exploring fractions, multiplication and division, and volume...610 unit 8 fractions advance...

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Lesson Guide

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Family Letters

CurriculumFocal Points

Common Core State Standards

610 Unit 8 Fractions

Advance PreparationBefore starting the lesson, decide where to set up the Fractions Museum in your classroom.

Teacher’s Reference Manual, Grades 1–3 pp. 59, 60, 69, 90–92, 159–161

Key Concepts and Skills• Count by 1s, 10s, and 100s.

[Number and Numeration Goal 1]

• Compare fractional parts using pattern blocks.

[Number and Numeration Goal 7]

• Make reasonable estimates for the total amount.

[Operations and Computation Goal 3]

• Make arrays on geoboards to model multiplication.


Key ActivitiesExploration A: Children compare pairs of pattern blocks as fractions and

multiples of each other.

Exploration B: Children form rectangles, or fences, on geoboards; they determine

the total number of pegs inside the fences and the dimensions of the rectangles.

Exploration C: Children use base-10 blocks to build structures; they estimate

and find the volumes of the structures in cubic centimeters.

Key Vocabularycubic centimeter � volume

MaterialsHome Link 8 �1

Exploration A: Per partnership:

Math Journal 2, pp. 187 and 188

Math Masters, p. 230

pattern blocks � Pattern-Block Template

Exploration B: Per group:

Math Journal 2, p. 189


geoboard � rubber bands � straightedge

Exploration C: Per group:


base-10 blocks (several cubes, longs, and flats; thousands cubes, if available) �

slate

Playing Beat the CalculatorMath Journal 2, p. 311

My Reference Book, pp. 124 and 125

1 calculator per child

Ongoing Assessment: Recognizing Student Achievement Use an Exit Slip (Math Masters, page 415). [Operations and Computation Goal 1]

Math Boxes 8�2Math Journal 2, p. 190

Home Link 8�2Math Masters, p. 233

READINESS

Naming Fractional Parts of a Regionpattern blocks � Pattern-Block Template

ENRICHMENTSolving Fraction PuzzlesMath Masters, p. 234

pattern blocks

EXTRA PRACTICE

Minute Math +Minute Math®+, p. 40

Teaching the Lesson

132

4

Ongoing Learning & Practice

Differentiation Options

Exploring Fractions, Multiplicationand Division, and Volume

Objectives To guide children as they explore the link of a fraction

amount to the size of the ONE, or whole; to prepare for multiplication

and division; and to introduce the concept of volume.

a

��

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Lesson 8�2 611

Small pizza

Large pizza

1 Teaching the Lesson

� Math Message Follow-Up WHOLE-CLASSDISCUSSION

On the board, draw and label two pizzas.

As children share their answers, it is important to emphasize that a fraction is not meaningful if they do not know what it is a fraction of; that is, what is the ONE, or the whole. A fraction is always “a fraction of something.”

� Comparing Pattern Blocks, WHOLE-CLASS ACTIVITY

One of Which Represents ONEHold up two pattern blocks, the larger of which is a multiple of the smaller. For example, hold a hexagon in one hand and a trapezoid in the other. Ask questions like the following:

● How many of these do you need to cover one of these

? 2

● If is ONE (the whole), what fraction of ONE

is ? one-half

● How do we write this fraction? 1 _ 2

Next, reverse the relationship between the pattern blocks. Ask:

● If is ONE, how much is ? 2

Display and ask about other pattern blocks. For example:

● If is ONE, is what number? 2

● If is ONE, is what fraction? 1 _ 2

Getting Started

Math MessageMartha ate

1

_ 4 of a large pizza. Juanita ate

1

_ 4 of a small pizza. Did they eat

the same amount of pizza?

Home Link 8�1 Follow-Up Use the items children brought from home to start a Fractions Museum. Encourage children to bring additional items to add to the museum. Suggest that they look for written fractions in ads and on food packages and kitchen utensils. Children might, for example, bring a half-pint or half-gallon milk carton and measuring spoons or cups.

