grade 4 supplement - the math learning center geoboards and geobands or geoboard app (online or...

GRADE 4 SUPPLEMENTSet A10 Number & Operations: Multiplying Whole Numbers by Fractions

Includes H Activity 1: Multiplying Fractions on the Geoboard A10.1

H Activity 2: Quick Sketches & Fraction Arrays A10.9

H Activity 3: Fraction Word Problems & Arrays A10.13

H Independent Worksheet 1: Pizza Party Planning A10.17

H Independent Worksheet 2: Brownie Dessert A10.19

H Independent Worksheet 3: Multiplying Fractions with Arrays A10.21

H Independent Worksheet 4: Company’s Coming! A10.23

Skills & Concepts H Add fractions with like denominators

H Write an equation showing a fraction a⁄b as the sum of a number of the unit fraction 1⁄b

H Write an equation to show a fraction as the sum of other fractions with the same denominator

H Demonstrate an understanding that a fraction a⁄b is a multiple of the unit fraction 1⁄b

H Write an equation showing that a fraction a⁄b is the product of a × 1⁄b

H Multiply a fraction by a whole number

H Solve word problems in which fractions are multiplied by a whole number

H Use a visual area model to multiply a whole number by a fraction

P0213

Bridges in Mathematics Grade 4 Supplement

Set A10 Number & Operations: Multiplying Whole Numbers by Fractions

The Math Learning Center, PO Box 12929, Salem, Oregon 97309. Tel. 1 800 575–8130.

© 2012 by The Math Learning Center

All rights reserved.

Prepared for publication on Macintosh Desktop Publishing system.

Printed in the United States of America.

P0213

The Math Learning Center grants permission to classroom teachers to reproduce blackline

masters in appropriate quantities for their classroom use.

Bridges in Mathematics is a standards-based K–5 curriculum that provides a unique blend

of concept development and skills practice in the context of problem solving. It incorpo-

rates the Number Corner, a collection of daily skill-building activities for students.

The Math Learning Center is a nonprofit organization serving the education community.

Our mission is to inspire and enable individuals to discover and develop their mathematical

confidence and ability. We offer innovative and standards-based professional development,

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visit us at www.mathlearningcenter.org.


© The Math Learning Center Bridges in Mathematics Grade 4 Supplement • A10.1

Set A10 H Activity 1

ACTIVITY

Multiplying Fractions on the GeoboardOverview Students re-examine problems from Bridges Unit 6: Ses-

sion 3, and consider how the relationship between areas

on a geoboard might be represented using multiplication.

Students then consider additional models on the geoboard

in which fractions are multiplied by a whole number.

Skills & Concepts H Write an equation showing a fraction a⁄b as the sum of

a number of the unit fraction 1⁄b (CCSS 4.NF.3)

H Express a fraction as the sum of other fractions with the

same denominator in more than one way (CCSS 4.NF.3b)

H Demonstrate an understanding that a fraction a⁄b is a

multiple of the unit fraction 1⁄b (CCSS 4.NF.4a)

H Write an equation showing that a fraction a⁄b is the

product of a × 1⁄b (CCSS 4.NF.4a)

H Use an area model to multiply a fraction by a whole

number (CCSS 4.NF.4b)

H Explore fraction relationships and operations using a

visual model

H Find equivalent fractions

Recommended TimingUse anytime after Unit 6 Session 3

You’ll need H Fractions on a Geoboard: Multiplying Fractions, pages

A10.5–A10.7 (run a class set.)

H Geoboards and geobands or Geoboard App (online

or iPad) for display

H Geoboards and geobands for students, optional

H Paper/pen for use on the display

H Colored pencils, optional

Note When you represent the symbolic form for a fraction,

please use a horizontal bar.

Instructions for Multiplying Fractions on the Geoboard1. Distribute copies of Fractions on a Geoboard: Multiplying Fractions and display the top portion. Give stu-dents time to build all the regions on their geoboards.

