124

1. Plan

3-4

124 Chapter 3 Real Numbers and the Coordinate Plane

3-4

New Vocabulary coordinate plane, y-axis, x-axis, quadrants, origin,

ordered pair, x-coordinate, y-coordinate

To graph points and to use the Pythagorean Theorem to find distances

in the coordinate plane

What You’ll Learn

� � �5 3 1 3, , ,

� �6 4 2 9, , ,

� �10 8 0 6, , ,

� �5 2 4 7, , ,

Graphing in the Coordinate Plane

Why Learn This?Mapmakers use a coordinate grid system

for maps. The coordinate plane is another

type of grid system. You can use coordinate

planes to help you find distances, design

projects, and read building plans.

A coordinate plane is a grid formed by the

intersection of two number lines. You can use

a coordinate plane to locate and name points.

An ordered pair (x, y) gives the coordinates of the location of a point. In

the graph above, point A has coordinates

You can graph a point when you know its coordinates.

A

1 2 3 4 5

B

C

D

Miami

Fort Myers

Sarasota

Key West

The y-axis

is a vertical

number line.

The x-axis

is a horizontal

number line.

The axes divide

the plane into

four quadrants.

O indicates the

origin, where the

axes intersect.A

4

2

-2

-4

-4 -2 2 4

Quadrant

III

Quadrant

IV

Quadrant

II

Quadrant

I

y

O x

( ).2, 4�

( , � )

The

the number of

horizontal units a point

is from the origin.

The

tells the number of

vertical units a point

is from the origin.

2 4

x-coordinate tells y-coordinate

1. Vocabulary Review How can you tell whether two numbers on a number line are opposites?

Order the integers in each set from least to greatest.

2.

3.

4.

5.

Lesson 1-2

3, 5, 1, 3� � �

9 4 6, 2, ,� �

� �8, 6, 0, 10

� �2, 7, 5, 4

See below.

Check Skills You’ll Need

1. They are the same distance from zero on a number line but on opposite sides of zero.

Objective

To graph points and to use the Pythagorean Theorem to find distances in the coordinate plane

Examples

1

Graphing Points

2

Finding Distance on a Coordinate Plane

Math Understandings:

p. 104C

Math Background

The

coordinate plane

is a grid created by a horizontal number line and a vertical number line—the

x

-axis and

y

-axis respectively. These number lines intersect at the

origin,

designated by the ordered pair (0, 0). Ordered pairs, such as (0, 0) and are used to locate points in the coordinate plane. The first number in the pair is the

x

-coordinate and the second number is the

y

-coordinate. The

x

-coordinate describes a point’s location to the left or right of the origin. The

y

-coordinate describes a point’s location up or down from the origin.

More Math Background:

p. 104C

Lesson Planning and

Resources

See p. 104E for a list of the resources that support this lesson.

( , )3 2�

Bell Ringer Practice

Check Skills You’ll Need

Use student page, transparency, or PowerPoint. For intervention, direct students to:

Integers and Absolute Value

Lesson 1-2Extra Skills and Word Problems

Practice, Ch. 1

Special Needs

Students graph points with a partner. Pair students who need help graphing points with those who can graph without assistance. Those students needing assistance can read the points to a partner who can then graph them.

L1

learning style: tactile

Below Level

Make sure students can identify the

x-

and

y

-coordinates in ordered pairs. Have them say, “

x

comma

y

” or “

x

-coordinate,

y

-coordinate” or “

x

comes before

y

” to help them graph the following points:

Check students’ graphs.

L2

A(4, 3), B(�3, 2), D(�1, �2).

learning style: verbal

phm07c3_te_0304.fm Page 124 Thursday, May 25, 2006 3:41 PM

125

3-4 Graphing in the Coordinate Plane 125

Graphing Points

Graph point on a coordinate plane.

1. Graph and on the same coordinate plane.

You can use the Pythagorean Theorem to find distances in the

coordinate plane.

Finding Distance on a Coordinate Plane

Multiple Choice The library is 5 miles north of your house. The post

office is 6 miles east of your house. To the nearest mile, how far is the

library from the post office?

7 mi 8 mi 9 mi 10 mi

Graph the three locations on a coordinate

plane. Place your home at the origin. Notice

that you can draw a right triangle. The

x-coordinate and the y-coordinate are the

lengths of the legs of the right triangle.

The answer is B.

2. Your school is 3 miles south of your house. The general store is

5 miles east of your school. To the nearest mile, how far is your house

from the general store?

