Day One Multiplying Fractions and Whole Numbers (using...

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Day One – Multiplying Fractions and Whole Numbers (using repeated addition and visual models)

Objective: The students will learn how to multiply a fraction by a whole number using repeated addition and visual models as strategies.

Standards Addressed:CCSS 4.NF.B.4a, bCCSS 5.NF.B.4a.

Skills: Multiplication of fractions by whole numbers.

Resources Needed: Paper/notebook for notes, pencil, copy of the Note Sheet (pg 6) for each student, copy of the Skill Practice Sheet A (pg 8) for each student, crayons (if desired for coloring the models, though pencil works just fine)

Lesson:

1) Begin by telling students that today we will be learning two different strategies to multiply whole numbers by a fraction. The first is repeated addition. The second will be using a visual model.

2) Remind students that multiplication is simply a way to add repeated numbers quickly. With that in mind, tell them that we are going to learn how to multiply fractions with a whole number.

3) Pass out the Note Sheet (pg 6) to each student (you can forgo the copies and just have them write the information in a notebook if you wish.) Once the top part of the Note Sheet is cut out, lead the students in the repeated addition model for the problem 3 x 1/5. Tell them that basically it means that 1/5 is added 3 times. So, when adding, it will look like this:

4) The visual model is slightly different, though really, not by much. Taking that same problem, have the students draw 3 rectangles. (I recommend rectangles, not circles, as circles are generally very difficult for students to draw and manipulate.) These three rectangles represent the 3 groups. Each group is then divided into fifths. Within each group, one of the bars is colored in.

5) To then add them together, all of the colored in parts need to be combined into one rectangle. Have the students draw another rectangle, divided into fifths. Then, model moving each of the individual squares over to form one cohesive colored in fraction strip.

© S.Moorman, 2014 http://teachinginroom6.blogspot.com

1/5 + 1/5 + 1/5 = 3/5

The note sheet has one more section on it (algorithm). That should be left blank for now. The students will go back to that tomorrow.

6) Cut out the bottom portion of the Note Sheet. This is a folded organizer that will be glued into the students’ notebooks. They should glue to portion with the words onto the notebook paper. The flaps will have the definitions for each strategy type on the front and an example on the inside. For this lesson, lead them through the definitions and then inputting an example for the Repeated Addition strategy and the Visual Model strategy. Leave the Algorithm part for tomorrow’s lesson.

7) Lead the students through several more problems. Using any teaching method you wish, scaffolding and releasing responsibility to students as possible. See the Practice Problem Sheet for samples to go through with your students. This sheet is intended for TEACHER use, not students. Use it as notes for yourself to guide your students. They should take notes in their notebooks based on the teaching notes you do with them.

Guided/Independent Practice: Pass out the Skill Practice Sheet A (pg 8) to the students. On this sheet, the students are asked to solve three multiplication problems using both the repeated addition strategy and the visual model strategy. Do the first one whole group. Then allow the students to complete the rest of the page either with a partner or alone.


Sample Multiplying Whole Numbers and Fractions using Repeated Addition and Visual Models to use as examples:

} }

6 x 1/4

1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 = 6/4 = 1 2/4 = 1 ½

1 2/4 = 1 ½

2/5 x 2

}2/5 + 2/5 = 4/5

4/5

3 x 5/8

}

1 7/8

5/8 + 5/8 + 5/8 = 15/8 = 1 7/8

2/3 x 4

} }

2/3 + 2/3 + 2/3 + 2/3 = 8/3 = 2 2/3


RepeatedAddition

Visual Model

Algorithm

Cut out the folded organizer below. Glue the first column down into your notebook. On the front of the loose flap, write the definition of the term. On the underside of the loose flap, draw a picture or diagram of the term.

3 X 1/5 =Repeated Addition Visual Model

Algorithm

Multiplyi

ng W

hole N

umbe

rs a

nd Fractions




1/5 + 1/5 + 1/5 = 3/5

=

RepeatedAddition

Adding the same number (in this case, a fraction) over and over again. Mimics

multiplication.

Visual Model

Using fraction strips or some other relatedpicture to model the math that is taking place.

Algorithm

Algorithm

Multiplyi

ng W

hole N

umbe

rs a

nd Fractions


Name: ____________________________________________________________ Date: ______________

Multiplying Fractions: Repeated Addition and Visual ModelsSkill Practice Sheet A

Use repeated addition AND a visual model to solve each problem. Simplify the final answer whenever possible.