NAME DATE

Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Blackline Run a class set.

Fractions on a Geoboard Multiplying FractionsIn the illustration below, the whole geoboard has an area of 1.

A

B

C D E

1 Look at the visual models below and think about what you notice about the relationships between the regions on the geoboard. How could you express your observations using multiplication? Explain your thinking in words and record your observations in a multiplication equation.

Visual Model Explanation Multiplication Equation

a

B

C C

b

C C

C C

A

c

(continued on next page)

D D D D

D D D D

A

18 =2 x 1

4There are 2 C’s in Part B

2. Write C + C = B on the whiteboard. Have student pairs discuss if the equation is true. Ask for a volunteer to share. Then ask student pairs to consider using multiplication to show the relationship be-tween regions B and C.


A10.2 • Bridges in Mathematics Grade 4 Supplement © The Math Learning Center

3. After students have had a chance to share observations with a partner, ask for volunteers to share with the class.

Teacher Jesse, where do you see multiplication in the relationship between C and B?

Jesse Well, if two Cs equal B, that’s like saying that 2 × C = B.

Lauren And since C is 1⁄8, you could write 2 × 1⁄8 = 1⁄4.

Max I see it as 1⁄4 = 2 × 1⁄8

4. Complete information in the first row as students write on their own sheets.

NAME DATE



A

B

C D E



a

B

C C

b

C C

C C

A

c


D D D D

D D D D

A

18 =2 x 1

4There are 2 C’s in Part B

5. Look at the next visual model. With student input, review ways that the relationship between D and A could be represented with addition. Compile a list of equivalent equations on the board.

116

116 = 1

2+1

16 +116 +1

16 +116 +1

16 +116 +

416

416 = 1

2+

216

216 = 8

16+2

16 +216 +

Teacher Does anyone know another way we could write 8⁄16?

Mia One half. You can see it on the geoboard. 8 is half of 16.

Activity 1 Multiplying Fractions on the Geoboard cont.



6. Ask students to help you write an explanation such as “There are 8 D’s in section A.” By writing the explanation, students will begin to more explicitly see fraction equations as a sum of the number of a unit fraction.

7. Then ask each student to record one way that relationship between D and A could be expressed using multiplication. Share with partners. Then ask for volunteers to share the multiplication equation.

Marin If 1⁄16 + 1⁄16 + 1⁄16 + 1⁄16 + 1⁄16 + 1⁄16 + 1⁄16 + 1⁄16 = 8⁄16, it’d be a lot easier to write 8 × 1⁄16 = 8⁄16.

Josh Mine was like Marin’s. But I said that 8 × 1⁄16 = 1⁄2. 8 is half of 16, like Mia said.

Marin 8⁄16 and 1⁄2 are the same thing.

Cybil How can 8 × 1⁄16 be 1⁄2?

Teacher Josh, could you show us on the geoboard?

Josh Sure. Look at the 8 D’s that are shaded in. That makes 8 sixteenths. But the entire part that’s shaded in shows 1⁄2 of the whole geoboard. So 8 sixteenths is 1⁄2. Or 8 × 1⁄16 = 1⁄2.

8. Continue having students look at parts in relationship to the entire area of the geoboard. Ask them to write explanations and multiplication equations to represent the shaded portion of the geoboard.

NAME DATE



A

B

C D E



a

B

C C

b

C C

C C

A

c

D D D D

D D D D


18 =

There are 8 D’s in Part A

2 x 14

There are 4 C’s in Part A

There are 2 C’s in Part B

A

116 =8 x 8

16

116 =8 x 1

2

18 =4 x 1

2

NAME DATE



d

CEE

EE

e

B

f

C C

C

C

C

C

g

D D

D D


Fractions on a Geoboard Multiplying Fractions cont.

The whole geoboard can be split into 32 E’s. C can be split into 4 E’s.

