formula for the area of a parallelogram - everyday math - … · 2012-05-04 · • describe...

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Common Core State Standards

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Lesson 8�6 687

Advance PreparationFor Part 1, each student needs 2 short straws, 2 long straws, and 4 twist-ties. Pairs of straws should be

the same length. Place them near the Math Message.

Teacher’s Reference Manual, Grades 4–6 pp. 180–185, 221, 222

Key Concepts and Skills• Find the area of a rectangle.

[Measurement and Reference Frames Goal 2]

• Develop a formula for calculating the area

of a parallelogram.


• Calculate perimeter.


• Identify perpendicular line segments and

right angles.

[Geometry Goal 1]

• Describe properties of parallelograms.

[Geometry Goal 2]

Key ActivitiesStudents construct models of parallelograms

and use them to review properties of

parallelograms.

Students cut apart and rearrange

parallelogram shapes; they develop and use

a formula for the area of a parallelogram.

Ongoing Assessment: Informing Instruction See page 690.

Key Vocabularybase � height � perpendicular

MaterialsMath Journal 2, pp. 236–238

Study Link 8�5

Math Masters, p. 260

centimeter ruler � straws and twist-ties �

scissors � tape � index card or other

square-cor ner object � slate

Playing Fraction OfStudent Reference Book, pp. 244

and 245

Fraction Of Cards (Math Masters,

pp. 477, 478, and 480)


Students practice finding fractions

of collections.

Playing Angle Add-Up Math Masters pp. 507–509

per partnership: 4 of each of number

cards 1–8 and 1 of each of number

cards 0 and 9 (from the Everything

Math Deck, if available) � full-circle

protractor (transparency of Math

Masters, p. 439) � dry-erase markers �

straightedge

Students draw angles and then use

addition and subtraction to find the

measures of unknown angles.

Math Boxes 8�6Math Journal 2, p. 239

Students practice and maintain skills

through Math Box problems.

Ongoing Assessment: Recognizing Student Achievement Use Math Boxes, Problem 4. [Operations and Computation Goal 5]

Study Link 8�6Math Masters, pp. 261 and 262

Students practice and maintain skills

through Study Link activities.

ENRICHMENTConstructing Figures with a Compass and StraightedgeStudent Reference Book, pp. 114, 117,

and 118

compass � straightedge

Students construct figures with a compass

and straightedge.

ENRICHMENTSolving Area and Perimeter ProblemsMath Masters, pp. 263, 264, and 437

scissors � tape

Students explore ways of combining various

two-dimensional shapes to form new shapes.

Teaching the Lesson Ongoing Learning & Practice

132

4

132

4

Differentiation Options

Formula for the Areaof a Parallelogram

Objectives To review the properties of parallelograms; and

to guide the development and use of a formula for the area

of a parallelogram.

t

��

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688 Unit 8 Perimeter and Area

NOTE Height is the distance perpendicular

to the base of a figure. Any side of a

parallelogram can be the base. The choice

of the base determines the height.

heightheight

base

base

1 Teaching the Lesson

� Math Message Follow-Up WHOLE-CLASS ACTIVITY

Ask students to tell what they know about parallelograms, using their straw constructions as models, while you list the properties they name on the board. The list should include:

� A parallelogram is a four-sided polygon called a quadrangle or quadrilateral.

� Opposite sides of a parallelogram are parallel.

� Opposite sides of a parallelogram are the same length.

� Rectangles and squares are special kinds of parallelograms.

Have students form a rectangle with their straw constructions, and then ask them to pull gently on the opposite corners. They should get a parallelogram that is not a rectangle. Ask the following questions:

● Does the perimeter remain the same? yes

● Does the area remain the same? No, because although the length of the base stays the same, the height decreases, so the area decreases.

Draw a parallelogram on the board. Choose one of the sides, for example, the side on which the parallelogram “sits,” and label it the base. Explain that base is also used to mean the length of the base.

The shortest distance between the base and the side opposite the base is called the height of the parallelogram. Draw and label a dashed line to show the height. Include a right-angle symbol. Point out that the dashed line can be drawn anywhere between the two sides as long as it is perpendicular to (forms a right angle with) the base.

Remind students that rectangles are parallelograms whose sides form right angles. If you think of one side of a rectangle as its base, then the length of an adjacent side is its height.

Getting Started

Math MessageTake 2 short straws, 2 long straws, and 4 twist-ties. Use them to construct a parallelogram.

Study Link 8�5 Follow-Up Have partners compare answers and discuss how they found the missing side measure in Problems 5 and 6.

