Secondary Final Module Completed.pdf

7/30/2019 Secondary Final Module Completed.pdf

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Module on Dividing Fractions for Grades 6-9

Authors: Murray Key, Macon Co. R-IV

Georgia Jackson, Chillicothe R-IILana Dawkins, Rennick R-V

Brian Sherrow, Marceline R-V

Table of Contents

Pretest Form A

Lesson 1: Dividing Natural Numbers by Natural Numbers

and Dividing Proper Fractions by Natural Numbers

Lesson 2: Dividing Natural Numbers by Proper Fractions

Lesson 3: Dividing Proper Fractions by Proper Fractions

Lesson 4: Dividing Improper Numbers by Proper Fractions

and Mixed Numbers by Proper Fractions

Posttest Form B

Module development was partially funded by the Missouri Coordinating Board forHigher Education through the Eisenhower Professional Development Program.

Translations were partially funded by NSF ESIE SGER Project 0086580


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Math Module OutlineStrand: FractionsGrade Level Span: Grades 6-9Concept(s) Included in Module: Dividing proper fractions and mixed numbers

Author(s)/Districts: Murray Key (Macon Co. R-IV), Georgia Jackson (Chillicothe R-II)Lana Dawkins (Rennick R-V), Brian Sherrow (Marceline R-V)

Brief Statement of Basis for Selection of Strand/Concepts: The task of dividingfractions is not always easy for middle school students. Not only do they get the steps

of the algorithm incorrect, they also often do not understand why the steps they takework.

Module Resources:Adapted from:

Korean Mathematics, Grades 5-6. (2001). Edited by Janice Grow-Maienza, translated

by Sue Chung Nugent. Kirksville, MO: Truman State University. From Ministry of

Education. Arithmetic, Grades 1-6. Seoul, Korea: National Textbooks Inc, l993.Everyday Mathematics (Based upon work supported by the National Science

Foundation Under Grant No. ESI-9252984) Illinois: University of Chicago School

Mathematics Project. (Copyright 2002).

Standards Addressed:

Show-Me Goals:

Goal 3-2 - Students will demonstrate within and integrate across all content areas the

ability to develop and apply strategies based on ways others have prevented or solved

problems.

Missouri Frameworks for Curriculum Development:

Mathematics 1 - In mathematics, students in Missouri public schools willacquire a solid foundation which includes knowledge of addition, subtraction,multiplication and division; other number sense, including numeration and

estimation; and the application of these operations. Mathematics 5 - In mathematics, students in Missouri public schools will

acquire a solid foundation which includes knowledge of mathematical systems

(including real numbers, whole numbers, integers, fractions), geometry, and

number theory (including primes, factors, multiples).

NCTM Content Standard #: Understanding Numbers - 3

Understanding Meanings - 2


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PRETEST Form A

1. Divide 5

2. How many 1/3 in. segments are in 4 inches?

3. 14 =

4. 2/3 5/8 =

5. 4 1/3 1 =

6. James filled 4 bottles using 5 liters of water. How much water is in

each bottle? (Show your work).

7. Create a word problem that requires a mixed number be divided by a

proper fraction.

8. We ordered 4 pizzas. Each pizza is cut into 1/8s. If each person gets

3 slices, how many people can we feed with 6 pizzas? (Show your

work).

9. List the steps needed to solve each of the following problems, and

solve the problems. 3/4 3/8 2 1

10. Jan and Jim both solved the following problem. They got different

answers. Determine who is correct and explain the error made.Jan Jim

3 1/2 1 2/3 3 1/2 1 2/3

7/2 5/3 7/2 5/3

21/ 15 10/15 21/ 15 5/3

21 10 = 4/5

= 2 1/2


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Rubrics for Pretest A

Scoring Guide for Constructed Response Items #6

2 points -- student found correct answer and shows complete process

1 point -- student found correct answer, but lacked complete process or studentattempted to find the correct answer, but may or may not have made

computation errors leading to an invalid answer

0 points -- lacked valid attempt


2 points -- student creates a word problem with a mixed number divided by a properfraction

1 point -- student creates a word problem without a mixed number or without aproper fraction

Scoring Guide for Constructed Response Items #8, 9





Scoring Guide for Constructed Response Item #10

2 points -- student stated which student was correct and explained the error made

1 point -- student either stated which student was correct, or explained the error

made


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Lesson 1: Dividing Whole Numbers by Whole NumbersPresent the Following Problem to Small groups or pairs of students.