4 x 1/5 3 x 4/72/3 x 5

Repeated Addition Repeated Addition Repeated Addition

Visual Model Visual Model Visual Model


Name: ____________________________________________________________ Date: ______________



4 x 1/5 3 x 4/72/3 x 5

Repeated Addition

1/5 + 1/5 + 1/5 + 1/5 = 4/5

Repeated Addition

4/7 + 4/7 + 4/7 = 12/7 = 1 5/7

Repeated Addition

2/3 + 2/3 + 2/3 + 2/3 + 2/3 = 10/3 = 3 1/3

Visual Model

=

Visual Model

=

Visual Model

=



Multiplyi

ng W

hole N

umbe

rs a

nd Fractions

Repeated Addition Visual Model

RepeatedAddition

Visual Model

Algorithm

Algorithm


Name: ____________________________________________________________ Date: ______________



Repeated Addition Repeated Addition Repeated Addition

Visual Model Visual Model Visual Model


X = = = 5 2 5 x 2 10 11 3 1 x 3 3 3

X = = = 3

Day Two – Multiplying Fractions and Whole Numbers (using the standard algorithm)

Objective: The students will learn how to multiply a fraction by a whole number using the traditional algorithm.

Skills: Multiplication of fractions by whole numbers.

Resources Needed: Paper/notebook for notes, pencil, copy of the Note Sheet (pg 6) for each student that was used in Day One, copy of the Skill Practice Sheet B (pg 14) for each student

Lesson:

1) Begin by reminding students of the two strategies for multiplying fractions that were discussed in Day One’s lesson. Ask them to lead you through the two strategies orally, as a refresher for everyone.

2) Tell students that today they will be learning how to multiply fractions using the standard algorithm.

3) Ask them to turn to the Note Sheet they glued into their notebooks during the previous lesson. Using the same sheet, they will take notes on using the standard algorithm.

4) Write the following multiplication problem on the board: 3 x 1/5. Remind them how they solved this same problem in the previous lesson. Have a few students share out what was already done. For the standard algorithm, they basically take those same ideas and put them together. The standard algorithm for multiplying fractions is

multiply numeratorsmultiply denominators

5) Since 3 is a whole number, it needs to be “turned into” a fraction. By adding a 1 underneath the 3, it turns it into a fraction, while still keeping the value intact. You can prove this to the students by performing the “improper fraction division” to it. When you divide 3 by 1, you will get 3, which shows that it keeps it’s value of 3.

6) Multiply the fractions as follows:

7) Students should write the algorithm sample on their Note Sheet. Then write a definition for algorithm on the folded organizer from the previous day’s lesson as well.

8) Do a few more examples with the students.

Guided/Independent Practice: Pass out the Skill Practice Sheet B (pg 14) to the students. On this sheet, the students are asked to solve three multiplication problems using the standard algorithm and one other strategy of choice. Do the first one whole group. Then allow the students to complete the rest of the page either with a partner or alone.


3 1 3 x 1 31 5 1 x 5 5

X = =

7 3 7 x 3 21 11 4 1 x 4 4 45

Standards Addressed:CCSS 4.NF.B.4a, bCCSS 5.NF.B.4a.



1/5 + 1/5 + 1/5 = 3/5

=

RepeatedAddition

Adding the same number (in this case, a fraction) over and over again. Mimics

multiplication.

Visual Model

Using fraction strips or some other relatedpicture to model the math that is taking place.

Algorithm The traditional formula for solving a math problem.

Algorithm

Multiplyi

ng W

hole N

umbe

rs a

nd Fractions

3 1 3 x 1 31 5 1 x 5 5

X = =


Name: ____________________________________________________________ Date: ______________

Multiplying Fractions: Standard AlgorithmSkill Practice Sheet B

Use the standard algorithm AND any other alternate strategy to solve each problem. Simplify the final answer whenever possible.