132 =4 x 1

8

14 =2 x 2

4

18 =6 x 6

8

116 =4 x 4

16

The whole geoboard can be split into 4 B’s. Two shaded parts is or 2

412

The whole geoboard can be split into 8 C’s. 6 of the 8 parts are shaded.

The whole geoboard can be split into 16 D’s. 4 of 16 are shaded.

14 =2 x 1

2

34

14

or

or

B

9. As students feel comfortable, encourage them to continue the problems on their own. If they finish the models you’ve created, ask them to solve for an unknown on the challenge problems provided. Re-serve time at the end of the session for volunteers to prove their thinking.




INDEPENDENT WORKSHEET

See Pizza Party Planning on pages A10.17 and A10.18 for more practice with multiplying fractions by a whole number in the context of word problems.



NAME DATE



A

B

C D E



a

B

C C

b

D D D D

D D D D

A

c

C C

C C

A



NAME DATE



d

CEE

EE

e

B

B

f

C C

C

C

C

C

g

D D

D D




NAME DATE


CHALLENGE

2 Create a sketch that could be used to solve the following unknowns:

Problem Labeled Sketch Answer

a 3 × _______ = 3 4

b 5 × ______ = 5 16

c 7 × ______ = 7 8

d 2 × ______ = 1 4





ACTIVITY

Quick Sketches & Fraction Arrays

OverviewThe teacher models how to sketch an array for 1/4 × 3.

Together as a class, students use this quick-sketch method

to find the product of 2/3 × 5. Then, students work inde-

pendently on 3 more combinations.

Skills & Concepts H Write an equation showing a fraction a/b as the sum of

a number of the unit fraction 1/b (CCSS 4.NF.3)

H Express a fraction as the sum of other fractions with the

same denominator in more than one way (CCSS 4.NF.3b)

H Write an equation to show a fraction as the sum of other

fractions with the same denominator (CCSS 4.NF.3b)

H Add fractions with like denominators

H Demonstrate an understanding that a fraction a/b is a

multiple of the unit fraction 1/b (CCSS 4.NF.4a)

H Write an equation showing that a fraction a/b is the

product of a × 1/b (CCSS 4.NF.4a)

H Multiply a fraction by a whole number (CCSS 4.NF.4b)

Recommended TimingUse after Supplement Set A10, Activity 1

You’ll Need H Paper for sketching



Instructions for Quick Sketches & Fraction Arrays1. Ask students to imagine the outline of a 1/4 × 3 array. Following a moment of private think-time, ask a couple of students to describe what they see in their mind’s eye.

Teacher Yesterday we made rectangular regions on a geoboard to multiply fractions. I wonder if we could make a quick sketch to multiply fractions if we didn’t have a geoboard with us? What would a 1/4 × 3 array look like? We’ll share after some think-time.

Devon I see a rectangle with a tiny piece on the side and then a long piece across the top.

Misty Me, too. The little piece on the side is 1/4. In my mind I saw the piece on top divided into 3 sections.

Teacher Let’s draw this model. I’ll make an array just like we did for multiplying 2-digit numbers.

2. Draw a rectangle using the closest whole numbers, 1 × 3. Ask students how this could help you to make a 1/4 × 3 array. With student guidance, split the array into fractional subsections and label the dimensions.

Teacher When I begin thinking about my rectangle, I’m going to use the closest whole numbers. So if the problem is 1/4 × 3, I’m going to make a 1 × 3 rectangle.



3

1

Teacher Turn to your partner and discuss how I could use this array to help me find 1/4 × 3. Who would like to volunteer an idea?

Jodi You can just cut the rectangle into fourths.

Teacher Can you show me what you mean?

Jodi Sure. (Points on grid.) You can just draw a line to divide the length of your rectangle in half. That shows the two halves or 1/2 of 3. Then split each half to make fourths.

Teacher I’m going to add the lines that Jodi mentioned to our grid. In my drawing, where do you see 1/4 × 3? Take a moment to tell your partner where you see the 1/4 × 3 array. Who would like to share?