Mental Math and Reflexes Write the following sets of numbers on the board. Ask children to find the number that appears most often (the most popular number) for each set of numbers. Remind them to first put the sets in ascending or descending order.

2, 5, 8, 9, 0, 1, 1 1

11, 18, 9, 5, 5, 11, 5, 18, 18, 11,18, 9 18

56, 56, 89, 34, 35, 35, 82 35 and 56

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NOTE On journal pages 187 and 188, have

children notice that they divided the shapes

into equal parts (thirds and fourths) that are

congruent.

3. If = 1, then = .

4. If = 1, then = .

5. If = 1, then = .

6. If = 1, then = .

Pattern-Block Fractions continuedLESSON

8 �2

Date Time

1 _ 4

1 _ 4

1 _ 4

1 _ 4

184_208_EMCS_S_SMJ_G2_U08_57640X.indd 188 2/10/11 9:31 AM


Student Page

Pattern-Block FractionsLESSON

8 �2

Date Time

Use pattern blocks to help you solve each problem.

Use your Pattern-Block Template to show what you did.

Example:

If = 1, then = .

1. If = 1, then = .

2. If = 1, then = .

1 _ 3

1 _ 3

1 _ 3



Student Page● If is ONE, is what number? 3






� Exploration A: Comparing Pairs PARTNER ACTIVITY

of Shapes When One Shape Represents ONE(Math Journal 2, pp. 187 and 188; Math Masters, p. 230)

Review the example on journal page 187 with children. Ask them to cover the trapezoid with three triangle pattern blocks to reinforce the fact that a triangle pattern block is one-third of a trapezoid.

Do the first problem together. Children cover the larger shape with hexagon pattern blocks. Because it takes three hexagons to cover the larger shape, a hexagon is one-third of the larger shape.Children write the fraction 1 _ 3 in the answer space and use their Pattern-Block Templates to partition the larger shape.

Partners work together to complete the rest of the problems on journal pages 187 and 188. Circulate and offer help as needed.

When children have completed the journal pages and Problem 3 on Math Masters, page 230, bring the class together to share solutions. Emphasize that each of the smaller shapes on journal page 187 is represented by the fraction 1 _ 3 . The fractions mean different things, however, because the wholes are not the same size and shape. Similarly, each of the smaller shapes on journal page 188 is 1 _ 4 of the larger shape.

Summary: The size of a fractional part of a whole depends on the size of the whole. A fraction has no meaning unless we know what the whole is.

� Exploration B: Making Arrays SMALL-GROUP ACTIVITY

with Geoboard Fences(Math Journal 2, p. 189; Math Masters, p. 231)

Children form a variety of rectangles on their geoboards, using one rubber band for each rectangle. They draw four of their rectangles at the top of the journal page.

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BBBBBBBBBBBBBBBBBBBB ELEELELEMMMMMMMMMOOOOOOOOOBBBLBLBLBLBBBBLOOROROROORORORORORORORORO LELELELEEEEEELEMMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRGGGGGLLLLLLLLLLLLLVVINVINVINVINNNVINVINVINVINNVINVINVINVINV GGGGGGGGGGGOLOOOOLOOOLOLOO VVINVINLLLLLLLLLLVINVINVINVINVINVINVINVINVINVINVINVINVINNGGGGGGGGGGGOOOLOLOLOLOLOLOOO VVVVLLLLLLLLLLVVVVVVVVVOSOOOSOSOSOSOSOSOSOOSOSOSOOOOSOSOOSOSOSOSOSOSOSOOSOSOSOSOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVVLLLLLLLLVVVVVVVVVLLVVVVVVVVLLLLLLLLVVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSSS GGGGGGGGGGGGGGGGOOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIIIISOLVING

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Work with a partner.

Materials □ Math Journal 2, pp. 187 and 188

□ pattern blocks

□ Pattern-Block Template

Study the example at the top of journal page 187. Cover the with s.

You need 3 triangles, so a is 1 _ 3 of a .

1. Do Problem 1 on journal page 187.

Cover the larger shape with hexagon blocks.