AQ R212

, 3�

y

x−2

−2

4O

A

Step 1Start atthe origin.

Step 3 Move 3 units down.

Step 4 Draw a dot. Label it A.

Step 2 Move 2 1

2 units to the right.

R( )4, 2� SQ R�412

, 2

y6

Library

Post

Office

Home 4 6 x

4

2

c � 8

61

�

7 .8 10249676

c2 61

d Use the Pythagorean Theorem.

d Substitute.

d Simplify.

d Add.

d Find the positive square root of each side.

d Use a calculator.

a2 � b2 � c2

52 � 62 � c2

25 � 36 � c2

61 � c2

� �

Test Prep TipTest Prep TipBe careful to follow the order of operations when solving for a variable in the Pythagorean Theorem.

See back of book.

6 mi

Activity Lab

Use before the lesson.

Teaching Resources

Activity Lab 3-4:

Similar Lines

Guided Instruction

Example 1

Have students say

right two and a half,

and

down three

as they graph point

A.

Saying

left

or

right

with each first coordinate and then

up

or

down

(depending on the sign) with the second coordinate will help students remember which way to move first.

Additional Examples

Graph on a coordinate plane.

The mall is 5 miles north of the library. The roller skating rink is 12 miles east of the library. To the nearest mile, how far is the mall from the roller skating rink?

13 miles

Teaching Resources

• Daily Notetaking Guide 3-4• Adapted Notetaking 3-4

Closure

•

Explain how an ordered pair locates a point in the coordinate plane.

Sample: The first coordinate in an ordered pair tells the number of horizontal units the point is from the origin. The second coordinate tells the number of vertical units a point is from the origin.

P( , )�3 122

x

y

�4

�1

�3

4

3

1

2

�2

431 2�2�4 �1�3

P

L3L1

Advanced Learners

A rectangle has opposite vertices at points with coordinates (4, 3) and What are the coordinates of the other vertices?

L4

( , ).� �4 3( , ), ,� �4 43 ( 3)

learning style: verbal

English Language Learners

In problems requiring the use of the Pythagorean Theorem, make sure students can identify and label the origin, the legs, and the hypotenuse as they would be drawn on a coordinate plane. Ask:

Which length are you trying to find? Is that a leg or the hypotenuse?

learning style: verbal

2. Teach

phm07c3_te_0304.fm Page 125 Thursday, May 25, 2006 3:41 PM

126

Assignment Guide

Check Your Understanding

Go over Exercises 1–10 in class before assigning the Homework Exercises.

Homework Exercises

A

Practice by Example 11–23

B

Apply Your Skills 24–30

C

Challenge 31Test Prep and

Mixed Review 32–38

Homework Quick Check

To check students’ understanding of key skills and concepts, go over Exercises 16, 23, 25, 27, and 30.

126 Chapter 3 Real Numbers and the Coordinate Plane

Check Your UnderstandingCheck Your Understanding

Vocabulary Match each ordered pair with the appropriate quadrant.

1.

2. (3, 5)

3.

4.

( , )�4 2

( , )12 6�

( , )� �7 1

A. Quadrant I

B. Quadrant II

C. Quadrant III

D. Quadrant IV

Name the coordinates of each point in the graph.

5. A 6. B

7. C 8. D

9. E 10. F

Homework ExercisesHomework ExercisesFor more exercises, see Extra Skills and Word Problems.

Graph each point on the same coordinate plane.

11. 12. B(3, 4) 13. C(5, 0) 14.

15. 16. 17. 18. H(6, 2)

Name the point with the given coordinates in the graph at the right.

19. 20.

21. 22.

23. Softball A softball diamond has the shape of a square. The distance

from home plate to second base is about 85 ft. Find the distance a

player would run going from first base to second base.

24. Guided Problem Solving Find the length of

the hypotenuse to the nearest tenth.

• The length of is units.

• The length of is units.

• Using the Pythagorean Theorem, the length

of is the square root of

25. a. Graph each of these points on a coordinate plane:

b. Connect the points in order and describe the figure formed.

-4

y

O

A

C

FD

E

B

x

2

-2

-2

2 4

A( , )4 5� D( , )0 3�

E( , )�5 1 F( , )� �2 4 G( , )�2 0

2

-2

-2 2 4

y

O x

M CR

PJA

( , )3 2� ( , )� �3 2

Q R334

, 0 ( )�2, 1.5

y

O x

C

A

B

3

1

-1 1 4

AC �

BC �

AB � �2 2� .

