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Fifth Grade CurriculumFraction Multiplication & Division

Table of ContentsTopic Page

Multiplying Fractions

Background Information Part 1 Pictorial Multiplication – Putting Together Equal Sets

Part 2 Pictorial Multiplication – Find Part of a Whole/Set

12 - 5

5 – 10

Dividing Fractions

Part 3 – Multiplication Algorithm/Dividing a whole number by a fraction

Part 4 – Dividing a unit fraction by a whole number

Additional Activity:Mixed Practice Sort Directions

Division of Fractions Answer Key– Benjamin Buck’s Picnic

10 – 14

14 – 17

17

18 - 19

Multiplying & Dividing Fractions

Property of Cy-Fair ISD Elem. Math Dept. (5th Grade) 2014 – 2015

TEKS: 5.3 Number and operations. The student applies mathematical process standards to develop and use strategies and methods for positive rational number computations in order to solve problems with efficiency and accuracy. The student is expected to:(I) represent and solve multiplication of a whole number and a fraction that refers to the same whole using objects, and pictorial models, including area models. (Supporting)(J) represent division of a unit fraction by a whole number and the division of a whole number by a unit fraction such as 1/3 ÷ 7 and 7 ÷ 1/3 using objects and pictorial models, including area models (Supporting)(L) divide whole numbers by unit fractions and unit fractions by whole numbers (Readiness)

Materials: MATH_5_A_MULTIPLY DIVIDE FRACTIONS 2014_RES.NOTEBOOK MATH_5_A_MULTIPLY FRACTIONS PART 1 AND PART 2 RECORDING SHEET 2014_RESMATH_5_A_MULTIPLY FRACTIONS IMN ACTIVITY 2014_RESMATH_5_A_DIVIDE UNIT FRACTIONS PRACTICE 2014_RESfraction towers manipulatives, student white boards & markers, two color counters, IMN

Vocabulary: factor, times, product, multiple, multiply, repeated addition, reciprocal, unit fraction, divide, quotient,

Multiplying Fractions:Background: Earlier in the year students had experience multiplying decimals. Multiplying a whole number by a fraction will be similar. Fractions and the number(s) to the right of a decimal point represent a quantity less than 1 whole.

Multiplication of a whole number and a fraction can be represented in a story problem 2 different ways.

1. Putting together equal sets of fractions - An example of this is when

you have an expression such as 5 x . There are 5 groups of . This ties back to putting together equal sets and repeated addition. A story problem

might refer to 5 plates with pound of grapes on each plate.

2. A part of a whole/set - An example of this is when you have an

expression such as x 5. This represents of the whole/set. A story problem

might refer to placing of a 5 pound bag of grapes on a plate.

Property of Cy-Fair ISD Elem. Math Dept. (5th Grade) 2014 – 2015 1

Using the commutative property, the algorithm x 5 can be written as 5 x . The difficulty may lie in identifying the correct operation as multiplication in the second example listed above. We have separated multiplication into these to two parts based on the action/operation as they solve story problems.

Part 1 – Putting together Equal SetsThe focus on this day is reviewing and building on previous knowledge of Putting Together Equal Sets. As you do the SB slides with your students, you will be able to:

review vocabulary review/identify the action: Putting Together Equal Sets write equations and expressions

Discuss each step of the Four-Step Process as you solve each problem. You can do steps 1 and 4 orally. Steps 2 and 3 will be recorded on the student recording sheet.

Slide 1Title page

Slide 2Read the following problem:

Jeri has 3 bags of candy. Each bag weighs 2 pounds. How many pounds of candy does Jeri have?

Identify the Main Idea.Represent the story by drawing a model. Discuss the action taking place. (Putting Together Equal Sets)How could you find the number of pounds of candy that Jeri has? Multiply 3 groups of 2 pounds. 3 x 2Discuss the how/justify. Slide 3 Note: As you complete the next section with your class you will be guiding students to complete the chart for each of the 4 problems. These are the steps to go through:


As you go through each problem it is important that students are actively participating in the lesson through your questioning.