3

1

Megan I see it across the top row. The top row just shows 1/4, so you could write 1/4 next to the top row.

Teacher I’m going to erase the 1, label our array with 1/4, and shade in the top of my drawing to show the problem we’re discussing.

314

3. Have students pair-share how they see the area pieces to determine the product of 1/4 × 3. Is it the same as 3 × 1/4? Can they prove it?

Students may notice that they can multiply straight across. Record their thoughts.

Joe I know you can put any number over 1 and it’s still the same number. 3 over 1, that’s like just 3. So I put 3 over 1 and multiplied straight across. 3/1 × 1/4= 3/4

Liz Hey, that’s cool. I wonder if it works every time?

David You don’t even need the 1 under the 3. You can just look at 3 × 1/4 and in your head go (3 × 1)/4.

4. Write another multiplication combination on the board, 2/3 × 5, and repeat the process. This time, begin by having students make and share estimates, first in pairs and then as a class.

Joy I think the answer is going to be more than one. Like somewhere between 2 1/2 and 4. 1/2 of 5 would be 2 1/2 and I know 2/3 of 5 is more than that.

Activity 2 Quick Sketches & Fraction Arrays cont.



5. Ask students to determine the closest whole numbers for 2/3 and 5. Have students draw a 1 × 5 rect-angle, divide it into 3 fractional parts, label the drawing, and shade in the area. If students find 2/3 con-fusing, you may want to first label each row with 1/3, or thirds, then erase and label the top two rows with 2/3.

523

6. Again, have students pair-share to determine the area of 2/3 × 5. Then work with student input to la-bel the sub-regions, recording the product and one or more equations to show how students figured it out.

• Students may look for groups of fractions that equal 1:

523

13

1 1 1

33

1

1333

333

1

+ + + =13

1 133+ =1

3+ +

• Students may observe that each column shows 2/3 of 1 whole:

523

23

23 5 10

3× =103+ =+ +2

323

23

23+2

3

23

23

23

23

• They may also notice that 2/3 × 5 is the same as 1/3 × 10.

523

13

23 5× = (2 × 5) × 1

3

10 × 13

or

13

13

13

13

13

13

13

13

13

• Or 5 × 2/3 could be understood as 5 times 2 groups of 1/3. » n × (a/b) = (n × a)/b » In this case, 5 × 2/3 = (5 × 2)/3

7. If students need additional modeling, do another problem together: 4 × 1/3.

Additional Resource• On The Math Learning Center website, visit “A Play on Numbers” by Eugene Maier and read

Scene 12: Multiplying/Dividing Fractions. http://www.mathlearningcenter.org/media/gene/play-on-numbers/PON_S12_01.asp





See Brownie Dessert on pagesA10.19 and A10.20 for more practice with equivalent fractions and word problems.





ACTIVITY 3

Fraction Word Problems & Arrays

OverviewToday’s session extends work using array models to

multiply a whole number by a fraction. Students sketch,

discuss, and solve four story problems.

Skills & Concepts: H Write an equation showing a fraction a/b as the sum of

a number of the unit fraction 1/b (CCSS 4.NF.3)

H Write an equation to show a fraction as the sum of other

fractions with the same denominator (CCSS 4.NF.3b)

H Add fractions with like denominators

H Demonstrate an understanding that a fraction a/b is a

multiple of the unit fraction 1/b (CCSS 4.NF.4a)

H Write an equation showing that a fraction a/b is the

product of a × 1/b (CCSS 4.NF.4a)

H Multiply a fraction by a whole number (CCSS 4.NF.4b)

H Solve word problems in which fractions are multiplied

by a whole number

H Use a visual fraction area model to understand multi-

plication of fractions by a whole number

Recommended TimingUse after Supplement Set A10, Activity 2

You’ll need: H pens and paper to record on the display

H Pencil/paper for each student

H Story Problems: Multiplying Fractions page A10.16, run

a class set plus one for display



Instructions for Fraction Word Problems & Arrays1. Give each student a copy of Story Problems: Multiplying Fractions while you display the same Teacher Master. Read the first problem together, taking time to discuss and record students’ thoughts about what the problem is asking them to do. Pair-share estimates and then ask volunteers to explain their reasoning with the class. Encourage students to consider benchmark fractions.