5 x 2/7 8 x 1/43/4 x 2

Standard Algorithm Standard Algorithm Standard Algorithm

Alternate Strategy Alternate Strategy Alternate Strategy


1 37

5 x 2/7 8 x 1/43/4 x 2


Alternate Strategy

2/7 + 2/7 + 2/7 + 2/7 + 2/7 = 10/7 = 1 3/7

Alternate Strategy

1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 = 8/4 = 2

Alternate Strategy

3/4 + 3/4 = 6/4 = 1 2/4 = 1 ½

5 2 5 x 2 101 7 1 x 7 7

8 1 8 x 1 81 4 1 x 4 4

3 2 3 x 2 64 1 4 x 1 4

X = = X = = X = =

12 12

24

1 =

=

=

=

Name: ____________________________________________________________ Date: ______________



= = =


Name: ____________________________________________________________ Date: ______________




Alternate Strategy Alternate Strategy Alternate Strategy


Day Three – Multiplying Fractions and Fractions (using the algorithm and area models)

Objective: The students will learn how to multiply a fraction by another fraction using the standard algorithm and the area model.

Standards Addressed:CCSS 5.NF.B.4/5.NF.B4aSkills: Multiplication of fractions

by another fraction.

Resources Needed: Paper/notebook for notes, pencil, copy of the Note Sheet (pg 20) for each student, copy of the Skill Practice Sheet C (pg 21) for each student, two different colors of crayons.

Lesson:

1) Begin by telling students that today we will be learning two different strategies to multiply one fraction by another fraction. The first is the algorithm. The second will be using an area model.

2) Remind students that multiplication is simply a way to add repeated numbers quickly. With that in mind, tell them that we are going to learn how to multiply fractions with a fraction.

3) Pass out the Note Sheet (pg 20) to each student (you can forgo the copies and just have them write the information in a notebook if you wish.) Once the top part of the Note Sheet is cut out, lead the students through the algorithm model for the fraction noted. Tell them that it is EXACTLY the same process as multiplying whole numbers by fractions, without placing the 1 underneath. (since both numbers are actually already written in fraction form.) Since they already know this process from the day before, there is really no need to harp on it. They really do understand the algorithm formula and can apply it to fractions times fractions.

4) Do 2 or 3 more examples that the students can write in their journals.

5) The area model is slightly different, though really, not by much, from the visual model used in whole numbers multiplied by fractions. Taking that same problem, have the students draw 2 rectangles. (I recommend rectangles, not circles, as circles are generally very difficult for students to draw and manipulate.) These two rectangles represent the 2 fractions. The first rectangle should be divided VERTICALLY into thirds. Within each group, two of the bars are colored in. The second rectangle is divided HORIZONTALLY into fifths. Within each group, two bars are colored in.

6) Then, overlay each of the rectangles on top of each other. This creates an array (just like when doing a simple multiplication problem!) Keep the pieces colored the way they were when separated.


2/32/5

7) Using that model, you can now work on finding the answer to the multiplication sentence. The portion of the array that is taken up by BOTH colors, is the numerator.

22

2 x 2 = 4

8) Using the same model, the TOTAL AMOUNT OF BOXES is the denominator.

3

5 x 3 = 155

9) The entire fraction is then put together.

2/3 x 2/5 = 4/15

10) Do three more model examples with the students. See the “Example Sheet”. This is intended for your reference and to be used to show step by step examples with the students. If you wish to make them a copy you can, however, it is NOT intended that way.

11) Cut out the bottom portion of the Note Sheet. This is a folded organizer that will be glued into the students’ notebooks. They should glue to portion with the words onto the notebook paper. The flaps will have the definitions for each strategy type on the front and an example on the inside. For this lesson, lead them through the definitions and then inputting an example for the Area Model strategy and the algorithm strategy.

Guided/Independent Practice: Pass out the Skill Practice Sheet C to the students. On this sheet, the students are asked to solve three multiplication problems using both the repeated addition strategy and the visual model strategy. Do the first one whole group. Then allow the students to complete the rest of the page either with a partner or alone.


Sample Area Model Problems:

1/6 x 1/4

4

} }

=

1

1

6

}

1/24=

2/3 x 1/2

x

x =

2/61/3= =

3/4 x 3/8

x

= 9/32

3

3

4

8

1

2

2

3

}} }

}

}

}

}

}}

=


Area Model

Algorithm


2/3 X 2/5 =Area Model

Algorithm

Multiplyi

ng Fractions

with

Frac

tion

s


Area Model

A visual model that uses rectangles divided into rows and columns to demonstrate

multiplication.

AlgorithmThe traditional formula for solving a math

problem.