( ) ( ) ( ) ( ) ( )� � � � �2 5 3 1 1, 2 , , 3 , , 3 , , 0 , , 3 ,, , 3 , , 2 ,

, , , , ,

( ) ( )

( ) ( ) (

3 0

0 7 2 7 2

�

� � � � ��2).

For Exercises See Examples

11–22 1

23 2

B

A

D

C

( )�4 3, (4, 2)

( )� �5 2,

( )3 2, �

( )�3 0,

( )0 1, �

11–18. See back of book.

C M

AP

about 60 ft

7.1 units

25a–b. See margin.

Adapted Practice 3-4 L1

Name the coordinates of each point in the graph.

1. J 2. R

3. K 4. M

5. I 6. P

7. N 8. L

In which quadrant or on which axis is each point located?

9. (�3, �2) 10. (7, 0) 11. (4, 0) 12. (�3, �9)

13. (4, �7) 14. (7, �5) 15. (2, 9) 16. (�3, 2)

Practice 3-4 Graphing in the Coordinate Plane

17. Arnie plotted points on the graph below.He placed his pencil point at A. He canmove either right or down any wholenumber of units until he reaches point B.In how many ways can he do this?

18. Marika had to draw �ABC that fit several

requirements.

a. It must fit in the box shown.

b. The endpoints of have coordinates A(�2, 0) and B(2, 0).

c. Point C must be on the y-axis and its y-coordinate is an integer.

Name all the points that could be point C.

AB

(–5, 0) (–5, –9)

(–6, –3) (2, –2)

(0, 5) (–2, 7)

(5, 2) (2, 7)

P

Ox

y

6

4

2

�2

�4

�6

�6 �4 �2 2 4 6

L

FA

CN

B

MEK

D

R

H

GJ

I

10 ways

(0, 5), (0, 4), (0, 3), (0, 2), (0, 1),

(0, –1), (0, –2), (0, –3), (0, –4), (0, –5)

III x-axis x-axis III

IV IV I y-axis

x

y

•B

•A

x

y

2

–2–3

–1

1

3

1 2–1–2O

L3

3-4 • Guided Problem Solving

Student Page 126, Exercise 25:

a. Graph each of these points on a coordinate plane:(�2, �2), (�5, 3), (�3, 3), (�1, 0), (1, 3), (3, 3), (0, �2),(0, �7), (�2, �7), (�2, �2).

b. Connect the points in order and describe the figure formed.

Understand

1. How many points are you to plot?

2. Given an ordered pair, the x-coordinate is the number; the y-coordinate is the number.

3. What is it that you are asked to do after plotting the points?

Plan and Carry Out

4. What does the x-coordinate indicate?

5. What does the y-coordinate indicate?

6. Plot the points on the graph.

7. Which quadrant is the first point in?

8. Connect the points in order.

9. What shape is formed?

Check

10. Is every point plotted correctly to create the figure?

Solve Another Problem

11. Plot the following points on the grid at the right. Connect thepoints in order, connecting the last point to the first. What shapeis formed?(�3, �2), (0, 4), (3, �2), (�3, 2), (3, 2)

GPS

nine

firstsecond

third

the letter Y

Connect the points in order, and describe the

figure that is formed.

the number of horizontal units a point is from zero

the number of vertical

units a point is from zero

Yes, the letter Y is made.

y

x

87654321

�2�3�4�5�6�7�8

O–8 –6 –5–4 –3–2–7 1 2 3 4 5 6 7 8

Check students’ work.

y

x

5

4

3

2

1

�1

�2

�3

�4

�5

�5 �4 �3 �2 �1 1 2 3 4 5O

Check students’ work.

L3

3. Practice

25a–b. See back of book.

phm07c3_te_0304.fm Page 126 Thursday, May 25, 2006 3:41 PM

127

3-4 Graphing in the Coordinate Plane 127

26. On a graph, the points and are

connected in order to form a trapezoid. To the nearest tenth, what

is its perimeter?

27. Geography Degrees of longitude

and latitude indicate locations on

a map. The longitude of Chicago is

about and the latitude is

about Estimate the longitude

and latitude of St. Paul and Lincoln.

In which quadrant is each point located?

28. (x, y) if and 29. (x, y) if and

30. Writing in Math

Use coordinates to

write directions that will get the mouse

to the cheese in the maze at the right.

31. Challenge Graph and connect the

points

and (3, 2) in order. Then graph and

connect the points

and in order.

How are these two figures related?