Set A

10 Nu

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mb

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NAME DATE

Story Problems Multiplying Fractions

Estimate then draw an array to show your work.

Problem

Estimate Array and Answer

1 The principal hands out ice cream bars and popsicles at field day. The first class finished two bags of ice cream bars. Only 1

4 of that amount of ice cream bars were needed for the second class. How much ice cream did the second class eat?

2 I have 6 dogs. This week each dog gets 1

8 of a bag of dog treats. How much of the bag is eaten?

3 The cookie recipe calls for 1 3 cup

of sugar. I need to make a double batch of cookies. How much sugar do I need?

4 If everyone wants a 1 4 pound

hamburger, and we have six people in our family, how many pounds of hamburger do we need?



Teacher When you estimate, sometimes it’s helpful to think about benchmarks or numbers that are easy to imagine. In this problem I’m wondering if the answer will be closer to zero, one half, or one whole?

Scott It must be close to zero. 1/4 of 2 bags has to be a really small number.

Jacque It’s small, but I don’t think it’s that small. 1/4 of 1 bag is 1/4. So 1/4 of 2 bags has to be more. My estimate would be closer to one half.

2. Next, work with the class to sketch an array to represent the problem 1/4 × 2. Ask students to deter-mine the closest whole numbers. Have them draw a 1 × 2 rectangle, and coach you to divide it into 4 fractional parts, as they sketch the same model. Label the drawing, and shade in the area.

214

3. After students have had a chance to sketch the array, work with the class to connect to the original problem. What do the two dimensions represent? What does the area represent? Then ask students to find the area of the array they’ve drawn. Have students pair-share their thoughts before a volunteer shares with the entire class. Record multiplication equations to match student strategies.

Elsie I saw 2 groups of 1/4. So I just added 1/4 and 1/4. That’s 2/4.

Jeb I did the same thing. But I said it equals 1/2. Because 2/4 and 1/2 are equivalent.

Sarah I just multiplied. I know that you can put any number over a 1 and it’s still the same num-ber. So I said 2/1 × 1/4 and just multiplied them across. That’s 2/4.

Set A

10 Nu

mb

er &

Op

eratio

ns: M

ultip

lying W

ho

le Nu

mb

ers b

y Fraction

s Blacklin

e

NAME DATE

Story Problems Multiplying Fractions

Estimate then draw an array to show your work.

Problem

Estimate Array and Answer

1 The principal hands out ice cream bars and popsicles at field day. The first class finished two bags of ice cream bars. Only 1

4 of that amount of ice cream bars were needed for the second class. How much ice cream did the second class eat?

2 I have 6 dogs. This week each dog gets 1

8 of a bag of dog treats. How much of the bag is eaten?

3 The cookie recipe calls for 1 3 cup

of sugar. I need to make a double batch of cookies. How much sugar do I need?

4 If everyone wants a 1 4 pound

hamburger, and we have six people in our family, how many pounds of hamburger do we need?

214

12

• How are we seeing the area of this rectangle? » 1⁄4 + 1⁄4 = 2⁄4 » 1⁄4 + 1⁄4 = 1⁄2 » 1⁄2 × 14 = 2⁄4

4. With student input, label the area of each rectangle.

214

14

14

Activity 3 Fraction Word Problems & Arrays cont.



Activity 3 Fraction Word Problems & Arrays cont.

5. Repeat this process with the three other problems on the projector as students complete the prob-lems on their own sheets.


See Multiplying Fractions with Arrays on page A10.21 for more practice with estimating and sketching arrays for multiplying fractions by a whole number.


NA

ME

DA

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Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Blackline Run a class set, plus 1 for display.

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NAME DATE

Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Blackline For use anytime after Supplement Set A10, Activity 1.

