skill are you ready? 76 slopes of parallel and ... · objective determine when a pair of lines is...

Copyright © by Holt McDougal. 163 Holt McDougal Algebra 1All rights reserved.

Teaching Skill 76Objective Determine when a pair of lines is

parallel or perpendicular.

Remind students that the slope of a line determines how steep or how flat the line is. Draw a pair of parallel lines on the board. Ask: What would you say is true about the steepness of the two lines? (The steepness is the same.) Explain that this is true for every pair of parallel lines; the steepness is the same so the slopes are equal.

Work through the first example with students.

Next, draw a pair of perpendicular lines on the board and explain that the slopes are negative reciprocals. Give several examples of negative reciprocals. Tell students that it is easy to find the slope of a line written in slope-intercept form, so if the line is written in any other form, rewrite it in slope-intercept form. Then work through the second example with students.

PRACTICE ON YOUR OWNIn exercises 1–9, students determine whether a pair of lines is parallel or perpendicular.

CHECKDetermine that students know how to determine whether a pair of lines is parallel or perpendicular.

Students who successfully complete the Practice on Your Own and Check are ready to move on to the next skill.

COMMON ERRORSStudents may incorrectly identify two lines as being perpendicular when their slopes are reciprocals, rather than negative reciprocals.

Students who made more than 2 errors in the Practice on Your Own, or who were not successful in the Check section, may benefit from the Alternative Teaching Strategy.

Alternative Teaching StrategyObjective Graph a pair of lines to determine

whether they are parallel, perpendicular, or neither.

Some students may benefit from visualizing why equal slopes produce parallel lines.

Present the following equations: y � 3x � 4

y � 3x � 1

Have students graph the two lines on the same coordinate plane using the y-intercept and the slope. Remind students that a slope of 3 means “rise 3 and run 1” and that the y-intercept indicates where to place the beginning point. Students’ graphs should look like the following:

y

x

–2

–4

2

4

2 4–4 –2

Ask: What kind of lines are these? (parallel) Have students repeat the exercise for several sets of parallel lines. When complete, ask: What is true about every pair of parallel lines? (They have equal slopes.)

Next, have students graph the following lines on the same coordinate plane:

y � � 2 __ 3 x � 4 and y � 3 __ 2 x � 5. Point out

that the slopes of the two lines are negative reciprocals. Ask: What kind of lines are these? (perpendicular lines) Have students graph several sets of perpendicular lines to arrive at the conclusion that lines with negative reciprocal slopes are perpendicular.

Finally, have students graph a pair of lines that is neither parallel nor perpendicular. Explain that these lines are intersecting lines, but not perpendicular.

Are You Ready?Slopes of Parallel and Perpendicular Lines76

SKILL


Name Date Class

Parallel Lines Perpendicular Lines

If two lines are parallel, they have equal slopes. So, if two lines have equal slopes, they are parallel.

If two lines are perpendicular, they have slopes that are negative reciprocals. So, if two lines have slopes that are negative reciprocals, they are perpendicular.

y � 7x � 5Example 1: y � 7x � 9

Remember, when an equation is written in the form y � mx � b, the coefficient of x is the slope.

Since the slope of both lines is 7, the lines are parallel.

y � 3x � 5Example 2: x � 3y � 8 First, rewrite the second equation in slope-intercept

form: y � � 1 __ 3 x � 8 __ 3 . The slope of the first line is 3 and

the slope of the second line is � 1 __ 3 . Since 3 and � 1 __ 3 are

negative reciprocals, the lines are perpendicular.

Practice on Your OwnState whether the linear equations in each pair are parallel, perpendicular, or neither.

1.

y � 6x � 3 2.

y � 3x � 2 3.

8x � 2y � 3

y � � 1 __ 6 x � 7 2y � 6x � 6 x � 4y � �1

4.

3x � 2y � 5 5.

y � 5 � 6x 6.

y � 3x � 9

3y � 2x � �3 y � 6x � �1 y � 1 __ 3 x � 4

7.

y � x � 3 8.

y � 6 9.

3y � �x

y � �x � 5 x � �2 3x � y

CheckState whether the linear equations in each pair are parallel, perpendicular, or neither.

10. y � 5 � 7x

11. 2x � y � 5

12. x � 1 __ 3 y � 1

y � � 1 __ 7 x � 2 2y � �4x � 3 2y � 6x

13. y � 2

14. y � 1 __ 4 x � 3

15. x � 2y � 0

y � 7 � 0 2y � 8x � 1 y � 1 � �2x

Are You Ready?Slopes of Parallel and Perpendicular Lines76

SKILL


Teaching Skill 78Objective Generate ordered pairs for given

values of a function.

