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212 Unit 3 Linear Measures and Area

Advance PreparationFor Part 1, you will need enough 1-yard squares to lay along the length of one classroom wall. Make a transparency

of Math Masters, page 417 or draw a 10-by-10 grid on the board.

Teacher’s Reference Manual, Grades 1–3 pp. 78–84

Key Concepts and Skills• Use multiplication facts to find the areas

of rectangles. 

[Operations and Computation Goal 3]

• Draw and use arrays to find the areas

of rectangles. 

[Operations and Computation Goal 6]

• Count unit squares to determine the areas

of rectangles. 

[Measurement and Reference Frames Goal 2]

• Use arrays to write number models to find

the areas of rectangles. 

[Patterns, Functions, and Algebra Goal 2]

Key ActivitiesChildren discuss ways to find the area of a

room that is to be carpeted. They write

number models to show how to find areas

of rectangles.

Ongoing Assessment: Recognizing Student Achievement Use journal page 74. [Measurement and Reference Frames

Goal 2]

MaterialsMath Journal 1, p. 74

Student Reference Book, p. 156

Home Link 3�7

transparency of Math Masters, p. 417

(optional) � 1-yard squares of paper � tape

(optional) � yardstick (optional)

Simulating a Shopping TripMath Masters, p. 75

Student Reference Book, pp. 214,

216, and 217

play money

Children simulate a shopping trip to

practice calculating money.

Math Boxes 3�8Math Journal 1, p. 75

Children practice and maintain skills

through Math Box problems.

Home Link 3�8Math Masters, p. 74

Children practice and maintain skills

through Home Link activities.

READINESS

Modeling Area with a Geoboardgeoboard � rubber bands � pennies

Children use geoboards to find areas

of rectangles.

ENRICHMENTFinding the Areas of Squares and TrianglesMath Masters, pp. 76 and 418

square pattern blocks � ruler � scissors

Children find and compare the areas of two

triangles and a square.

EXTRA PRACTICE

Investigating Perimeter and AreaMath Masters, p. 416

Children draw different rectangles with areas

of 12 square centimeters and find the

perimeter of each.

Teaching the Lesson Ongoing Learning & Practice Differentiation Options

Number Models for Area

Objectives To guide children as they develop the concept of

area by measuring with identical squares; and to demonstrate

how to calculate the area of rectangles using number models.

a

�������

eToolkitePresentations Interactive Teacher’s

Lesson Guide

Algorithms Practice

EM FactsWorkshop Game™

AssessmentManagement

Family Letters

CurriculumFocal Points

Common Core State Standards

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Lesson 3�8 213

Area = length of base × height

rows × square yards

in each row = square yards

Links to the FutureThe activities in this lesson lay the foundation

for the formula for the area of a rectangle:

Area = length of base × height. Using the

formula to calculate the area of a rectangle is

a Grade 5 Goal.

Adjusting the Activity Have children tape the 1-yard squares together and roll them into a scroll.

The scroll can be useful if children want to find the areas of other large surfaces, such as the hall or various parts of the playground.

AUDITORY � KINESTHETIC � TACTILE � VISUAL

Getting Started

Math MessageSuppose you want to order carpet to cover the whole classroom floor. How would you find out how many square yards of carpet to buy? Be ready to talk about it.

Home Link 3�7 Follow-Up Children share their answers with a partner.

Mental Math and Reflexes Pose fact extensions like the following:

2 + 3 = 5 7 + 5 = 12 9 + 7 = 16

2 + 23 = 25 7 + 35 = 42 9 + 97 = 106

3 + 4 = 7 6 + 7 = 13 7 + 8 = 15

3 + 64 = 67 6 + 47 = 53 7 + 98 = 105

1 Teaching the Lesson

� Math Message Follow-Up WHOLE-CLASSDISCUSSION

To find out how many square yards of carpet to order, it is necessary to find the area of the floor. Discuss ways to do this. Someone may suggest tiling the entire floor with 1-yard squares. This might be difficult since furniture would get in the way.

� Using 1-Yard Squares to WHOLE-CLASS ACTIVITY

Estimate AreaTo find the area of the room with very little moving of furniture, try the following activity with the class:

Rules

1. Place 1-yard squares in a row along one wall of the room. Ask: How many squares are in the row?

2. Ask children to imagine a second row of 1-yard squares next to the first row. Ask: How many squares would be in the second row? How many squares are there in all in the 2 rows?