Transparency 1

Problem 1: Sam has two pizzas in the oven. Jill and Jordan want to share the pizzas with

him. How will they share the two pizzas equally? How much will each get?

Students will work 3-5 minutes on a solution to present to the class. Solutions are shared

and explained.

Transparency 2

Show a possible solution transparency

23 = 2 x 1/3 = x 1/

Transparency 3

Problem 2: The three kids, Sam, Jill, and Jordan, have of a liter of Dew to share.

What is the share of a full bottle of Dew for each person?

Students will again be given 1-3 minutes to solve the problem. Students will share andexplain their solutions.

Continuation of Transparency 3

Show a possible solution on transparency or on the board. are shaded

3/12 is shaded purple

3/12 is purple

O = O x 1


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Student Worksheet 1

Fill in the with the right number.

4 3 = 4 x 7 8 = 7 x

4/5 6 = 4/5 x 2/3 7 = 2/3 x

5/7 4 = 5/7 x 11/12 5 = 11/12 x =

Discuss the above problems, and ask if the students see a pattern in the above problems.If not, point out that the is a reciprocal of the number being divided by at the left

side of the = sign.

Assign the problem worksheet to reinforce learning.


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Transparency 1

Problem 1

Sam has two pizzas in the oven. Jill and

Jordan want to share the pizzas with him.

How will they share the two pizzas equally?

How much will each get?


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Transparency 2

23 = 2 x 1/3

= x 1/


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Transparency 3

Problem 2:

The three kids, Sam, Jill, and Jordan, have of a liter of Dew to share.

What is the share of a full bottle of Dew for each person?


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Here is a possible solution to problem 2.

is shaded

We will further divide the above into 3 equal parts.

3/12 is shaded purple

3/12 further reduces to . Each person will get 1/3 of the bottle of dew, which

would be of a full bottle of Dew.

Look at another method of showing this calculation.

O = O x 1


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Student Worksheet 1

Fill in the with the right number.

4 3 = 4 x 7 8 = 7 x

4/5 6 =4/5 x 2/3 7 = 2/3 x

5/7 4 = 5/7 x 11/12 5 = 11/12 x =

Do you notice any patterns in the boxed numbers?

How does the square relate to the division on the left?

Can you think of a term that is used to relate numbers and fractions

such as 4 = ? What is that term?

Now make up a problem of your own.


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Student Worksheet 2

Divide the following:

1. 1/4 3 = 2. 5/3 2 =

3. 1/2 5 = 4. 5/8 2 =

5. 7/9 8 = 6. 5/11 6 =

7. 5/6 9 = 8. 9/5 7=

9. 1/3 4 = 10. 5/12 7 =

11. 8/15 5 = 12. 9/20 8 =


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Lesson 2: Dividing natural numbers by proper fractions

Objectives:

*Students can find the quotient of natural numbers / proper fractions from using specific

materials.*Students understand the calculation principle of (natural numbers) / (proper

fractions).*Students can calculate (natural numbers) / (proper fractions) using a calculation format.

Lesson Resources:

Korean Mathematics Textbook 5-2 pgs. 26-27


Show-Me Goals #:

Goal 3-2 - Students will demonstrate within and integrate across all content areas

the ability to develop and apply strategies based on ways others haveprevented or solved problems.

Missouri Frameworks for Curriculum Development:

Mathematics 1 - In mathematics, students in Missouri public schools will acquirea solid foundation which includes knowledge of addition, subtraction,

multiplication and division; other number sense, including numeration and

estimation; and the application of these operations.

Mathematics 5 - In mathematics, students in Missouri public schools will acquirea solid foundation which includes knowledge of mathematical systems (including

real numbers, whole numbers, integers, fractions), geometry, and number theory

(including primes, factors, multiples).