2/3 X 2/5 =Area Model

Algorithm

Multiplyi

ng Fractions

with

Frac

tion

s

2/3 x 2/5 = 4/15

=

2

2

3

5}}}}

= 4/15


Name: ____________________________________________________________ Date: ______________

Multiplying Fractions: Area Model and AlgorithmSkill Practice Sheet C

Use the standard algorithm AND the area model to solve each problem. Simplify the final answer whenever possible.

Problem Standard Algorithm Area Model

1/2 x 3/5

4/5 x 3/8

1/6 x 1/3

5/8 x 1/9

2/3 x 1/7


Name: ____________________________________________________________ Date: ______________

Multiplying Fractions: Area Model and AlgorithmSkill Practice Sheet C

Use the standard algorithm AND the area model to solve each problem. Simplify the final answer whenever possible.

Problem Standard Algorithm Area Model

1/2 x 3/5

4/5 x 3/8

1/6 x 1/3

5/8 x 1/9

2/3 x 1/7

1 x 3 32 x 5 10

4 x 3 12 35 x 8 40 10

1 x 1 16 x 3 18

5 x 1 58 x 9 72

2 x 1 23 x 7 21

= =

=

=

=

= =

1

3

5

2

=

4

38

5

3

6

1

1

=

=

1

2

7

3© S.Moorman, 2014 http://teachinginroom6.blogspot.com

Day Four – Multiplying Fractions in Context (Word Problems)

Objective: The students will solve real world problems involving multiplication of fractions.

Standards Addressed:CCSS 4.NF.B.4CCSS 5.NF.B.4, 5.NF.B.6

Skills: Multiplication of fractions by whole numbers, multiplication of fractions by fractions, Word problems involving multiplication of fractions.

Resources Needed: Paper/notebook for notes, pencil, copy of all Task Cards #1-16 (pgs 25-28) for each group of students, copy of Task Card Recording Sheet for each student

Lesson:

1) Begin by telling students that today they will be applying the strategies that they have learned for multiplying fractions, using a set of task cards and real world problems.

2) Pass out the Task Card Recording Sheet to each student. Then, choose one of the task cards to demonstrate how to solve the problems on the recording sheet. Students should show ALL of their work, regardless of strategy, inside the space provided. There are no explicit instructions on cards #1-12 for solving the problems. The students need to decide which strategy would work best for them.

3) Break the students into groups based on the number of Task Card sets that you copied. If you have 4 sets copied, you should break your students into 4 groups. 8 sets of Task Cards would call for 8 groups, etc…

4) Give each group one set of Task Cards. They should work together to solve each Task Card on the recording sheet.

Guided/Independent Practice: The students should work together, as a group, until all of the Task Cards are complete. Then review together as a class.

NOTE: After this lesson is complete, these task cards would work well as a review station in a math center.


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1 2 3 4

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Name: ____________________________________________________________ Date: ______________

Multiplying Fractions In ContextTask Card Recording Sheet

Use whichever strategy you choose to solve the task card problems. Solve each problem within the matching number space below. Simplify the final answer whenever possible.


1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

Name: ____________________________________________________________ Date: ______________

Multiplying Fractions In ContextTask Card Recording Sheet

Use whichever strategy you choose to solve the task card problems. Solve each problem within the matching number space below. Simplify the final answer whenever possible.

Students should show all work inside the correct

number box.

8 chocolates 8 baseball cards 25 dogs 9 red erasers

12 ribbons 7 chocolates 20 cookies 18 minutes

1/6 of the cake 1/6 of cup of oil 1/12 of the cars 3/8 of the teachers

¾ + ¾ + ¾ + ¾+ ¾ = 15/4 = 3 ¾

5/1 x ¾ = 15/4 = 3 ¾

¾ x 1/6 7 x 1/3


Day Five – Multiplying Fractions Test Prep(Multiple Choice and Constructed Response)

Objective: The students will apply what they learned about multiplying fractions to answer test prep questions.

Skills: Multiplication of fractions by whole numbers, multiplying fractions by fractions, word problems, constructed response.

Resources Needed: Copy of Multiplying Fractions Test Prep Multiple Choice (pg 32) for each student, copy of Multiplying Fractions Test Prep Constructed Response (pg 34) for each student.

Lesson:

1) Begin by reminding students of all they have learned about multiplying fractions. Review the three basic strategies for solving multiplication of fraction problems. Students can look back in their notes from the previous 4 days.

2) Today, students will be using all they know about multiplying fractions and doing some test prep as well.