Read the following problem:

1. Jeri has 3 bags of candy. Each bag weighs pound. How many pounds of candy does Jeri have?

Identify the Main Idea. Have students use their fraction towers to illustrate this story. They

should have 3 halves in front of them. Discuss the action taking place. How could you find the number of pounds of candy that Jeri has?

Put together 3 groups of . Place 2 halves together to make a whole. Now students will have 1 whole and 1 half.

Model- Let’s draw our model. How many pieces should there be in each model? Why? Each model should have 2 pieces because the

denominator in is 2. There are 2 equal pieces in each whole. Complete the model section of the recording sheet. Alternate directions or colors as you shade each fraction (see answer key). This will help students distinguish each part that is being shaded. This is especially helpful when there are smaller fraction pieces.

*Models may vary.

Number line – This is another way to represent the story. How many wholes are on this number line? 4

Count the lines with the students 1…2…3…4. How many equal parts should each whole be divided into? 2. How do you know? The denominator is 2, so there should be 2 equal parts. In order to show this expression on a number line, we will show “hops” for each fraction. How many times will we have to “hop” on the number line? 3 How

long will each “hop” be? because our expression is asking for 3

groups of .


0 1 2 3

Addition Expression – What is a way this can be written as an

expression using repeated addition? + + . Multiplication Equation - What is a way this can be written as an

expression using multiplication? 3 x = 1Record the equation in the box on the recording sheet. Discuss the how/justify.

Slides 4 -6Repeat this process with your students on slides 4-6. All of these problems have the action of Putting Together Equal Sets. We have included an answer key for your reference. It is important to keep the action in mind as you write the expression and equation, not the order in which the numbers appear in the story. In Part 1, the whole number comes first because it reflects the action of

Putting Together Equal Sets (3 groups of ). Glue record sheet in IMN. Discuss each step of the Four-Step Process as you solve each problem. You can do steps 1 and 4 orally. Steps 2 and 3 will be recorded on the

student recording sheet.

Slide 7 - Closure3 × 0.25 and 3 × ¼

Compare and contrast the 2 algorithms. As you discuss include:

Alike multiplying a whole number by a number with a value less than 1 the products have the same value


representation of the same action (3 groups of the same value) Putting Together Equal Sets Both ¼ and 0.25 are benchmarks

Different one has a fraction and the other is a decimal answers could be in different forms (decimal and fractions)

IMN ActivityDo Set A after completing the SB slides. See the IMN document for detailed instructions.

Part 2 – Finding a Part of a Whole/SetThe focus on this day is reviewing and building on previous knowledge of Finding a Part of a whole/set. As you do the SB slides with your students, you will be able to:

review vocabulary review/identify the action: Finding a Part of a Whole/Set write equations and expressions

Slide 8Practice writing division as a fraction. Reveal each division expression and ask students to write them on their white boards.

Examples on SB:

“Write this division statement as a fraction. 4÷2.” Students should write .




“Write this division statement as a fraction. 20÷4.” Students should write . “Solve the division problem.” Students should write 20÷4 = 5

Continue this process for a few minutes to ensure your students understand that fractions are another way to write a division problem. Suggestions include: six thirds, five ninths, nine thirds, eight fourths, twelve fourths, three fifths.

Slide 9Ask students to arrange 6 counters in an array on their white board, all turned to the same side color. (Option: you may ask students to draw the circles. Additionally, you could have students record notes in their IMN instead of using white boards.)


42

64342

10204

Ask students to draw lines to show three equal parts. Draw lines on the slide. Ask students to write an expression to indicate what they have just done. Students should write 6÷3. Encourage students to rewrite the division sentence

as a fraction ( )

How many equal parts are there? 3

If I want to show of this set, how many counters would I flip over? 2

(or shade in, if students drew them on the boards.) Tap the two circles on the left and they will shade in. Encourage students to talk to their partner to see if they got the same answer, then come back as a whole group.

Students should indicate that each group is one third of all the counters, so we would turn over 1 group of two counters. Students should also see that 6 divided by 3 tells us that there are 2 in each group. I have 6 counters and we have them divided into 3 groups so, what is

of 6? 2

On the board, write: of 6 = 2. So, we have shaded a part of this set of 6 counters. This action is Finding a Part of a Set/Whole.