3. Point out that to find the area of the whole floor, you need to find out how many rows of 1-yard squares would cover the floor. Ask: How would you find the number of rows? Sample answers: You could place 1-yard squares next to each other along the adjacent wall: Each square represents 1 row. Or you could simply measure the adjacent wall with a yardstick: Each yard represents 1 row.

4. After children have figured out the number of rows, have them estimate the area of the floor. rows with square yards in each row gives a total of square yards.

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214 Unit 3 Linear Measures and Area

Measurement

Find the area of this rectangle.Each square is 1 square foot.• There are 4 rows of squares.• Each row has 10 squares.• So there are 4 × 10 squares,

or 40 squares in all.

The area is 40 square feet.

Some surfaces are too large to cover with squares. It would take too long to count a large number of squares.

To find the area of a rectangle, you do not need to count all of the squares that cover it. The example below shows a shortcut for finding the area.

Summary To find the area of a rectangle:

1. Count the number of rows.

2. Count the number of squares in each row.

3. Multiply: (number of rows) × (number of squares in each row)

Find the area of each rectangle.

3. Which area is larger, 1 square yard or 1 square meter?Check your answers on page 341.

1. 2.

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Student Reference Book, p. 156

Student Page

10 × 9 rectangle

10 rows9 square yards in each row

10 × 9 = 90Answer: The area is 90 square yards.

S

� Finding the Areas of WHOLE-CLASSDISCUSSION

Rectangles(Student Reference Book, p. 156; Math Masters, p. 417, optional)

Show children how to draw a 10-by-9 rectangle (10 squares down, 9 squares across) on the 10-by-10 grid you prepared for the lesson. Tell the class that this rectangle represents the floor of a room for which you want to order carpet. Shade a row of squares along the shorter side of the rectangle. Say that each square represents 1 square yard.

● How many squares are there in this row? 9 squares

● How many equal rows of squares are there inside the rectangle? 10 equal rows

● How many squares are there in all? 90 squares

Write 10 × 9 = 90, and tell the class that you can use this number model to represent the area of the rectangle. The number model uses the multiplication symbol.

Children who used Second Grade Everyday Mathematics have had experience with arrays, multiplication, and the multiplication symbol. Remind children that × is read as times or multiplied by.

● How much carpet should you order? At least 90 square yards

Repeat this routine with other rectangles. In discussing solutions, use such phrases as an x-by-y rectangle and x rows with y squares in each row, for a total of z squares. Summarize the activity by reviewing Student Reference Book, page 156 with the class.

� Finding the Area of WHOLE-CLASSDISCUSSION

Rectilinear Figures(Student Reference Book, pp. 156A and 156B;

Math Masters, p. 417)

Draw a rectilinear figure on the 10-by-10 grid, as shown below on the left. Explain to children that this figure represents an outdoor reading area that you want to cover with rubber tiles. Say that each tile measures 1 square yard and that each square on the grid represents 1 square yard.

A

B

NOTE Some children might be curious

about the symbols for the paper units

they are using or for other commonly

used units of area.

square foot sq ft ft2

square yard sq yd yd2

square inch sq in. in.2

square centimeter sq cm cm2

square meter sq m m2

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1. Draw a 5-by-7

rectangle.

Number of rows:

Squares in a row:

Area = square

units

Number model:

× =

2. Draw an 8-by-8

rectangle.

Number of rows:

Squares in a row:

Area = square

units

Number model:

× =

3. Draw a 3-by-9

rectangle.

Area = square

units

Number model:

× =

More Areas of RectanglesLESSON

3 �8

Date Time

Draw the rectangle on the grid. Then fill in the blanks.

4. Follow these steps to find the area of this shape.

1. Divide the shape into 2 rectangles.

2. Find the area of each rectangle.

3. Add the areas to find the total area of the shape.

Area of shape = square units

Sample answers:

5

7

35

5 7 3 9 27

8

8

64

8 8 6435

52

27

Sample answer:

� � �

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Math Journal 1, p. 74

Student Page

Name Date Time

1. List the items you are buying in the space below. You must buy at

least four items.

If you buy the same item 2 times, list it 2 times.

Item Sale Price

2. Estimate how many dollar bills you will need to pay for your items.

3. Give the clerk the dollar bills.

4. The clerk calculates the total cost. You owe $ .

5. The clerk calculates the change you should be getting. $

6. Record your change. Use Î, Â, Í, ‰.

LESSON

3�8 A Shopping Trip

Answers vary.

Use the Stationery Store Poster in your Student Reference Book, page 214.