Test Prep and Mixed ReviewTest Prep and Mixed Review Practice

32. What are the coordinates of point P at the right?

33. Oscar grew 3 inches each year for 5 years, until he

was 18 years old. What additional information is

necessary to find Oscar’s height at age 18?

the average height of an 18-year-old

how quickly he grew from age 0 to age 6

how tall Oscar was when he was 13

Oscar’s weight when he turned 18

34. Sarah walks across a rectangular field as shown.

Which is the closest to the distance she walks?

100 ft 70 ft

90 ft 50 ft

Find each square root. Where necessary, round to the nearest tenth.

35. 36. 37. 38.

( ) ( ) ( )4 7 9 5, 2 , , 2 , , ,� � � ( )2 5, �

88� W,

42� N.St. Paul

ChicagoLincoln

105W 100W50N

45N

40N

95W 90W 85W

x � 0 y � 0 x � 0 y � 0

1 3-1-3

3

-3

y

x

( , ) , , ,3 2 2 2 3 7, ( 2), ( 7), ( ),� �

( , ) ,3 2 2, ( 2),� � �

( 7), ( ),� � �2 3 7, , ( , )3 2�

y

x

2�2 O

�2

2P( . , )1 5 2� ( , )�2 1.5

( . , )�1 5 2 ( , )2 1.5�

40 ft

60 ft

path

50 � 16

7 0 18.

Multiple Choice

lesson quiz, PHSchool.com, Web Code: asa-0304

nline

Visit: PHSchool.comWeb Code: ase-0304

17.2 units

93 45 97 41� � � � W, N; W, N

Quadrant IV Quadrant I

See back of book.

See back of book.

C

H

C

7.1 �0 4. 2.6 0.4

For Exercises See Lesson

35–38 3-1

Lesson Quiz

1.

Graph the points and

C

(2, 1.5) on the same coordinate plane.

See graph below.

2.

Graph the points and

H

(6.5, 5) on the same coordinate plane.

See graph above.

3.

Find the length of the hypotenuse of

7.5 units

4.

On a soccer field, one goalpost is 25 yards west of a second goalpost. The gymnasium is 20 yards north of the second goalpost. How far is the gymnasium from the first goalpost?

32 yards

A B( , ), ( , ),2 3 12 �

y

�4

�1

4

3

6

5

1

2

�2

�3

�6

�5

4 3 x 6 5 1 2 �2 �4 �1 �3 �6 �5

B

F G

C

H

A

F G( , ), ( . , ),2 1 6 5 1� �

�FGH.

Alternative Assessment

Pairs of students draw a right triangle with horizontal and vertical legs, and each of its endpoints in different quadrants. They list the coordinates of the endpoints. They then find the length of the hypotenuse.

Test Prep

Resources

For additional practice with a variety of test item formats:• Test-Taking Strategies, p. 151• Test Prep, p. 155• Test-Taking Strategies with Transparencies

4. Assess & Reteach

Enrichment 3-4 L4

You can graph a point on a coordinate plane. Use an ordered pair(x, y) to record the coordinates. The first number in the pair is thex-coordinate. The second number is the y-coordinate.

To graph a point, start at the origin, O. Move horizontally accordingto the value of x. Move vertically according to the value of y.

Example 1: (4, �2) Example 2: (�3, 2)Start at O, move right 4, then down 2. Start at O, move left 3, then up 2.

Graph each ordered pair on the coordinate plane. Label each pointwith its letter. Then connect the points in order from A to S. Connectpoint S with point A to complete a picture.

A(7, 7) J(�4, �6)

B(6, 3) K(�7, �7)

C(6, 2) L(�5, 1)

D(5, 1) M(�6, 2)

E(7, �7) N(�6, 3)

F(4, �6) P(�7, 7)

G(1, �2) Q(�1, 3)

H(0, �4) R(0, 4)

I(�1, �2) S(1, 3)

Reteaching 3-4 Graphing in the Coordinate Plane

�2

4

2

y

y-axis

x-axis x

O 1 2 3 4

(4, �2)

�2�1�3�4

�4

4

2

y

xO1 2 3 4

(�3, 2)

�1�2�3�4

�2

�4

8

6

2

4

�2

�4

�6

�8

y

SR

Q

A

BNCM

DL

EKF

GI

HJ

P

xO1 2 3 4 5 6 7 8�1�6�7�8 �2�3�4�5

L2

phm07c3_te_0304.fm Page 127 Thursday, May 25, 2006 3:41 PM