How is this action shown in an equation? Multiplication

How would this be written as an equation? x 6 = 2.


63

13

13

13

13

Label the Smart Board, write: x 6 = 2

Continue by asking: How many counters would I flip over (or shade in) to

show of this set? Students should indicate that since they know that one third of six is equal to 2, two thirds of six would be equal to four.

On the board, write of 6 = ____. I have 6 counters and we have them divided into 3 groups. I shaded

in two groups. What is of 6? 4

of 6 = 4

What action happened when I shaded the four circles? Finding a Part of a Set/Whole

How is this action shown in an equation? By using the operation of multiplication

Write the equation on the board. It should look like this:

of 6 = 4

x 6 = 4


13

23

23

23

23

23

What is three thirds of 6? On the board, write of 6 = 6. Students should be able to answer, “6.” Have students turn to a partner and explain why. Let’s write this as an equation.

of 6 = 6

x 6 = 6

Slide 10Use the circles on the board to facilitate the conversation.

Continue the conversation using of twelve. Students may use colored counters or draw them on the white boards. Have students draw lines to represent division.

Ask students to make an array using 12 counters all on the same color. Ask them to divide the array into fourths by drawing lines.

How many counters are in each fourth? 3

Have students write the division equation as a fraction. What is one fourth of 12? 3

On the board, write: of 12 = 3. What action is taking place? Finding a Part of a Set/Whole

How is this action shown in an equation? By using the operation of multiplication

Write the equation on the board. It should look like this:

of 12 = 6

x 12 = 6

One fourth of 12 is equal to 3. What fraction of 12 is equal to 6

counters/circles? Students should see that it is . Remind students that since one fourth is equal to 3, and six is twice as much, we can double one fourth to get two fourths. Write on the board:


33

33

33

14

12 34

14

1414

24

of 12 = 6.

What fraction does equal to when simplified? One-halfI have 12 counters and we have them divided into 4 groups. I shaded

in two groups. What is of 12? 6 What action is taking place? Finding a Part of a Set/WholeHow is this action shown in an equation? MultiplicationWrite on the board:

of 12 = 6

x 12 = 6

Slide 11Use the models on the board to facilitate the conversation. Tell students there are two number sentences that can describe this model. Encourage your students to come up with a division equation to describe the model. Tap on the “Equation 1” box to reveal the division equation.

After verifying their equations, ask students to write a multiplication equation, or number sentence, that fits the model. Students may not immediately connect the “of” we have been using to mean “multiplication.” Encourage them to think about the action of Finding a Part of a Set/Whole. Tap on the “Equation 2” box to reveal the multiplication equation.

Note: Multiplication can be interpreted as “groups of,” or in the case of fractions, “part of.” However, please do not say that “of” ALWAYS means multiply. While this is frequently the case with fractions, it can be misleading. Students should always analyze the context of the problem using the problem solving process and identify the action before deciding the problem indicates multiplication, and not make their decisions just because the word “of” is present (or not present).

Slide 12Ask your students to write the two equations that fit the model on their white boards. Have them show you their equations to verify that they are correct.

Slide 13 Use the model on the board to facilitate the conversation. Ask students how this model is different than what we have seen so far. Remind students that a multiplication problem can be reversed and the answer will stay the same. See if students can generate both equations to represent this model. On the smart file, the shaded pieces can be pulled together to create 2 wholes.


24

24

( )

Slides 14 – 17Guide students through these slides as you did the previous day. See the answer key for an example of how the recording sheet can be filled out. Glue recording sheet in IMN.


The last slide is an example of the action that was taught on the previous day:

Putting Together Equal Sets. It is written with a similar story as the slide that precedes it so that you are able to compare and contrast them. Keep in mind that the focus is to help students identify that both of the stories would be solved with multiplication.

IMN ActivityDo Set B after completing the SB slides. See the IMN document for detailed instructions.

Part 3 - Multiplication Algorithm/Dividing whole number by a unit fractionSlide 18 - Recap, review and make connectionsStudents will need their Multiplying Fractions Part 1 recording sheet

Ask students to describe the model shown.


