16.4 Relate Figures, Fractions, and Area Figures, Fractions, and Area How can you divide ﬁgures...

For the student For the teacher

Teacher Edition

Math on the Spot Video Tutor Online Assessment

System

Soar to Success Math Online Intervention

iTools Virtual Manipulatives

Digital ManagementCenter organizes program resources by TEKS!

Interactive Student Edition provides students

with an interactive learning environment!

Resources

e

Texas Essential Knowledge and Skills

Geometry and Measurement—3.6.E Decompose two congruent two-dimensional figures into parts with equal areas and express the area of each part as a unit fraction of the whole and recognize that equal shares of identical wholes need not have the same shape

MATHEMATICAL PROCESSES3.1.C Select tools, technology, and techniques3.1.E Create and use representations

Are You Ready?Access Prior KnowledgeUse the Are You Ready? 16.4 in the Assessment Guide to assess students’ understanding of the prerequisite skills for this lesson.

Vocabulary

Go to Multimedia eGlossary at thinkcentral.com

MaterialsPattern blocks, color pencils, ruler

16.4INVESTIGATE

Relate Figures, Fractions, and Area How can you divide figures into parts with equal areas and write the area as a unit fraction of the whole?

Essential Question?

Lesson OpenerMaking ConnectionsInvite students to tell you what they know about kites.

Have you ever flown a kite? What shape was your kite? Where did you fly it?

Using the Digital LessonModel the shape of the kite with different colored pattern blocks, as in the problem. Have students make observations about the different shapes and parts of the kite shape.

Learning TaskWhat is the problem the students are trying to solve? Connect the story to the problem.

• What are you being asked to find? (How much of the kite is green)

• What figures can you see in the kite? (Triangles and parallelograms)

• What does Doc want to use to describe the figures in the kite? (Fractions)

Literacy and MathematicsChoose one or more of the following activities.

• Have students design a kite and write three reasons why their design is creative, original, and interesting.

• Have students work in pairs to clarify information by asking and answering questions about the problem.

How can you divide f igures into parts with equal areas and write the area as a unit

fraction of the whole?

Lesson 16.4 531A

Mathematical ProcessesMath Talk

InvestigateInvestigateInvestigate HandsOn

Make ConnectionsMake Connections

Essential Question?

Materials ■ pattern blocks ■ color pencils ■ ruler

Connect Use what you know about combining and

separating pattern blocks to explore the relationship

between fractions and area.

A. Trace a hexagon pattern block.

B. Divide your hexagon into two parts with equal area.

What new figures have you drawn?

________

C. Write the fraction that names each part of the

whole you divided. _Each part is 1 _

2 of the whole figure’s area.

D. Write the fraction that names the whole area. _

Name

How can you divide figures into parts with equal areas and write the area as a unit fraction of the whole?

Relate Figures, Fractions, and Area16.4

Geometry and Measurement—3.6.E

MATHEMATICAL PROCESSES3.1.C, 3.1.E

Math IdeaEqual parts of a whole have equal area.

The rectangle at the right is divided into four parts with equal area.

• Write the unit fraction that names each part of the divided whole. _

• What is the area of each part? ___

• How many equal shares does it take to make one whole? _

• Does each equal share of the whole have the same shape? _

• Is the area of each equal share the same? Explain how you know.

trapezoids or pentagons

1 _ 2

2 _ 2

Yes; possible explanation: I can count the unit squares. There are 5 unit squares in

each 1 _ 4 part, so the area of each 1 _

4 part is the same.

1 _ 4

5 square units

4

No

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

Module 16 531

English Language Learners Language SupportELLLeveled Activities ELPS

Beginning: Activity 20 1.A.1, 3.G.2, 4.C.3

Intermediate: Activity 3 2.D.2, 2.E.3, 3.F.2

Advanced: Activity 58 2.C.2, 4.C.3, 4.F.9

Advanced High: Activity 59 1.F, 3.E, 3.H.3, 4.C.3

thinkcentral.com for the ELL Activity Guide containing these leveled activities.