NCTM Content Standards:

Number and Operations

Understand numbers, ways of representing numbers, relationships amongnumbers, and number systems

Understand meanings of operations and how they relate to one another Compute fluently and make reasonable estimates

Problem Solving:

Build new mathematical knowledge through problem solving Solve problems that arise in mathematics

Apply and adapt a variety of appropriate strategies to solve problems Reflect on the process of mathematical problem solving

Communication

Organize and consolidate mathematical thinking through communication Communicate mathematical thinking coherently and clearly to peers and teachers


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Connections

Recognize and apply mathematics in context outside of mathematicsRepresentation

Create and use representations to organize, record and communicate mathematical

ideas

Real Life Problem:Present the Following Problem to Small groups or pairs of students.

This container holds 2 cups of water. We need to put 2/3 cups of water in each plant.

How many plants can we water with this 2 cups of water?


and explained.

Demonstration of Process(es)/Exploration:

Show a possible solution transparency. (Teacher Transparency 2-1)

2c - 2/3 c - 2/3 c- 2/3 c = 0.

Number line showing 0 to 2.

Let students understand the principle to find the quotient by doing division ofnumerators of fractions, that is, the division of natural numbers.

Also, let them understand the principle of contents of material provided

in a textbook nest.That is, let them think 6/3 _ 2/3 on the number line in the following way.

1/3 2/3 3/3 4/3 5/3 6/3

0 1 2

Let them know that when they divide a natural number by a proper fraction,they multiply the natural numbers after changing the numerator and

denominator of the divisor.

2 2/3 = 2 x 3 = 6 = 3 / = x /2 2

Here, let them understand the principle that the division of fractions with samedenominators can be found from division of numerators of fractions, that is, thedivision of two natural numbers.


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2= 6 so 6 2 = 3 = 33 3 3 1

Real Life Problem 2:

Jim eats 2/5 lb of jelly beans every day. How many days will it take him to eat 4 pounds?

Students will again be given 1-3 minutes to solve the problem. Students will share and

explain their solutions.


Show a possible solution on transparency or on the board. (Teacher Transparency 2-2)4lbs = 10 days

4 2/5 = 4x 5/2= 20/2 =10

How would we do this calculation without inverting the divisor fraction?20 2 = 10 = 10

5 5 1

Where did the 20/5 come from? Students explain.

Calculation practiceAsk students to create a problem similar to the ones we solved in class. Allow students to pose

these to the class to become skillful in calculation. Students show solutions on the board and

answer questions that arise about the problem. (Or use the following problems.)

5 2/3 = 7 4 = 5 1/3 8 = 16 3 3/5 = 5 5 = 20

Allow students to independently complete Student Worksheet 2-1.


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TeacherTransparency 2-1

This container holds 2 cups of water. We need to put 2/3 cups of waterin each plant. How many plants can we water with this 2 cups of water?

Possible solutions

2 c - 2/3 c - 2/3 c - 2/3 c = 0.

1/3 2/3 3/3 4/3 5/3 6/3

0 1 2

2 2/3 = 2 x 3 = 6 = 32 2

or

2 = 6 so 6 2 = 3 = 3

3 3 3 1


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Teacher Transparency 2-2

Jim eats 2/5 lb of jelly beans every day.

How many days will it take him to eat 4pounds?

Possible solutions

4lbs = 10 days

4 2/5 = 4 x 5 = 20 = 102 2


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Student Worksheet 2-1

Skill DevelopmentDividing Natural Numbers by Proper Fractions

Name______________________

1.) Grandma Jenny is making a quilt. She has 2 yards of green material. A quilt

block takes of a yard of material. How many green quilt blocks can she make?

2.) Penny and Denny have 6 new notebooks. They want to give 2/3 of them to their

cousins. How many will they give away?

3.) Lenny has 8 gallons of gas for the mower. Each time he mows, he uses of agallon. How many times can he mow before he runs out of gas?

4.) 5

3/5 = 6.) 9

2/3 =

5.) 7 = 7) 12 =

Write in words how you would solve this problem: 8 5/6 .

Now you create a real-life problem to match this equation. Please provide the solution as

well.