18 2 8 4 24

Label the model and review fraction basics. The numerator represents a number of equal parts and the denominator represents how many of those parts make up a unit or a whole. For example, in the fraction 3

4 , the numerator, 3, tells us that the fraction represents 3 equal parts and the denominator, 4, tells us that 4 parts make up a whole. (Use model shown to explain to students) Pull down the shade to reveal 5 wholes. Can whole numbers be expressed as fractions? YesHow would we write 5 as a fraction? In the above fraction the 4 represented the number of parts to make the whole. If we look at our model

below, how many parts are in one whole? One And how many are shaded? One Label the fraction: 1

1 Do this for all five wholes shown on the board. Now we have labeled the expanded form for our model. Let’s add!

Let’s take a look at how this can help us to multiply fractions. Refer back to Multiplying Fractions Part 1 Student Recording Sheet from the previous day. Guide students to solve the addition expression. 12 + 1

2 + 12 = 3

2

Guide students to solve the multiplication equation. In order to multiply fractions, they both need to be in fraction form. First, let’s set up our equation. How do we express the whole number 3 as a fraction? 3

1


Continue setting up your equation by multiplying by one half. Have students record in the multiplication equation on their recording sheet. 3

1 × 12

You can see you are multiplying the numerators and multiplying the denominators. Solve for product. Make sure you simplify your answer. Let your students solve #2-4 on their recording sheet using multiplication algorithm. Dividing FractionsSlide 19Use these slides to review with students recognize that dividing by a whole number and multiplying by its reciprocal are the same thing. Slide 19 defines the word “reciprocal.”

Reciprocals are best defined as two numbers whose product is 1. To help kids see this for whole numbers, we put 1 over the number to create a reciprocal. For example:

and 8 are reciprocals because (One eighth of 8 is 1.)

and 4 are reciprocals because

(One fourth of 4 is 1.)

Remind students that a whole number can be written as a fraction over 1, so if we “flip” or “invert” the whole number or “flip” or “invert” the unit fraction, we can go back and forth between the reciprocals in a pair.

See if students can generate other examples of reciprocal pairs, and then ask students if they can think of a “generalization” to accompany this new vocabulary word. Example: Any number multiplied by its reciprocal equals 1

Divide Whole Numbers and Unit Fractions (Benjamin Buck)Throughout the first part of the lesson, money is used to represent a fractional part of a whole. Money is most often represented as a decimal value, but it will be good practice to convert the decimal to a fraction for the purposes of this lesson.

Slides 20 - 24 discuss dividing a whole number by a unit fraction.Unit Fraction is a fraction with numerator of 1.

Students will need white boards or IMN to record in. Discuss each step of the Four-Step Process as you solve each problem. You can do steps 1 and 4 orally. Steps 2 and 3 will be recorded on the student recording sheet.


18

1 8 18

14

1 4 14

Slide 20Guide students through the scenario and ask students to talk to a partner to come up with the expression and write it on their white boards. Use the model and divide the 10 squares into to 5 groups of 2. Some students may also

write the expression as .

Encourage students to write the phrase “How many groups of___are in ___?” at the bottom of the slide on their white board, filling it in with the appropriate numbers. Remind students that division can be thought of as “How many groups of 2 are in 10?”

Slide 21 Guide students through the scenario and write the fraction that represents 50

cents as . Discuss the action of take away equal sets.

Ask students to talk a partner to come up with the expression and write it on their white boards. Ask students to think about whether their answer will be larger or smaller than 10. Students should see that the answer will be larger than 10.

When you tap the model, it will show the 10 pieces each split in half.

Encourage students to write the phrase “How many groups of___are in ___?” at the bottom of the slide on their white board, filling it in with the appropriate numbers. Remind students that division can be thought of as “How many

groups of are in 10?” Kids may understand this better without the “groups

of.” You may choose to re-phrase as “How many s are in 10?”

Slide 22Guide students through the scenario and write the fraction that represents 5

cents as . Discuss the action of take away equal sets.