Explain to students that function tables are used to generate ordered pairs in the form (x, y). These ordered pairs can be used to graph a function.

Review with students how to evaluate a function for a given value of x. Remind students that they should be particularly careful when substituting negative values into a function.

Direct students’ attention to the example. Point out that the x-values are called inputs. Ask: In this example, what are the inputs? (�2, �1, 0, 1 and 2) Explain that the inputs are substituted into the function, and the y-values that result are called the outputs. The ordered pairs are the points (x, y).

Work through the example, making sure students understand how the function is being applied to the input values to generate the outputs. Then have students complete the practice exercises.

PRACTICE ON YOUR OWNIn exercises 1–8, students generate ordered pairs by completing function tables for given inputs.

CHECKDetermine that students know how to complete function tables.

Students who successfully complete the Practice on Your Own and Check are ready to move on to the next skill.

COMMON ERRORSStudents may not perform the operations of a function’s rule correctly and may arrive at an incorrect output for a given input.

Students who made more than 2 errors in the Practice on Your Own, or who were not successful in the Check section, may benefit from the Alternative Teaching Strategy.

Alternative Teaching StrategyObjective Generate ordered pairs for given

values of a function.

Some students may need additional practice evaluating functions.

Remind students that when you evaluate a function, you apply a rule to an input value to generate an output. Understanding the rule that is being applied is critical to completing a function table.

Write the following function table on the board and have students copy it on their paper.

x y � y

�2 y � 3( ) � 1 � � 1 �

�1 y � 3( ) � 1 � � 1 �

0 y � 3( ) � 1 � � 1 �

1 y � 3( ) � 1 � � 1 �

2 y � 3( ) � 1 � � 1 �

Have students consider the “rule” that is being applied to the input values, �2, �1, 0, 1, and 2. Have a volunteer try to write an algebraic expression that represents the rule. Guide the volunteer if needed. (y � 3x � 1) Then have students write the function in the center column of the table and complete the function table.

Have students work in pairs for the next part of the exercise. Instruct each student to write down a function (rule) in the form of an algebraic expression (e.g. y � 2x � 6), but not show their partner. Each student should then create a function table similar to the one above for their function. They should use boxes, rather than variables, as above. When they have created their tables, they should exchange papers, try to figure out their partner’s rule, and then complete the table. Repeat the exercise until you feel comfortable that students understand how to complete function tables.

Are You Ready?Function Tables78

SKILL

A function table is a set of solutions to an equation. When given a set of x-values, substitute the values into the equation to find the corresponding y-values. Use the x- and y-values to generate ordered pairs.

Example: Generate ordered pairs for the function y � 5x � 2 for x � �2, �1, 0, 1, 2.

x y � 5x � 2 y

�2�1012

y � 5(�2) � 2 � �10 � 2 � �8y � 5(�1) � 2 � �5 � 2 � � 3y � 5(0) � 2 � 0 � 2 � 2y � 5(1) � 2 � 5 � 2 � 7y � 5(2) � 2 � 10 � 2 � 12

�8�32712

x y

�2�1012

�8�327

12

Practice on Your OwnGenerate ordered pairs for each function for x � �2, �1, 0, 1, 2.

1. y � 3x � 1 2. y � �4x � 5 3. y � x 2 4. y � x 2 � 3

x y

�2�1012

x y

�2�1012

x y

�2�1012

x y

�2�1012

5. y � 1 __ 2 x � 3 __ 2 6. y � � 1 __ 2 x � 4 7. y � (x � 2 ) 2 8. y � (x � 2 ) 2

x y

�2�1012

x y

�2�1012

x y

�2�1012

x y

�2�1012

CheckGenerate ordered pairs for each function for x � �2, �1, 0, 1, 2.

9. y � 4x � 3 10. y � �3x � 10 11. y � 1 __ 2 x � 1 __ 2 12. y � (x � 3 ) 2

x y

�2�1012

x y

�2�1012

x y

�2�1012

x y

�2�1012


Name Date Class

Are You Ready?Function Tables78

SKILL


Name Date Class

A perfect number is a number which is the sum of its own positive factors (other than itself). For example, the following numbers are perfect.