3) Pass out the first multiple choice test prep page. Have students either do it by themselves or with partners. They should show all of their work. (You can also assign this as an actual assessment if you wish.)

4) Go over the Multiple Choice test as a class. The review will help them to really cement the ideas in their minds.

5) Pass out the Constructed Response page. You can give this as an assessment, or have them work in partners to talk it all out. It is up to you how you wish to proceed with this page.

6) Review the Constructed Response page as a class.


Standards Addressed:CCSS 4.NF.B.4a, bCCSS 5.NF.B.4a., 5.NF.B.6

Name: ____________________________________________________________ Date: ______________

Multiplying Fractions: Test PrepMultiple Choice

Directions: Use whichever strategy you choose to solve the problems below. Simplify the final answer whenever possible.

1. Mrs. Raymer was baking a cake. The recipe called for 3/4 cup of butter. She only wanted to put 1/2 of that amount in her cake. How many cups of butter did Mrs. Raymer put in her cake?

A. 3/4 cups

B. 3/2 cups

C. 1/4 cups

D. 3/8 cups

2. Thomas has 40 chocolate bars. If 2/5 of them contain almonds, how many of the chocolate bars have almonds in them?

A. 8 bars

B. 16 bars

C. 20 bars

D. 25 bars

3. 5/6 x 8 =

A. 6 2/3

B. 4 1/8

C. 3 5/6

D. 6 1/2

4. Which visual model represents 3 x 1/4?

A. B.

C. D.

5. Which area model would help you solve 3/5 x 1/2?

A. B.

C. D.


Name: ____________________________________________________________ Date: ______________

Multiplying Fractions: Test PrepConstructed Response

Use whichever strategy you choose to solve the problems below.Simplify the final answer whenever possible.

Solve.

5 x 1/6

Karla was making a quilt. She sewed 10 squares together in each row. If each square was 3/4 of a foot long, how long

was each row on the quilt? Explain your solution using pictures, words, and numbers.

Solve.

3/8 x 7

Gregory is creating a painting for a client. The client wants 1/2 of the painting to be all red. The other half should be painted blue. The client also wants 1/2 of the red

to have yellow polka dots, while 4/5 of the blue should have orange stripes. What fraction of the entire painting does NOT have orange stripes or yellow polka dots? Explain your

solution using pictures, words, and numbers.


Name: ____________________________________________________________ Date: ______________

Multiplying Fractions: Test PrepConstructed Response

Use whichever strategy you choose to solve the problems below.Simplify the final answer whenever possible.

Solve.

5 x 1/6

Karla was making a quilt. She sewed 10 squares together in each row. If each square was 3/4 of a foot long, how long

was each row on the quilt? Explain your solution using pictures, words, and numbers.

Solve.3/8 x 7

Gregory is creating a painting for a client. The client wants 1/2 of the painting to be all red. The other half should be painted blue. The client also wants 1/2 of the red

to have yellow polka dots, while 4/5 of the blue should have orange stripes. What fraction of the entire painting does NOT have orange stripes or yellow polka dots? Explain your

solution using pictures, words, and numbers.

The portion of the painting that is red without polka dots is ½ of ½. That is ½ x ½, which equals ¼ of the painting.

The portion of the painting that is blue without stripes is 1/5

of ½. That is 1/5 x 1/2, which equals 1/10 of the painting.

If you add the portion of the painting that has no polka dots to the portion of the painting that has no stripes, you would have ¼ + 1/10, which equals 7/20 of the painting.

The answer is 7/20 of the painting.

3/43/4

3/43/4

3/43/4

3/43/4

3/43/4

3/4 + 3/4 + 3/4 + 3/4 + 3/4 + 3/4 + 3/4 + 3/4 + 3/4 = 3/4 +

30/4 = 7 2/4 = 7 ½ feet

Each of the squares is ¾ of a foot long. By putting 10 of them together, that is a repeated addition problem of ¾ ten different times. When ¾ is

added 10 times, it comes out to 30/4, which simplifies to 7 ½ feet .

3/8 + 3/8 + 3/8 + 3/8 + 3/8 + 3/8 + 3/8 = 21/8 = 2 5/8

1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 5/6

5/1 x 1/6 = 5/6

3/8 x 7/1 = 21/8 = 2 5/8


Day One Multiplying Fractions and Whole Numbers (using...

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