7. Linda wants to buy a box of pens and a box of pencils. How much

will she save by buying them on sale?

Regular price Sale price Difference

pens $ $ Regular total $

pencils $ $ Sale total $

Total cost $ $ Amount saved $ 1.904.486.38

4.481.491.89

6.382.994.49

Try This

Math Masters, p. 75

Teaching Master

Lesson 3�8 215

Journal page 74 �Problems 1–3

Ask children to think about how they might find the number of rubber tiles needed to cover an outdoor reading area of this shape. Expect that some children may view the rectilinear figure as having the same area as the 8-by-9 rectangle surrounding it, and may suggest multiplying 8 times 9 to get the area. Point out that this strategy will not work because the shape is not a rectangle; it is not made up entirely of equal rows of squares. Explain that the figure can be partitioned into 2 nonoverlapping rectangles and model how to do it as shown. You may want to label one rectangle A and the other B or outline each rectangle with a different color. For Rectangle A, ask:

● How many square yards are there in one row? 9 square yards

● How many equal rows of square yards are there inside the rectangle? 4 equal rows

● Write 9 × 4 = 36 and remind children that this number model can be used to represent the area of Rectangle A.

In the same manner, determine the area of Rectangle B. 24 square yards

Next, ask children to calculate the area for the whole figure. 60 square yards. Have them share their strategies with the class. Sample answer: I added the area of Rectangle A and the area of Rectangle B together to find the area of the whole figure. Some children may suggest finding the area of the 9-by-8 rectangle and then subtracting the cutout area. 9 × 8 = 72; 3 × 4 = 12; 72 – 12 = 60. Praise this strategy, but keep the focus of the activity on partitioning the figure into rectangles.

● How many square-yard rubber tiles are needed to cover the entire outdoor reading area? 60 tiles

Summarize the activity by reading Student Reference Book, pages 156A and 156B with the class.

� Practicing Finding the Areas INDEPENDENTACTIVITY

of Rectangles(Math Journal 1, p. 74)

Algebraic Thinking Children write number models and find the areas of rectangles.

Ongoing Assessment:Recognizing Student Achievement

Use journal page 74, Problems 1 through 3 to assess children’s progress with

using strategies to calculate the area of rectangular shapes. Children are making

adequate progress if they are able to count the square units to determine the

area. Some children may use multiplication to determine the area.

[Measurement and Reference Frames Goal 2]

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216 Unit 3 Linear Measures and Area

Name Date Time

AreaHOME LINK

3�8

Today we discussed area as an array, or diagram. An array is a rectangular arrangement of objects in rows and columns. Help your child draw an array of the tomato plants in Problem 3. Use that diagram to find the total number of plants.

Please return this Home Link to school tomorrow.

FamilyNote

64 65

Mr. Li tiled his kitchen floor.

This is what the tiled floor looks like.

1. How many tiles did he use? tiles

2. Each tile cost $2. How much did all the tiles

cost? $

3. Mrs. Li planted tomato plants. She planted

5 rows with 6 plants in each row. Draw a diagram

of her tomato plants.

Hint: You can show each plant with a large dot or an X.

4. How many tomato plants are there in all?

plants

Write these problems on the back of this page. Fill in a unit box. Write a

number model for your ballpark estimate. Use any method you wish to solve

each problem. Show your work.

5. 548 � 59 �

6. � 616 � 57

7. � 571 � 264307

673

489

30

160.00

80

Sample answer:

Practice

Unit550 � 60 � 490

620 � 60 � 680

570 � 270 � 300

Math Masters, p. 74

Home Link Master

5. Describe 2 events that you are

certain will not happen today.

Answers vary.

3. When I left home, I had $4.00. I

spent $0.73 at the fruit stand and

$2.59 at the grocery store. How

much did I spend in all?

$3.32

How much do I have left?

$0.68

Date Time

2. Find the perimeter.

2 cm

2 cm

3 cm

5 cm Fill in the circle for the best answer.

A. 13 cm

B. 12 cm

C. 8 cm

D. 4 cm

4. John’s dad gave him 107 pennies.

Now John has 392 pennies. With

how many pennies did John start?

6. Fill in the empty frames.

Rule

�0 21 00

0 00

1. Circle the best unit of measurement.

distance to Spain

miles centimeters inches

width of a crayon

miles centimeters feet

length of your journal

miles yards inches

Math BoxesLESSON

3 �8

141 142 148 149

250–253

150 151

254 255

92 56200 201

Number model: 392 − 107 = ?