6 � 1 � 2 � 328 � 1 � 2 � 4 � 7 � 14496 � 1 � 2 � 4 � 8 � 16 � 31 � 62 � 124 � 248

The next perfect number is 8128. What is the fifth perfect number?

To find the answer, solve each equation and write your answer in the blanks provided at the bottom of the page.

1. 7x � 12 � 9 2. x __ 3 � 4 � �3

3. 25 � 3x � 10 4. �2 � x __ 5 � 3

5. 4(2x � 1) � �4 6. 7(2 � x) � �7

7. 25x � 10x � 40 � 5 8. 15 � x __ 6 � 14

1.

2.

,

3.

4.

5.

,

6.

7.

8.

EnrichmentPerfect Numbers

CHAPTER

5

SKILL 76 ANSWERS:

Practice on Your Own

1. perpendicular

2. parallel

3. perpendicular

4. neither

5. parallel

6. neither

7. perpendicular

8. perpendicular

9. perpendicular

Check

10. perpendicular

11. parallel

12. parallel

13. parallel

14. neither

15. perpendicular

SKILL 77 ANSWERS:


1. x � 10

2. x � 8

3. x � 80

4. x � 20

5. x � 50

6. x � 30

7. x � 7 __ 3

8. x � 10

9. x � 8 __ 9

10. x � 1 __ 3

11. x � 36

12. x � 12

Check

13. x � 21

14. x � 10

15. x � 26

16. x � 12 ___ 11

17. x � 8

18. x � 15 ___ 2

19. x � 4 __ 5

20. x � 10

SKILL 78 ANSWERS:


1. x y

�2�1012

�5�2147

2. x y

�2�1012

13951

�3

3. x y

�2�1012

41014

4.

x y

�2�1012

1�2�3�21

5. x y

�2�1012

0.51

1.52

2.5

6. x y

�2�1012

54.54

3.53

7. x y

�2�1012

014916

8. x y

�2�1012

169410


Answer Key continued

Check

9. x y

�2�1012

�5�13711

10. x y

�2�1012

�4�7

�10�13�16

11. x y

�2�1012

�1.5�1

�0.50

0.5

12. x y

�2�1012

2516941

SKILL 79 ANSWERS:

Practice on Your OwnNumbers 1–12 are plotted by letters.

A

F

JB

E

D

K

CG

M

H

L

x

–10

–5

5

10

–5–10 5 10

y

CheckNumbers 13–18 are plotted by letters.

A

DC

E

F B

y

x

–5

5

–5 5

SKILL 80 ANSWERS:


1.

2.

3.


Answer Key continued


Chapter 1: All Kinds of Numbers

T2

W

E

N

T

Y

H I

H

T1

W O

R

D

R T3

H

O

U N D R E D T H SH7

T

E

E V E N

E

E

F

IF6

F T H S

8

E

V

E

N

T

H

SHT14

H

R

E17

I G H T

Y

X

I

S15

E

NE

N

T

HTRUOF16

L

A

H12

T11

T13

F

S9

N

U

H

T4 5

S

A

N

D

TROF10

I

F

T

Y

Y

H

Y

Chapter 2: Order Up!

1. 4(3 � 1) � 6 � 2 � 11

2. (1 � 4 ) 2 � 5 � 6 � 3 � 8

3. (10 � 6 ) 2 � 12 � 4 � 8 � 11

4. 8 � 6 � 4 � 10 � 2 � 12 � 5

5. 5 � 1 � 4 (6 � 3) � 2 � 10

6. 5(6 � 1 � 4 � 2 3 ) � 50

Chapter 3: Magic Square

8 –5 –6 5

–3 2 3 0

1 –2 –1 4

–4 7 6 –7

The magic sum is 2.

Chapter 4: Coordination

Greatest Common Factor

Chapter 5: Perfect Numbers

33,550,336

Chapter 6: Multiply It Out

1. �6, T

2. �1, I

3. 11, E

4. 3, R

5. 10, U

6. 1, S

7. 9, Y

8. �8, B

9. �7, O

10. 8, D

11. �11, P

12. 0, V

Distributive Property

Chapter 7: Fraction, Decimal, and Percent Mania

Boy do I have problems!

Chapter 8: The Power of Pi

3.14159265358979323846

1. 81

2. 64

3. 41

4. 625

5. 49

6. 512

7. 256

8. 75

Enrichment Answer Key

skill are you ready? 76 slopes of parallel and ... · objective determine when a pair of lines is...

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