Ask students to talk to a partner to come up with the expression and write it on their white boards. Ask students to think about whether their answer will be larger or smaller than 10. Students should see that the answer will be larger than 10.


10 2

102

12

110 2

12

12

120

110 20

When you tap the model, it will show the 10 pieces each split in 20ths.

Encourage students to write the phrase “How many groups of ___are in ___?” at the bottom of the slide on their white board, filling it in with the appropriate numbers. Remind students that division can be thought of as “How many

groups of are in 10?” Kids may understand this better without the “groups

of.” You may choose to re-phrase as “How many s are in 10?”Slide 23Previous slides were division using models. Now we are going to use the algorithm. Use this slide to summarize. What do kids notice about what happens when they divide a whole number by a fraction? Remind them about reciprocals. “Dividing by a number and multiplying by its reciprocal will give us the same answer.” Pull the shade down to reveal each expression and its reciprocal.

Kids should be able to look at the patterns to recognize the standard algorithm. The standard algorithm indicates dividing by a number and multiplying by its reciprocal yield the same answer.

Slide 24Students should complete the four problems on their white boards. Be sure to discuss how they got their answers. Use the reciprocal to help you solve.

4 x 21 = 8 4 × 3

1 = 12 4 × 51 = 20

Benjamin Buck has $5. How many boxes can he make if each box has a

quarter in it?

Optional: Students can complete #1-3 from the Division Practice. These

problems are located at the end of the Smart Board for you to model if needed.


5 × 41

= 20

120

120

110 2 5 10 52110 20 10 2 202110 200 10 20 20020

1 1 14 4 42 38 12 5 20

15 204

Part 4 – Dividing a unit fraction by a whole numberDiscuss each step of the Four-Step Process as you solve each problem. You can do steps 1 and 4 orally. Steps 2 and 3 will be recorded on the student recording sheet.

Slide 25

Guide students through the scenario and ask them to identify the action share

a set equally and generate the expression . Some students may also write

the expression as

.

Encourage students to write the “How big is each group when ____ is evenly split ____ ways?” phrase at the bottom of the slide on their white board, filling it in with the appropriate numbers. Encourage students to see that division can be thought of as “How big is each group when 1 is evenly split 2 ways?”

Slide 26Guide students through the scenario and ask them to identify the action share

a set equally and write the fraction , and then generate the expression on their whiteboards.

Encourage students to write the “How big is each group when ____ is evenly split ____ ways?” phrase at the bottom of the slide on their white board, filling it in with the appropriate numbers. Remind students that division can be thought

of as “How big are the group when is evenly split 2 ways?”

Ask students to think about whether their answer will be larger or smaller than

. They should see that it will be smaller than the dividend.

When you tap the model, you can see that one tenth, when divided into two pieces, is only one twentieth of the whole.

Slide 27Guide students through the scenario and ask them to identify the action share

a set equally and write the fraction , and then generate the expression

on their whiteboards.


1 2

12

110

1 210

110

110

1100

1 2100

Encourage students to write the “How big is each group when ____ is evenly split ____ ways?” phrase at the bottom of the slide on their white board, filling it in with the appropriate numbers. Remind students that division can be thought

of as “How big is each group when is evenly split 2 ways?”

Ask students to think about whether their answer will be larger or smaller than

. They should see that it will be even smaller than the previous answer.

Use the model to illustrate that dividing one one-hundredth by two creates an even smaller piece.

Slide 28Use this slide to see what kids notice about what has happened when they divide a fraction by a whole number. Remind them about reciprocals. “Remember from yesterday that we learned that dividing by a number and multiplying by its reciprocal will give us the same answer.” Pull the shade down to reveal each expression and its reciprocal.

Encourage kids to see the change taking place in the denominator as the size of the quotient gets smaller. This should further their understanding of why the standard algorithm works.

Slide 29Students should complete the four questions on their white boards. Be sure to discuss how they got their answers. Use the reciprocal to help you solve.