Mental Math and ReflexesPose multiplication facts and problems. Suggestions:

3 ∗ 7 = 21

4 ∗ 9 = 36

8 ∗ 5 = 40

9 ∗ 6 = 54

90 ∗ 8 = 720

10 ∗ 90 = 900

60 ∗ 70 = 4,200

80 ∗ 30 = 2,400

8 ∗ 52 = 416

4 ∗ 63 = 252

9 ∗ 76 = 684

88 ∗ 5 = 440

688-692_EMCS_T_TLG1_U08_L06_576906.indd 688688-692_EMCS_T_TLG1_U08_L06_576906.indd 688 2/2/11 10:28 AM2/2/11 10:28 AM

Areas of ParallelogramsLESSON

8�6

Date Time

135

1. Cut out Parallelogram A on Math Masters, page 260.

DO NOT CUT OUT THE ONE BELOW. Cut it into

2 pieces so that it can be made into a rectangle.

Parallelogram A Tape your rectangle in the space below.

base = 6 cm length of base = 6 cm

height = 2 cm width (height) = 2 cm

Area of parallelogram = 12 cm2 Area of rectangle = 12 cm2

2. Do the same with Parallelogram B on Math Masters, page 260.

Parallelogram B Tape your rectangle in the space below.




1 cm2

Sample answer:

Sample answer:

219-247_EMCS_S_MJ2_G4_U08_576426.indd 236 2/1/11 1:47 PM

Math Journal 2, p. 236

Student Page

3. Do the same with Parallelogram C.

Parallelogram C Tape your rectangle in the space below.




4. Do the same with Parallelogram D.

Parallelogram D Tape your rectangle in the space below.




5. Write a formula for the area of a parallelogram.

A = b ∗ h

Areas of Parallelograms continuedLESSON

8�6

Date Time

Sample answer:

Sample answer:

base

he

igh

t



Student Page

Lesson 8�6 689

Links to the Future

Tell students that in this lesson they will use the formula for the area of a rectangle to develop a formula for the area of a parallelogram.

The use of a formula to calculate the area of a parallelogram is a Grade 5 Goal.

� Developing a Formula for WHOLE-CLASS ACTIVITY

the Area of a Parallelogram(Math Journal 2, pp. 236 and 237;

Math Masters, p. 260)

Point out that Parallelogram A on journal page 236 is the same as Parallelogram A on Math Masters, page 260.

Guide students through the following activity:

1. Cut out Parallelogram A from the master.

2. Cut the parallelogram into two pieces along one of the vertical grid lines.

3. Tape the pieces together to form a rectangle.

cut

4. Tape this rectangle in the space next to the parallelogram in the journal.

Discuss the relationship between the parallelogram and the rectangle formed from the parallelogram.

● Why must the parallelogram and the rectangle both have the same area? The rectangle was constructed from the parallelogram. Nothing was lost or added.

5. Record the dimensions and area of the parallelogram and the rectangle. Length of base of parallelogram and length of base of rectangle = 6 cm; height of parallelogram and width (height) of rectangle = 2 cm; area of each figure = 12 cm2

Have students repeat these steps with Parallelograms B, C, and D, working on their own or with a partner.

Bring students together to develop a formula for the area of a parallelogram. These are three possible lines of reasoning:

� The area of each parallelogram is the same as the area of the rectangle that was made from it.

� The area of the rectangle is equal to the length of its base times its width (also called the height).

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Areas of Parallelograms continuedLESSON

8 � 6

Date Time

6. Draw a line segment to show the height of Parallelogram DORA.

Use your ruler to measure the base and height.

Then find the area.

base � cm

height � cm

Area � cm2

7. Draw the following shapes on the grid below:

a. A rectangle whose area is 12 square centimeters

b. A parallelogram, not a rectangle, whose area is 12 square centimeters

c. A different parallelogram whose area is also 12 square centimeters

8. What is the area of:

a. Parallelogram ABCD? b. Trapezoid EBCD? c. Triangle ABE ?

cm2 cm2 cm261824

2045

D A

O R

Sample answers:a.

b.

c.

A

B C

E D


Student Page

2 cm

6 cm

abou

t 2.2

cm

Perimeter = about 16.4 cm

Area = 12 cm2

2 cm

6 cm

2 cm

Perimeter = 16 cm

Area = 12 cm2

� The length of the base of the parallelogram is equal to the length of the base of the rectangle. The height of that parallelogram is equal to the width (height) of that rectangle. Therefore, the area of the parallelogram is equal to the length of its base times its height. Using variables:

A = b ∗ h

where b is the length of the base and h is the height.

Have students record the formula at the bottom of journal page 237.

Ongoing Assessment: Informing Instruction

Watch for students who think that the perimeter of each parallelogram and

rectangle pair is also the same. Point out that although the height and base are

the same measure, the height of a parallelogram is only used in computing

its perimeter when the parallelogram is a rectangle or square. (See margin.)