6 4/5


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Lesson 3: Dividing proper fractions by proper fractions

Objectives:

*Students can find the quotient of division whose divisor is a unit fraction on the

number line.

*Students understand the calculation principle of proper fractions.*Students know the calculation format of proper fractions and can calculate them.

Lesson Resources:Korean Mathematics Textbook 5-1 pg. 104 - 105

Standards Addressed:Show-Me Goals #:

Goal 3-2 - Students will demonstrate within and integrate across all content areasthe ability to develop and apply strategies based on ways others have

prevented or solved problems.Missouri Frameworks for Curriculum Development:

Mathematics 1 - In mathematics, students in Missouri public schools will acquirea solid foundation which includes knowledge of addition, subtraction,multiplication and division; other number sense, including numeration and


Mathematics 5 - In mathematics, students in Missouri public schools will acquirea solid foundation which includes knowledge of mathematical systems (including

real numbers, whole numbers, integers, fractions), geometry, and number theory



Number and Operations:

Understand numbers, ways of representing numbers, relationships amongnumbers, and number systems Understand meanings of operations and how they relate to one another Compute fluently and make reasonable estimates

Problem Solving:

Build new mathematical knowledge through problem solving Solve problems that arise in mathematics Apply and adapt a variety of appropriate strategies to solve problems Reflect on the process of mathematical problem solving

Communication: Organize and consolidate mathematical thinking through communication Communicate mathematical thinking coherently and clearly to peers and teachers

Connections:

Recognize and apply mathematics in context outside of mathematics


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Representation:

Create and use representations to organize, record and communicate mathematicalideas

Real Life Problem: Five containers each contains 2/3 cup of water. If the water in each

container is poured in to a big container, what is the total water in the one? Students willbe put in groups. Each group will get one 2/3 cup and we will solve together as a whole

class.

Independent Practice: Student Worksheet 3-1

Note(s) for Teachers: If there is enough time, have students partner check Worksheet 3-1

and then write answers on board.

Students will be given a fraction problem to solve as a group or with partners.

Transparency 1Laura purchased 2/3 of a yard of ribbon in order to make bows. Each bow takes 1/6 of a

yard of ribbon. How many 1/6 yard bows can be made?

* This section below not on transparency.

Discussion: Allow 5-10 minutes for small groups and/or partners to solve the above

problem.At the end of the problem-solving time, ask different groups how they solved the

problem, and what answer they arrived at.

Transparency 1

Give this example on Transparency 1 as a possible solution.Here is a possible solution to the problem.The box below shows 2/3 of a yard of ribbon.

Once we have the two thirds of ribbon, we need to further divide into 1/6 sections.

As you can see, the answer is 4 bows. Because 4 sections are shaded.


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Let the students find the quotient of the above problem by multiplying by 6/1 which is the

reciprocal of 1/6.

If done this way 2/3 1/6 = 2/3 x 6/1 = 12/3 = 4

A number line example of this would be as follows:

2/3

1/3(2/6)

0 1/6 1/3 3/6 2/3 5/6 6/6

1/3 = 2/6 = Purple arrow 2/3=4/6 =Yellow Arrow

The number of bows we can make is 4.

Go back to the original problem: 2/3 1/6 = 4Does this make sense? How did we get 4?

Think of this algorithm for division of fractions:

A/B C/D = A/B x D/C

Using our example the numbers would look like this:

2/3 1/6 = 2/3 x 6/1 = 12/3 = 4

Notice that 6/1 is the reciprocal of 1/6. Now on worksheet number 1, try to use the

algorithm to solve the problems.

Worksheet 1

Division of Fractions Algorithm

Use the algorithm above to solve the following problems.

A/B C/D = A/B x D/C

1. 3/8 5/6 = 6. 4/7 2/3 =2. 3/10 3/8 = 7. 9/10 1/3=3. 5/8 1/4 = 8. 7/8 4/9=4. 5/12 1/3 = 9. 3/4 7/8=5. 5/9 1/3 = 10. 3/8 2/3 =


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Transparency 1

Laura purchased 2/3 of a yard of ribbon in order to make bows. Each bow takes 1/6 of a

yard of ribbon. How many 1/6 yard bows can be made?