Benjamin Buck has a fourth of a pound of diamond dust. What fraction of a

pound will be in each box if he wants to make 5 boxes? 14 ×

15 =

120


1 ×

12

14 ×

12 = 1

8 1

4 × 1

3 = 1

12

1100

1100

1 11 2 10 22 21 1 1 1 1210 20 10 2 201 1 1 1 12100 200 100 2 200

1 1 12 31 1 18 1 44 4 2 164

1 154 20

Slide 30Use this slide to compare and contrast the similarities and differences between how the two types of division problems are alike and how they are different.

Students should recognize that the equations on the top generate answers larger than the divisor and equations on the bottom generate answers smaller than the dividend. Students should also recognize that dividing with fractions is really just multiplying by a reciprocal.

Slide 31 – Closure½ ÷ 2 0.5 ÷ 2

Compare and contrast the 2 algorithms. As you discuss include:

Alike dividing a number with a value less than 1 by a whole number the quotients have the same value representation of the same action Share a set equally Both ½ and 0.5 are benchmarks

Different one has a fraction and the other is a decimal answers could be in different forms (decimal and fractions)

Slides 32 - 37Finish Benjamin Bucks Picnic Activity. These problems are included in the Smart Board for you to model for your students if needed. Use these slides for guided, partner, and independent practice along with the student document, which may be used at teacher discretion. An answer key is provided at the end of this lesson document.


Additional Activity:Mixed Operation Sort


MATH_5_A_MULTIPLY DIVIDE FRACTIONS MIXED PRACTICE SORT 2014_RES

Run one-sided partner/table copies of the problems. Each table group will place the symbols (×÷+-) on their table with dry erase. Cut problems into individual strips for students to sort them. Students will perform steps 1 and 2 of the 4 Step Process to determine the action and place the story problem under the correct symbol (operation). After monitoring and checking for understanding, you can determine the number of problems you would like the students to finish solving.

Part 3 & 4: Benjamin Buck’s Picnic - Division of Fractions - KEY

1. At the picnic, Benjamin has four ears of corn on his tray. He places an ear of corn on each plate. How many plates will get corn?

Which of the following could be used to determine the number of plates that will receive corn? (Circle one)

A. B. C. D.

Benjamin will be able to put corn on ____8____ plates.

2. Benjamin has 3 pounds of ground turkey meat to make turkey burgers. He wants to

make each turkey burger pound.

Which of the following could be used to determine the number of turkey burgers Benjamin can make? (Circle one)

A. B. C. D.

How many burgers can Benjamin make from 3 pounds of ground turkey? ____12________


3. Benjamin’s sister made eight trays of brownies to share at the picnic. When they come out of the oven, each brownie tray is cut into fifths as shown.

Which of the following could be used to determine the number of individual brownies Benjamin’s sister has to share? (Circle one)

A. B. C. D.

How many individual brownies can Benjamin’s sister get from 8 trays? _____40_______

4. One of Benjamin’s three dogs grabbed a bag with cup baby carrots from the table. All three of the dogs consumed an equal amount of the carrots. What amount of carrots did each dog eat?

5. At the picnic, a bowl contains 15 cups of potato salad. A serving size of potato salad is

cup. How many servings of potato salad are in the bowl?

6. On of Benjamin’s friends brought a pound bag of candies. He shares the candies evenly among four children. How many pounds of candies did each child get?

7. To make a cheese dip, one of Benjamin’s friend’s uses pound of cheddar cheese. If 6 friends share the cheese dip evenly, how much cheddar cheese does each person eat?

Matching - Match each scenario with the correct expression (not all will be used).

__G_8. Benjamin’s aunt has 10 yards of ribbon to make bows for the tables. Each bow requires

yard ribbon. How many bows can she make?

__ F _9. Benjamin’s cousin has 4 sacks of crawfish. Each serving of crawfish is of a sack.

How servings of crawfish are there?


__A__10. Benjamin plans to make jump-ropes. He has 40 feet of rope, and each jump-rope is

10 feet long. How many jump-ropes can he make?

__C__11. A recipe for salsa calls for 4 cups of diced tomatoes. Benjamin’s uncle only owns

one measuring cup and it holds cup. How many cups of diced tomatoes will Benjamin’s uncle have to put into the recipe?


A. B. C. D.

E. F. G.

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