Set A10 H Independent Worksheet 1


Pizza Party Planning

A fourth grade class won a pizza party for contributing the most paper in the recycling contest at school. Medium pizzas were cut into 8 slices and large pizzas were cut into 12 slices.

1 Mariah ate 2 slices of a large pizza. What fraction of the pizza did she eat? Draw a sketch to show your thinking.

2 Carlos said that Mariah ate 1 6 of a pizza. Explain why you agree or disagree.

3 Mariah’s table group seats four students. Each student ate 2 slices of a large pizza. Write a multiplication equation that shows what fraction of a pizza was eaten.



Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Blackline

NAME DATE

4 Tony ate 3 slices of a medium pizza and his friend Conner ate 4 slices of the same pizza.

a Write two equivalent fractions to tell how much pizza Conner ate.

b Write an equation to show what fraction of the pizza they ate together.

5 Lionel’s table group drank 1 1 2 liters of juice with their pizza. How many milliliters

did they drink?

Independent Worksheet 1 Pizza Party Planning (cont.)


NAME DATE




Brownie Dessert

A fourth grade class earned a brownie dessert party for the highest attendance in one grading period. Smaller pans of brownies were cut into 9 pieces and large pans were cut into 16 pieces.

1 Tori ate 2 pieces of brownie from the smaller pan. What fraction of the brownie did she eat? Draw a sketch to show your thinking.

2 Jorge ate one piece more than Tori. Write two equivalent fractions to show how much Jorge ate.

3 Ford’s table group seats five students. Each student ate 2 pieces from the large brownie pan. Write an equation that shows what fraction of the large brownie pan was eaten.

4 April ate 1 piece from the large pan and her friend Karissa ate 4 slices from the same pan.

a Write two fractions to tell how much of the large brownie pan Karissa ate.




NAME DATE

b What fraction of the large brownie pan did they eat together?

5 Frederick had 2 of the brownies from the large pan. His brother said that he ate 1

8 of the pan of brownies. Explain why you agree or disagree.

CHALLENGE

6 In a 12-egg carton, 1 6 equals 2 eggs. Use the grids below to help you imagine

Independent Worksheet 2 Brownie Dessert (cont.)

and draw cartons where:

a 1 6 is 3 eggs b 5

6 is 25 eggs

c How did you decide on the sizes of the cartons for a and b?


NAME DATE




Multiplying Fractions with Arrays

1 Sketch each array below. Then write a multiplication equation to show how you found the product.

Sketch This Array Equation

a 3 8 × 3

b 2 6 × 2

c 1 7 × 5

d 1 5 × 10

2 Write a story problem to match one of the combinations above.


NAME DATE



Company’s Coming!

1 Jamie has three people in her family. For three people, they make a recipe for Cranberry Sauce that calls for the following ingredients:• 1(12ounce)bagoffreshcranberries• 3

4 cup orange juice• 2

3 cup. brown sugar• 1

4 cup white sugar

a Jamie’s grandparents are joining them for dinner. Double Jamie’s recipe for Cranberry Sauce. Show your work and the new ingredient list below.

b Jamie’s aunt, uncle and cousin are now coming for dinner as well. Triple the original recipe. Show your work and the new ingredient list below.





NAME DATE

Independent Worksheet 4 Company’s Coming! (cont.)

2 Jamie also needs to plan for the meal. The following portions are for one person. To serve 8, how much of each food is needed? Write an equation to show your thinking.

Note: remember to label the units in your equation.

Food 1 Serving Show Your Work and Write an Equation for 8 Servings

a Vegetables 1 2 cup

b Turkey 3 ounces

c Gravy 1 4 cup

d Potatoes 1 2 cup

e Stuffing 1 3 cup

f Green Salad 3 4 cup

g Cranberry Sauce

1 4 cup

h Apple Pie 1 8 pie

grade 4 supplement - the math learning center geoboards and geobands or geoboard app (online or...

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