Here is a possible solution to the problem.The box below shows 2/3 of a yard of ribbon.

Once we have the two thirds of ribbon, we need to further divide into 1/6 sections.

As you can see, the answer is 4 bows. Because 4 sections are shaded.


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Transparency 2

Now lets try finding the quotient of the above problem by multiplying by 6/1

which is the reciprocal of 1/6.

If done this way 2/3 1/6 = 2/3 x 6/1 = 12/3 = 4

A number line example of this would be as follows:

2/3

1/3(2/6)

0 1/6 1/3 3/6 2/3 5/6 6/6

1/3 = 2/6 = Purple arrow 2/3=4/6 =Yellow Arrow

The number of bows we can make is 4.

Go back to the original problem: 2/3 1/6 = 4Does this make sense? How did we get 4?

Think of this algorithm for division of fractions:

A/B C/D = A/B x D/C

Using our example the numbers would look like this:

2/3 1/6 = 2/3 x 6/1 = 12/3 = 4

Notice that 6/1 is the reciprocal of 1/6. Now on worksheet number1, try to use the algorithm to solve the problems.


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Worksheet 1

Division of Fractions Algorithm

A/B C/D = A/B x D/C

Use the algorithm above to solve the following problems.

1. 3/8 5/6 = 6. 4/7 2/3 =

2. 3/10 3/8 = 7. 9/10 1/3=

3. 5/8

1/4= 8. 7/8

4/9=

4. 5/12 1/3 = 9. 7/8=

5. 5/9 1/3 = 10. 3/8 2/3 =


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Lesson 4: Improper fraction divided by proper fraction; Mixed number

divided by proper fraction

Objectives:

*Students know the principle of (improper fraction) - (proper fraction) and do the

calculation.*Students know the principle of (mixed number) - (proper fraction) and do the

calculation.*Students can solve word problems that the division of fraction is applied to.

Lesson Resources:Korean Mathematics Textbook 5-2 pgs. 28-34


Show-Me Goals:

Goal 3-2 - Students will demonstrate within and integrate across all content areasthe ability to develop and apply strategies based on ways others have prevented or

solved problems.

Frameworks for Curriculum Development:

Mathematics 1 - In mathematics, students in Missouri public schools will acquirea solid foundation which includes knowledge of addition, subtraction,

multiplication and division; other number sense, including numeration and


Mathematics 5 - In mathematics, students in Missouri public schools will acquirea solid foundation which includes knowledge of mathematical systems (includingreal numbers, whole numbers, integers, fractions), geometry, and number theory



Number and Operations

Understand numbers, ways of representing numbers, relationships amongnumbers, and number systems

Understand meanings of operations and how they relate to one another Compute fluently and make reasonable estimates

Problem Solving:

Build new mathematical knowledge through problem solving Solve problems that arise in mathematics

Apply and adapt a variety of appropriate strategies to solve problems Reflect on the process of mathematical problem solving

Communication

Organize and consolidate their mathematical thinking through communication Communicate their mathematical thinking coherently and clearly to peers and

teachers


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Connections

Recognize and apply mathematics in context outside of mathematics

Representation

Create and use representations to organize, record and communicate mathematical

ideas

Real Life Problem:Present the Following Problem to Small groups or pairs of students.

Kyle rides his bike 5/4 miles to school. He can ride 3/8 mile in one minute. How longwill it take him to get to school?


and explained.

Demonstration of Process(es)/Exploration:Show a possible solution transparency (Transparency 4-1)

5/4 3/8 = 5/4 x 8/3 = 40/12 = 10/3 = 3 1/3

5/4 3/8 = 10/8 3/8 = 10 3 = 3 1/3

After solutions have been explained, Point out that the method they learned in the

previous class still works.

Remind them that when the quotient is an improper fraction, we change it into a mixed

number and further simplify it as far as we can.

If students have not been exposed to cancellation, now may be the time for it.

Real Life Problem 2:

Present the following problem to small groups or pairs of students.

Shayla has a cord that is 2 2/3 ft long. She wants to cut it into pieces that are 7/9 ft. How

many pieces can she get?


and explained.