Answer: 285 pennies

75

Start EndChange

? 392107

or ? + 107 = 392

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Math Journal 1, p. 75

Student Page

2 Ongoing Learning & Practice

� Simulating a Shopping Trip PARTNER ACTIVITY

(Math Masters, p. 75; Student Reference Book, pp. 214, 216, and 217)

This activity was introduced in Lesson 1-11. To buy items on the shopping trip, children can use any of the following sources:

� the ad display suggested by Home Link 1-10

� the Stationery Store Poster on Student Reference Book, page 214

� either of the two Stock-Up Sale Posters in the Student Reference Book, pages 216 and 217

Working in partnerships, children first select the items they want to buy; record the information on Math Masters, page 75; and estimate the number of bills to give the shopkeeper. Then they take turns acting out their shopping trip. After each child completes the shopping trip, partners work together to complete the Try This problem.

� Math Boxes 3�8 INDEPENDENTACTIVITY

(Math Journal 1, p. 75)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 3-6. The skill in Problem 6 previews Unit 4 content.

� Home Link 3�8 INDEPENDENTACTIVITY

(Math Masters, p. 74)

Home Connection Children solve number stories involving equal rows of things. They also solve addition and subtraction problems by estimating and then finding an exact answer.

3 Differentiation Options

READINESS SMALL-GROUP ACTIVITY

� Modeling Area with 5–15 Min

a GeoboardTo provide concrete experience with area, have children use geoboards to model and calculate the area of a rectangle. Have them use a rubber band to make various rectangles on a geoboard. When counting the number of squares in a row or rows in the rectangle, have children place a penny in each of the squares.

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Name Date Time

1. Cut out the square.

2. Use square pattern blocks

to find the area

of the square.

3. Cut the square into

2 equal triangles.

4. Find the area

of each triangle.

Names:

1. Area of square: sq. in.

2. Area of first triangle: about sq. in.

3. Area of second triangle: about sq. in.

4. How do the areas of the two triangles compare?

They are about the same.5. How does the area of one triangle compare to the area of the

original square?

It is about half the area of the square.

8

8

16

LESSON

3 � 8 Finding and Comparing Areas

Math Masters, p. 76

Teaching Master

418

Name Date Time

One-Inch Grid

Math Masters, p. 418

Teaching Aid Master

Lesson 3�8 217

5 × 4 = 20

Example: 5-by-4 rectangle (See margin.)

● How many squares are there in a row? 4 squares

● How many equal rows of squares are there inside the rectangle? 5 equal rows

● How many squares are there in all? 20 squares

Discuss strategies for finding the total number of squares.

Ask children to describe their rectangles without showing them to the class. For example, “My rectangle has 4 squares in a row. There are 5 equal rows. The area of my rectangle is 20 squares.” After children have described a rectangle, ask the other children to make the same rectangle on their geoboards.

ENRICHMENT PARTNER ACTIVITY

� Finding the Areas of 15–30 Min

Squares and Triangles(Math Masters, pp. 76 and 418)

To further explore area concepts, have children find and compare the areas of triangles and squares. Partners cut a 4-inch square from Math Masters, page 76. Using square pattern blocks, inch grid paper (Math Masters, page 418), and rulers, partners work together to find the area of the square. Then they cut the square into 2 equal triangles and find the area of each triangle. They answer the questions on Math Masters, page 76.

EXTRA PRACTICE SMALL-GROUP ACTIVITY

� Investigating Perimeter and Area 15–30 Min(Math Masters, p. 416)

To provide additional experience with finding the areas and perimeters of rectangles, children draw as many different rectangles as they can with areas of 12 square centimeters. Distribute several copies of Centimeter Grid Paper (Math Masters, page 416) to each small group for children to draw on. Remind them to label their rectangles with the lengths of each side. Then have them find and record the perimeter of each rectangle. After children complete this task, bring the class together to go over their answers.

There are three possible rectangles with an area of 12 square centimeters whose sides have whole-number lengths: 1 cm × 12 cm, 3 cm × 4 cm, and 2 cm × 6 cm. The perimeters of these rectangles are 26 cm, 14 cm, and 16 cm. Some children may suggest other rectangles, such as 1 1 _ 2 cm × 8 cm. The perimeter of this rectangle is 19 cm.

Over the next few days, encourage children who show interest in this activity to draw as many rectangles as they can for other specified areas. You may want to make a bulletin board of the results and discuss patterns that children observe.

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