ELPS 1.B.1, 2.E.3, 3.F.2

Strategy: Model LanguageMaterials: pattern blocks

• Teachers model language to teach pronunciation.• Show each pattern block and tell students its proper name.• Have students repeat the name for each block.• Trace a pattern block and divide it into equal parts. Have students

name the shape of each part.

Visual Small Group

Investigate Be sure students have partitioned the hexagon tracing into equal parts.

Point out that since the denominator of the fraction should show the total number of equal parts, and there are 2 parts in all, the fraction named in Part D should be 2 _ 2 .

• How do you know the two figures have the same area? Possible answer: I can cover the area with two trapezoid pattern blocks. The blocks are the same size and shape.

• What is another way you can be sure the figures have the same area? Possible answer: cut out the figure and fold it in half; if the sides match, they are the same size.

Go DeeperTo extend students’ thinking, ask what would happen if they divided the hexagon into three figures with equal area. Have them tell what fraction names the area of each part of the divided hexagon and what fraction names the whole area.

Make ConnectionsRemind students that a unit fraction will always have a numerator of 1.

• What unit fraction names 1 part of the divided whole? 1 _ 4

Guide students to see how counting the squares in each part has the same effect as covering each part with pattern blocks to determine whether the areas are equal, even though the shapes are different.

531 Module 16

Share and ShowShare and Show

Mathematical ProcessesMath Talk

Problem SolvingProblem Solving

1. Divide the trapezoid into 3 parts with equal area. Write

the names of the new shapes. Then write the fraction

that names the area of each part of the whole.

Draw lines to divide the figure into equal parts that show the fraction given.

2. 3. 4.

5. 6. 7.

Draw lines to divide the figure into parts with equal area. Write the area of each part as a unit fraction.

8. Write MathWrite Math If the area of three is equal

to the area of one , the area of how many equals

four ? Explain your answer.

9. Apply Show how you can divide the hexagon

into four shapes with equal area.

Each part is _ of the whole shape’s area.

16

12

18

Explain how you know the areas of all the parts

are equal.

3 equal parts 6 equal parts 4 equal parts

Possible explanation: I can cover the area with 3 triangle pattern blocks. The blocks are the same size and shape, so the areas of the parts are equal.

3 triangles; 1 _ 3

Possible drawings are shown.


Check students’ drawings.

1 _ 3 1 _

6 1 _

4

12 blue rhombuses; Possible explanation: I can multiply 3 × 4 = 12.

1 _ 4

532

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

Go to Go to thinkcentral.com for additional enrichmentactivities in the Enrich Activity Guide.

Enrich

Hexagon

16 1

616

16

16

16

Materials: pattern blocks, paper, scissors

• Have students trace a pattern block and cut it into equal parts. Then have them label each piece with the unit fraction that names each part of the divided whole.

• Ask students to trade their pieces with a partner. Partners should use the pieces to reassemble the whole and name the original figure.

• Challenge students to divide the figures into as many equal parts as possible before trading.

Visual / KinestheticPartners

Share and ShowThe first problem connects to the learning model. Have students use the MathBoard to explain their thinking.

• How do you know that your figures in Exercise 1 have the same area? Possible answer: I can cover the area with 3 triangle pattern blocks.

1

2

3

a student misses the checked exercises

Quick Check

IF

THENDifferentiate Instruction withRtI Tier 1 Lesson 81

Math Talk Use Math Talk to focus on students’ understanding of partitioning a figure into parts with equal areas.

Mathematical Processes

Use the checked exercises for Quick Check. Students should show their answers for the Quick Check on the MathBoard.

Problem SolvingProblems

Problem 8 requires students to use proportional reasoning using given relationships between objects. Students can use drawings to represent the relationship. For each yellow hexagon, students should draw three blue rhombuses.

Problem 9 requires students to partition the hexagon into four equal parts. Emphasize that all parts should be equal when they draw the lines to divide the figure.

Lesson 16.4 532

10. Multi-Step Sense or Nonsense?

Alexis said the area of 1 _ 3 of the trapezoid is greater than the

area of 1 _ 6 of the hexagon because 1 _

3 > 1

_ 6 . Does her statement

make sense? Explain your answer.

Write a statement that makes sense.