Show a possible solution transparency (Transparency 4-2)


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2 2/3 = 8/3

2 2/3 7/9 = 8/3 7/9 = 8/3 x 9/7 = 72/21 = 24/7= 3 3/7or

2 2/3 7/9 = 8/3 7/9 = 72/ 27 21/27 = 72 21 = 3 3/7

Calculation practiceAsk students to create problems similar to these for the class to solve to improve

calculation skills. Or use the following:

5/8 7/6 7/8 3/5 4/5

For the calculation process, when we change division into a multiplication, let them dothe calculation with changing numerator and denominator, simplifying if possible, and

expressing the results into simplified fraction and mixed numbers.

Let students study the division when the dividend is a mixed number. In this case, letthem do the calculation after changing the mixed number into an improper fraction.

1 2/3 3/8 4 6 5/8

They have studied how to change a mixed number into an improper fraction and becausethe calculation process of doing the division with fraction as a divisor, make sure that

they change the mixed number into an improper fraction correctly, and exchange

numerator and denominator of a fraction and multiply it correctly.

Allow students to independently complete Student worksheet 4-1.


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Transparency 4-1

Kyle rides his bike 5/4 miles to school.

He can ride 3/8 mile in one minute.

How long will it take him to get toschool?

5 3 = 5 x 8 = 40 = 10 = 3 1/34 8 4 3 12 3

5 3 = 10 3 = 10 3 = 3 1/34 8 8 8


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Transparency 4-2

Shayla has a cord that is 2 2/3 ft long. She wants to cut it into pieces that are 7/9 ft. How

many pieces can she get?

Possible Solutions

2 2/3 = 8/3

2 2 7 = 8 7 = 8 x 9 = 72 = 24= 3 33 9 3 9 3 7 21 7 7

2 2 7 = 8 7 = 72 21 = 72 21=3 33 9 3 9 27 27 7


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Student Worksheet 4-1

Skill DevelopmentDividing Improper Fractions and Mixed Numbers by Proper Fractions

Name______________________

1.) 4/3 2/3 2.) 8/5 3/4

3.) 1 4/7 4.) 3 2/3 5/6

5.) The area is 10 2/3 ft2

and the height is 4/9 ft . Then, what is the length of base in a

parallelogram?

6.) We have 4 cups of chocolate chips. A batch of our super duper cookies requires cup of chocolate chips. How many batches can we make?

7.) Mojo is chained to the tree. He has 2 yards of chain to roam. It takes 2/5 yard togo around the tree. How many times can he go around the tree until he runs out ofchain?

Draw a picture showing that 4 = 25/4.

Write in words how you would solve this problem: 3 1/3 4/5.

Now you create a real-life problem to match this equation. Please provide the solution as

well. 2 1/8 .


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POSTTEST Form B

1. Divide 5

2. How many in. segments are in 5 inches?

3. 12 5/6 =

4. 11/12 5/8 =

5. 3 7/8 1 =

6. Four boys evenly split 3 gallons of ice cream. How much will

each get? (Show your work).

7. Create a word problem that requires a mixed number be divided by a

proper fraction.

8. I ordered pizzas. Each pizza is cut into 1/6s. Each person will get 4

slices. How many people can I feed with 6 pizzas? (Show yourwork).

9. List the steps needed to solve each of the following problems, and

solve the problems. 2/3 1/6 3 2 .

10. Jan and Jim both solved the following problem. They got different

answers. Determine who is correct and explain the error made.3 1/8 1 3 1/8 1

25/8 6/4 25/8 5/4

25/8 12/8 25/8 10/8

25 12 25 10

2 1/12 2 Jan Jim


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Rubrics for Pretest B







2 points -- student creates a word problem with a mixed number divided by a properfraction

1 point -- student creates a word problem without a mixed number or without aproper fraction

Scoring Guide for Constructed Response Items #8, 9





Scoring Guide for Constructed Response Item #10

2 points -- student stated which student was correct and explained the error made

1 point -- student either stated which student was correct, or explained the error

made


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Documents

Secondary Final Module Completed.pdf