� Solving Area Problems PARTNER ACTIVITY

(Math Journal 2, p. 238)

Algebraic Thinking Work with the whole class on Problem 6, journal page 238. Students can place an index card (or other square-corner object) on top of the shape, align the bottom edge of the card with the base, and then use the edge of the card to draw a line for the height. They will need a centimeter ruler to measure the length of the base and the height.

index card

heig

ht

Drawing the height of a parallelogram

Have partnerships complete Problems 7 and 8.

� Problem 7 illustrates the fact that shapes that do not look the same can have the same area.

� Problem 8b lends itself to a variety of solution strategies. Some students may have partitioned the trapezoid into a rectangle flanked by two triangles. The rectangle covers 12 grid squares. If one triangle were cut apart and placed next to the other triangle to form a rectangle, the pair would cover 6 squares. The rectangle and two triangles cover 12 + 6 = 18 cm2.

E D

CBProblem 8b


Math Boxes LESSON

8�6

Date Time

4. Add or subtract.

a. 3 _ 16

+ 7 _ 16

=

b. 2 _ 3 + 1 _

6 =

c. = 9 _ 10

- 3 _ 10

d. = 3 _ 4 - 3 _

8

3

_ 8

6

_ 10 , or 3

_ 5

5

_ 6

10

_ 16 , or

5

_ 8

1. Dimensions for actual rectangles are

given. Make scale drawings of each

rectangle described below.

Scale: 1 cm represents 20 meters.

a. Length of rectangle: 80 meters

Width of rectangle: 30 meters

b. Length of rectangle: 90 meters

Width of rectangle: 50 meters

2. What is the area of the parallelogram?

Number model: 7 ∗ 3 = 21

Area = 21 in2

3. A jar contains

8 blue blocks,

4 red blocks,

9 orange blocks, and

4 green blocks.

You put your hand in the jar and without

looking pull out a block. About what

fraction of the time would you expect to

get a blue block?

8

_ 25

5. Multiply. Use a paper-and-pencil algorithm.

6,142 = 83 ∗ 74

145

135 45

55 57 18 19

a.

b.

3"

7"

�



Student Page

Name Date Time

STUDY LINK

8�6 Areas of Parallelograms

Find the area of each parallelogram.

1. 2.

Number model: Number model:

Area = square feet Area = square centimeters

3. 4.

Number model: Number model:

Area = square feet Area = square centimeters

6 ft

4 ft 65 cm

72 c

m

Try This

The area of each parallelogram is given. Find the length of the base.

5. 6.

Area = 26 square inches Area = 5,015 square meters

base = inches base = meters

8 cm

3 cm

2 in.

?

59 m

?

135

9'

4'

8 ∗ 3 = 24

65 ∗ 72 = 4,680

4 ∗ 9 = 36

6 ∗ 4 = 24

24

4,680

85

36

24

13

247-277_EMCS_B_MM_G4_U08_576965.indd 261 2/1/11 2:17 PM


Study Link Master

Lesson 8�6 691

� Problem 8c can be solved without using a formula for the area of a triangle. The parallelogram area minus the trapezoid area is the triangle area. 24 - 18 = 6 cm2

2 Ongoing Learning & Practice

� Playing Fraction Of PARTNER ACTIVITY

(Student Reference Book, pp. 244 and 245; Math Masters, pp. 477–480)

Students play Fraction Of to practice finding fractions of collections. See Lesson 7-3 for additional information.

� Playing Angle Add-Up PARTNER ACTIVITY

(Math Masters, pp. 439 and 507–509)

To further explore the idea that angle measures are additive, have students play Angle Add-Up. See Lesson 7-9 for more information.

� Math Boxes 8�6 INDEPENDENTACTIVITY

(Math Journal 2, p. 239)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 8-8. The skill in Problem 5 previews Unit 9 content.

Ongoing Assessment: Math Boxes

Problem 4 �Recognizing Student Achievement

Use Math Boxes, Problem 4 to assess students’ ability to solve fraction addition

and subtraction problems. Students are making adequate progress if they are

able to solve Problems 4a and 4c, which involve fractions with like denominators.

Some students may be able to solve Problems 4b and 4d by using equivalent

fractions with like denominators, using manipulatives, or drawing pictures.

[Operations and Computation Goal 5]

� Study Link 8�6 INDEPENDENTACTIVITY

(Math Masters, pp. 261 and 262)

Home Connection Students calculate the areas of parallelograms on Math Masters, page 261.

NOTE Math Masters, page 262 should be completed before Lesson 9-1,

in which students share and discuss examples of percents they have collected.