11. What if you divide the hexagon into 3 equal parts?

Use Math Language to write a sentence that compares

the area of each equal part of the hexagon to each equal

part of the trapezoid.

Divide the hexagon into six

equal parts.

Which pattern block represents 1 _ 6

of the whole area?

____

Divide the trapezoid into three

equal parts.

Which pattern block represents

1 _ 3

of the whole area?

____

Name

Check students’ drawings.

triangle triangle

No; possible explanation: Alexis just looked at the fractional parts 1 _ 6 and 1 _

3 . All the triangle

Possible answer: the area of 1 _ 3 of the hexagon is greater than the area

of 1 _ 3 of the trapezoid.

Possible answer: the area of 1 _ 3 of the trapezoid is equal to the area of 1 _

6 of the hexagon.

parts have the same area, but since the wholes are not the same size, you cannot

compare the fractions.

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

Module 16 • Lesson 4 533

1

© H

ou

gh

ton

Mif

flin

Har

cou

rt P

ub

lish

ing

Co

mp

any

Name

3.6.E

You can separate a plane figure into equal parts to explore the relationship between fractions and area.

Divide the rectangle into 6 parts with equal area. Write the fraction that names the area of each part of the whole.

Step 1 Draw lines to divide the rectangle into 6 parts with equal area. Use the grid to help you.

Step 2 Write the fraction that names each part of the divided whole.

Think: Each part is 1 part out of 6 equal parts.

Each part is 1 __ 6 of the whole figure’s area.

Step 3 Write the fraction that names the whole area.

Think: There are 6 equal parts.

The fraction that names the whole area is 6 __ 6

.


1. 4 equal parts

Each part is of the whole figure’s area.

2. 8 equal parts

Each part is of the whole figure’s area.

Relate Figures, Fractions, and AreaOBJECTIVE Partition figures into parts with equal areas and express the area

as a unit fraction of the whole.

LESSON 81

1 __ 8

1 __ 4


Geometry and Measurement 161 © Houghton Mifflin Harcourt Publishing Company

Name

E77Enrich

Enrich 77

Secret Message

Draw lines to divide the figure into the given number of

parts of equal area. Write the fraction that names each part

of the whole.

A 10 equal parts

F 4 equal parts

H 8 equal parts

I 6 equal parts

M 12 equal parts

N 2 equal parts

S 5 equal parts

T 9 equal parts

U 3 equal parts

Write the fractions in order from least to greatest.

Use the letter for each fraction to write the secret message.

1 __ 12

M

1 __ 10

1 _ 9

1 _ 8

1 _ 6 1 _

5 1 _

4 1 _

3 1 _

2

1 __ 10

1 _ 4

1 _ 8

A T H I S F U N

1 _ 6

1 __ 12

1 _ 2

1 _ 5

1 _ 9

1 _ 3


Enrich 77RtI Tier 1 Lesson 81

Math on the Spot Video Tutor

Through the Math on the Spot Video Tutor, students will be guided through an interactive solving of this type of H.O.T. problem. Use this video to also help students solve the H.O.T. problem in the Interactive Student Edition. With these videos and the H.O.T. problems, students will build skills needed in the TEXAS assessment.

MV

Math on the Spot videos are in theInteractive Student Edition and atthinkcentral.com.

COMMON ERRORSError Students may draw lines of division that do not result in equal parts.

Example

Springboard to Learning Use pattern blocks to show that the hexagon cannot be covered using the figure created by the partition. Remind students that unless the parts are equal, you cannot write a unit fraction to represent the area of a part.

CE

ProblemFor Problem 10, ask students to compare the figures that result from dividing the hexagon into 6 equal parts with the figures that result from dividing the trapezoid into 3 equal parts.

• Are any of the triangles larger than the others? no

• Why can the triangle areas be described with different fractions if they are the same size? Possible answer: the fractions compare the area of one part to the area of the whole. The original shapes of the wholes were different, so the denominators will be different.

1

2

3

533 Module 16

Mathematical Processes

Daily Assessment TaskDaily Assessment Task

TEXAS Test Prep 15. Rav divided the rectangle at the right into six parts with

equal area. What is the area of each part?

A 6 square units

B 24 square units

C 4 square units

D 10 square units

Fill in the bubble for the correct answer choice.