The number of hexagons helps you find the fraction to write as the answer.

Use your Pattern-Block Template to divide the larger shape into hexagons.

2. Do the rest of the problems on journal pages 187 and 188 in the same way.

3. Get together with other partners in the class. Check one another’s work.

Each smaller shape on page 187 is 1 _ 3 of the larger shape.

Are all the smaller shapes the same size? no Each smaller shape on page 188 is 1 _ 4 of the larger shape.

Are all the smaller shapes the same size? no

LESSON

8�2

Name Date Time

Pattern-Block Fractions

py

gg

p

227-251_EMCS_B_MM_G2_U08_576949.indd 230 2/9/11 12:34 PM


Teaching Master

Lesson 8�2 613

1. 2.

3. 4.

How many pegs How many rows How many in Fence in all? of pegs? each row?

1.

2.

3.

4.

Geoboard FencesLESSON

8 �2

Date Time

Fences drawn and table entries vary.



Student Page

NOTE To vary Exploration B, have children

estimate the total number of pegs within each

fence before they count the number of pegs.

Discuss strategies children might use to make

their estimates. After they count the actual

number of pegs within each fence, have them

compare that number with their estimates.

Next, children find the total number of pegs within each fence (including the pegs touching the rubber band), the number of rows of pegs, and the number of pegs in each row. Children record this information in the table at the bottom of the journal page.

NOTE Exploration B reinforces the work with arrays done at the end of Unit 6.

� Exploration C: Finding the SMALL-GROUP ACTIVITY

Volumes of Base-10 Structures(Math Masters, p. 232)

Two or three group members take turns building structures with base-10 blocks. Each group member estimates the total number of centimeter cubes (cubic centimeters) in the structure. Children write their estimates on slates.

As they take the structures apart, children count the cubes and compare the actual totals to their estimates. They report the volume in the form of, “This structure has a volume of cubic centimeters.” To support English language learners, discuss both the mathematical and everyday uses of the word volume, emphasizing its mathematical meaning.

NOTE Exploration C helps prepare children for work in Unit 9.

2 Ongoing Learning & Practice

� Playing Beat the Calculator PARTNER ACTIVITY

(Math Journal 2, p. 311; My Reference Book,

pp. 124 and 125)

See My Reference Book, pages 124 and 125 for directions. Children play Beat the Calculator to develop their recall of addition facts. They should record the facts for which they can beat the calculator by making a check mark in the box of the corresponding fact. When they have three check marks next to a particular fact, they write the sum in that box to indicate that they know the fact from memory. The Caller should select problems at random from the Fact Power Table.

Ongoing Assessment: Exit Slip

Recognizing Student Achievement

Use an Exit Slip (Math Masters, page 415) to have children record the facts in

boxes for which they have at least one check mark. Children are making

adequate progress if they record all the facts in the unshaded area of the Fact

Power Table. Some children may record additional facts.


ELL



Work in a small group.

Materials � Math Journal 2, p. 189 � geoboard

� 4 rubber bands � straightedge

Each group member does Steps 1–6:

1. Make a rectangle on a geoboard. Use1 rubber band. Think of the rubber bandas a fence.

2. Draw your rectangle (fence) on the firstgeoboard on journal page 189.

3. Count the number of pegs inside your rectangle (fence). Include the pegs that touch the rubber band.

4. Fill in the table at the bottom of the journal page. Include

� the number of pegs inside the fence.

� the number of rows of pegs inside the fence.

� the number of pegs in each row.

5. Make 3 more rectangles (fences) on your geoboard. Draw eachrectangle (fence) on the journal page.

6. Fill in the table for your other three fences.

LESSON

8�2

Name Date Time

Geoboard Fences

Follow-Up

Compare your table to those of the other members of your group.

Do any members of your group have the same total number of pegsinside a fence (but with a different number of rows)? What else isdifferent? Can you tell why this happened?


Teaching Master

Work in a group.