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LESSON

8�6 Perimeter and Area

263

150


Teaching Master

LESSON

8�6

Name Date Time

Perimeter and Area continued

131133–136

Cut out and use only the shapes in the top half of Math Masters,

page 263 to complete Problems 1–5.

1. Make a square out of 4 of the shapes. Draw the square on the centimeter dot

grid on Math Masters, page 437. Your picture should show how you put the

square together.

2. Make a triangle out of 3 of the shapes. One of the shapes should be the

shape you did not use to make the square in Problem 1. Draw the triangle

on Math Masters, page 437.

3. Find the area of the following:

a. the small triangle cm2

b. the square cm2

c. the parallelogram cm2

4. a. What is the perimeter of the large square

you made in Problem 1? cm

b. What is the area of that square? cm2

5. What is the area of the large triangle you

made in Problem 2? cm2

6. Cut out the 5 shapes in the bottom half of Math Masters, page 263 and add

them to the other shapes. Use at least 6 pieces each to make the following

shapes.

a. a square b. a rectangle

c. a trapezoid d. any shape you choose

Tape your favorite shape onto the back of this sheet. Next to the shape, write

its perimeter and area.

32

64

32

16

16

8

Try This

Answers vary.


Teaching Master

3 Differentiation Options

ENRICHMENT INDEPENDENTACTIVITY

� Constructing Figures with 30+ Min

a Compass and Straightedge(Student Reference Book, pp. 114, 117, and 118)

To apply students’ understanding of the properties of parallelograms, have them construct parallelograms and perpendicular line segments as described on pages 114, 117, and 118 of the Student Reference Book.

ENRICHMENT PARTNER ACTIVITY

� Solving Area and 30+ Min

Perimeter Problems(Math Masters, pp. 263, 264, and 437)

To apply students’ understanding of area and perimeter, have them explore different ways of combining various 2-dimensional shapes to form new shapes.

Possible solutions to Problem 6:

Name Date Time

Dot Paper

pyg

gp


Teaching Aid Master


Name Date Time

507

Copyright

© W

right

Gro

up/M

cG

raw

-Hill

132

4Angle Add-Up

Materials □ number cards 1–8 (4 of each)

□ number cards 0 and 9 (1 of each)

□ dry-erase marker

□ straightedge

□ full-circle protractor (transparency of Math Masters, p. 439)

□ Angle Add-Up Record Sheet (Math Masters, p. 509)

Players 2

Skills � Drawing angles of a given measure

� Recognizing angle measures as additive

� Solving addition and subtraction problems to find the measures

of unknown angles

Objective To score the most points in 3 rounds.

Directions

1. Shuffle the cards and place the deck number-side down on the table.

2. In each round, each player draws the number of cards indicated

on the Record Sheet.

3. Each player uses the number cards to fill in the blanks and form

angle measures so the unknown angle measure is as large

as possible.

4. Players add or subtract to find the measure of the unknown angle

and record it in the circle on the Record Sheet. The measure of the

unknown angle is the player’s score for the round.

5. Each player uses a full-circle protractor, straightedge, and marker

to show that the angle measure of the whole is the sum of the angle

measures of the parts.

6. Players play 3 rounds for a game. The player with the largest total

number of points at the end of the 3 rounds wins the game.

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Name Date Time

508

Copyrig

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Wrig

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raw

-Hill

132

4Angle Add-Up Example

Example: In Round 1, Suma draws a 2, 7, 1, and 5. She creates the angle

measures 51° and 72° and records them on her record sheet.

Round 1:

Draw 4 cards. 5 1 ° +

7 2 °

=

°

m∠ABD m∠DBC m∠ABC

Using addition, Suma finds the sum of the measures of angles ABD and DBC.

She records the measure of angle ABC on her record sheet and scores

123 points for the round.

Round 1:

Draw 4 cards. 5 1 ° +

7 2 °

=

°


Suma uses her full-circle protractor to show that m∠ABD + m∠DBC = m∠ABC.

123

12

6

11

5

10

4

1

7

2

8

39

degrees

A

B

D

C


Name Date Time

509

Copyright

© W

right

Gro

up/M

cG

raw

-Hill

132

4Angle Add-Up Record Sheet

Game 1

Round 1:

Draw 4 cards. ° +

°

=

°


Round 2:

Draw 2 cards. ° +

°

=

90°


Round 3:

Draw 2 cards.

°

+

°

=

180°


Total Points =

Game 2

Round 1:

Draw 4 cards. ° +

°

=

°


Round 2:

Draw 2 cards. ° +

°

=

90°


Round 3:

Draw 2 cards.

°

+

°

=

180°


Total Points =


formula for the area of a parallelogram - everyday math - … · 2012-05-04 · • describe...

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