12. Inez made a stained glass design with

the rectangles shown at the right. Which

rectangles have the same shape and the

same area?

A A and C C A and D

B B and D D B and C

13. Paulo buys a mirror. The frame is made of 6 same-size

trapezoids. What unit fraction of the frame is each

trapezoid?

A 1 _ 3

C 1 _ 8

B 1 _ 6

D 1 _ 2

14. Multi-Step Anastasia is painting one wall of her room.

She divides the wall into 8 squares with the same area.

She paints half of the squares light green and half of the

squares dark green. What fraction of the wall is one dark

green square?

A 1 _ 2

C 1 _ 4

B 1 _ 8

D 1 _ 1

AD

B C

534

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

THENIF

YES

NO

Daily Assessment Task 1

2

3

Differentiated Centers Kit

LiteratureThe Whole PictureStudents read the book and model fractional parts.

ActivitiesClassification ActStudents complete orange Activity Card 18 by classifying two-dimensional shapes based on their attributes.

ActivitiesFish for FractionsStudents complete orange Activity Card 11 by playing a Go Fish game in which they match fraction symbols, words, and pictures.

TEXAS Test Prep CoachTest Prep Coach helps teachers to identify common errors that students can make.

In the Test Prep exercise, if students selected:

A They counted the number of equal parts.

B They multiplied the number of equal parts by the number of unit squares in each part.

D They added the number of equal parts and the number of unit squares in each part.

Essential Question? WriteMathWriteMath

How can you divide figures into parts with equal areas and write the area as a unit fraction of the whole? Possible answer: I can trace pattern blocks and then draw lines that divide the figure into equal parts. Then I can write the area of each part as a fraction by using 1 as the numerator and the number of equal parts as the denominator.

• Enrich 77

• Homework and Practice Lesson 16.4

• Soar to Success MathWarm-Up 5.10

Can students divide figures into parts with equal areas and write the area as a unit fraction of the whole?

Lesson 16.4 534

TEXAS Test PrepLesson CheckLesson Check

7. Georgie has a rectangular window

that is divided into 6 same-size

squares. What fraction of the

window is 1 square?

A 1 _ 2

B 1 _ 4

C 1 _ 3

D 1 _ 6

8. Marcy has a square window that

is divided into 8 same-size squares.

What fraction of the window is

1 square?

A 1 _ 3

B 1 _ 4

C 1 _ 8

D 1 _ 2

9. Rory divides the rectangle into

7 parts with equal area. What is the

area of each part?

A 4 square units

B 5 square units

C 7 square units

D 6 square units

10. Multi-Step Perry makes a quilt

with same-size squares. He makes

2 red squares, 2 blue squares,

2 yellow squares, and 2 green

squares. What fraction of the quilt

are the red squares?

A 1 _ 4

B 1 _ 2

C 1 _ 8

D 1 _ 3

11. Multi-Step Doreen makes a

painting with 12 same-size squares.

She has painted one-half of the

squares with a design. How many

squares has Doreen painted?

A 1

B 4

C 3

D 6

Fill in the bubble completely to show your answer.

536

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

Problem SolvingProblem Solving

Homeworkand Practice

Name

Draw lines to divide the figure into equal parts that show the fraction given.

1. 2. 3.


4.

4 equal parts

6. Sanchez puts a window in his house

made of 2 same-size triangles. What

unit fraction of the window is

each triangle?

5.

3 equal parts

Relate Figures, Fractions, and Area16.4

Geometry and Measurement—3.6.E MATHEMATICAL PROCESSES 3.1.C, 3.1.E



1 _ 4

1 _ 2

1 _ 3

Module 16 • Lesson 4 535

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

1 _ 2

1 _ 4

1 _ 6

555555555

Homework and PracticeUse the Homework and Practice pages to provide students with more practice on the concepts and skills of this lesson.

535-536 Module 16

16.4 Relate Figures, Fractions, and Area Figures, Fractions, and Area How can you divide ﬁgures...

Documents

Transcript of 16.4 Relate Figures, Fractions, and Area Figures, Fractions, and Area How can you divide ﬁgures...