b; }​34   ;​ +; }​43   ​; –2; }​43   ;​ –2
2. y -intercept; 1; 1; 21; 2} ​21 ;​ 21; 0;

21; rise; 21; 1; 2; 2; 22; 2; 22;

y
4
3
2
1

24 23 22 1 2 3 4x

22
23
24

Practice:

1. b; b; b; 8; 1; 8; 2; 8; 2
2. slope 5 }​15   ​, y-intercept 5 23
3. slope 5 27, y-intercept 5 24
4. slope 5 2​}32   ,​ y-intercept 5 }​21   ​
5. slope 5 1, y-intercept 5 210

6. 3; 0; 3; run; 24; 4; down; right; 0; 3

7. y

4
3
2
1

24 23 22 21 1 2 3 4x

23
24

8. y 4x
4
24 23 22 3
2
1

O 12

23
24

Answer Key

9. y
8
26 6
4
2

O 2 4 6 8x

28

10. y

8
6
4
2

28 26 24 O 2 4 6 8 x

24
26

11. y

8
6
4
2

28 26 24 22 2 4 6 8x

24
26
28

12. y
4
24 3
2
1

22 O 1 2 3 4 x

22
23
24

Answer Key

13. y

4
3

1 1 2 3 4x

23 22 O

22
23
24

14. y

3 1 2 3 4x
2
1

24 23 22 O

22
23
24

15. slope; y-intercept; slope-intercept
16. Sample answer: It would be.ctions and decimals.