Materials � base-10 blocks: cubes, longs, flats; a big cube(thousands), if available

� slate for each person

1. Two group members use the blocks to build a structure. They should work quickly so others can have a turn later.

2. Each small cube has a volume of 1 cubic centimeter.

Each group member writes an estimate of the total number of cubes (cubic centimeters) in the structure on her or his slate.

3. Together, count the cubes as the builders take the structure apart.

The total number of cubes equals the volume of the structure in cubic centimeters.

Record the result like this:

“This structure has a volume of cubic centimeters.”

4. Compare the actual number of cubes to the estimates of the group members.

5. Change builders. Repeat Steps 1–4. Continue until everyone hashad a turn. As you build the structures, think of ways to improveyour estimates.

6. Write a group report about your estimates and the actualvolumes of the structures.

LESSON

8�2

Name Date Time

Volumes of Base-10 Structures

This structure has a volume of520 cubic centimeters.


Teaching Master

� Math Boxes 8�2 INDEPENDENTACTIVITY

(Math Journal 2, p. 190)

Mixed Practice Math Boxes in this lesson are linked with Math Boxes in Lessons 8-4 and 8-6. The skills in Problems 5 and 6 preview Unit 9 content.

Writing/Reasoning Have children draw, write, or verbalize their answers to the following: Explain how you knew how many dots to color green in Problem 4. Sample

answer: I counted all the dots and split them into two equal groups. I colored one group green.

� Home Link 8�2 INDEPENDENTACTIVITY

(Math Masters, p. 233)

Home Connection Children write fractions to name parts of regions.

3 Differentiation Options

READINESS INDEPENDENTACTIVITY

� Naming Fractional Parts 15–30 Min

of a RegionTo provide experience naming fractional parts of a whole using a concrete model, have children create a design with one type of pattern block. Then use the Pattern-Block Template to draw the design. The design is the ONE. Children write what fraction one pattern block represents of the whole design. Example:

1-6

1-6

1-6

1-6

1-6

1-6

Have children describe the color, shape, and features of their designs they have made including naming the fractional part.


Math BoxesLESSON

8 � 2

Date Time

1. Fill in the missing numbers.

179190

199189

200

157

177

156

167

3. Put a line under the digit in theones place.

479 364

1,796 5,079

6. � 1 sq cm

Area � sq cm 4

2. 1 hour � minutes

�12� hour � minutes

�14� hour � minutes 15

3060

4. Color �12� of the set green.

5. Measure the length of this line.

about cm

about in.25

86

69

10


Student Page

Lesson 8�2 615

Name Date Time

In today’s lesson, your child compared pattern blocks, one of which represents ONE whole.As you work on this activity with your child, keep in mind that the shape below the ONE is afractional part of the whole shape. Remind your child that the size of a fractional part of awhole depends on the size of the whole.

Please return this Home Link to school tomorrow.

FamilyNote

HOME LINK

8�2 Fractions of Shapes

1. If this shape is ONE,

then is what fraction of the shape?


then is what fraction of the shape?


then is what fraction of the whole shape?

Solve.

4. 75 5. 56 6. 84 7. 91� 26 � 45 � 48 � 67

158132101101

Practice

Try This

�16�

�23�

12

Unit

�12�


Home Link Master

1. The first figure is �12� of the whole. What fraction of the same whole

is each of the other figures? Write the fraction inside the figure.



LESSON

8�2

Name Date Time

Fraction Puzzles

12

16

13

1

13

23 2

31

Try This



4. Create your own fraction puzzle. Answers vary.

13

122

31


Teaching Master

ENRICHMENT INDEPENDENTACTIVITY

� Solving Fraction Puzzles 5–15 Min

(Math Masters, p. 234)

To explore naming fractions of regions using a concrete model, have children use pattern blocks and follow directions on Math Masters, page 234, to solve the fraction puzzles.

EXTRA PRACTICE SMALL-GROUP ACTIVITY

� Minute Math + 5–15 Min

To offer children more experience with finding parts of a whole, see the following page in Minute Math+: p. 40.


exploring fractions, multiplication and division, and volume...610 unit 8 fractions advance...

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