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Lesson Plan: 6.NS.A.1 Division with Fractions(This lesson should be adapted, including instructional time, to meet the needs of your students.)

Background InformationContent/Grade Level The Number System/Grade 6

Unit/Cluster Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Essential Questions/Enduring Understandings Addressed in the Lesson

How does division of fractions relate to multiplication of fractions?Is division related to repeated subtraction as multiplication is related to repeated addition?How is multiplying or dividing whole numbers similar to multiplying or dividing fractions?

Division breaks quantities into groups of equal size.Division is related to repeated subtraction as multiplication is related to repeated addition.The size of the divisor determines the size of the quotient.

Standards Addressed in This Lesson 6.NS.A.1: Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.

It is critical that the Standards for Mathematical Practice are incorporated in ALL lesson activities throughout the unit as appropriate. It is not the expectation that all eight Mathematical Practices will be evident in every lesson. The Standards for Mathematical Practice make an excellent framework on which to plan your instruction. Look for the infusion of the Mathematical Practices throughout this unit.

Lesson Topic Division with Fractions

Relevance/Connections 6.NS.B.2: Compute fluently with multi-digit numbers and find common factors and multiples.6.NS.B.3: Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm.

6.RP.A.2: Understand the concept of a unit rate abassociated with a ratio a:b with b≠0, and use rate

language in the context of a ratio relationship.

Student Outcomes Students should be able to divide fractions using different methods. Make and use models for division of fractions.

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Use the inverse relationship between multiplication and division to understand how to divide by a fraction.

Prior Knowledge Needed to Support This Learning

Whole number operationsModeling fraction addition, subtraction, multiplicationFraction addition, subtraction, multiplication using standard algorithmsUnderstanding of division as creating equal groupsComponents of time – number of minutes in an hour, number of minutes in a quarter hour

Students interpret and compute quotients of fractions and solve word problems involving division of fractions by fractions. This completes the extension of operations to fractions.

Method for determining student readiness for the lesson

Teachers should make up a few division problems with whole numbers scenarios for the students. An example could be: Adria bought a bag of gummy bears which contained 72 gummy bears. Adria and her 7 friends want to share these equally. How many gummy bears will each of them receive?

Learning Experience

Component DetailsWhich Standards for Mathematical Practice does this address? How is each Practice used to help students develop proficiency?

Warm Up Min is making lemonade to sell at a lemonade stand. She uses an 8-cup pitcher to make the lemonade. She’ll serve the

lemonade in 14 cup sized servings. How many servings will

she be able to make from 8 cups?Motivation Give each group of students varying lengths of string, yarn,

adding machine tape, or other material that can be folded or partitioned easily. Make sure that every student has a different length of material. Ask students to complete the following task:Distribute your material equally among three people.Discuss with students how they accomplished the task.

Activity 1 UDL Components: Principle I: Representation is present in the activity. Prior

Students should be able to identify quantities in a practical situation and use the appropriate

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Learning Experience


UDL Components Multiple Means of

Representation Multiple Means for

Action and Expression

Multiple Means for Engagement

Key QuestionsFormative AssessmentSummary

knowledge is activated through the Warm-up and Motivation. Visual diagrams, with color for clarification and emphasis are provided throughout to model various examples of division with fractions.

Principle II: Expression is present in the activity. Students are encouraged to work with a variety of materials, such as the face of an analogue clock, fraction bars, and model drawings, among others to aid in categorizing and systematizing the various examples of division with fractions.

Principle III: Engagement is present in the activity. The information in the tasks for this activity is authentic and student-centered.

A. Jamal has 1 1

4 hours to finish his three homework assignments. If he divides his time evenly, how much time can he give to each?

During this activity students will explore different ways to represent the division using a variety of manipulatives. For example, a student may choose a

clock and recognize that 1 1

4 hour is 60 minutes + 15 minutes and those 75 minutes must be divided by 3 assignments. Another student may select a ruler,

draw a line that is 1 1

4 inches and attempt to divide the line into three equal portions.

Pose the question. How is this question similar to the motivation activity?

diagram to represent the situation. (SMP # 4)

Students should be able to reasoning and create a coherent representation of the problem. (SMP #2)

Students should be able analyze a problem and plan a pathway that finds a solution to the problem. (SMP #1)

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Learning Experience


Put students in groups of 2-3 students.o Students in the groups will explore ways to

solve the problem.o Monitor students during the activity. Pose

questions to clarify and extend thinking.o Students will share out as a large group –

strategies and materials used to solve the problem.

B. Jamal has been given one more homework assignment to

finish during the 1 1

4 hours. How does this additional assignment change the

problem? Will Jamal have more or less time to complete each

chore? Ask students, in their groups, to model the new

problem and be prepared to share. Monitor groups and ask questions to clarify and extend thinking.

NOTE: If a group has not shared the solution using fraction models, share the following strategy with the class.

Establish as a unit. It represents one

hour. In order to represent 1 and

14 hours use two

units and break each into 4 equal parts. Shade 1

14 to

represent the number of hours needed to be shared among the 4 homework assignments.

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Learning Experience


You want students to recognize that one unit can be divided into 4 equal sections and be distributed among the 4 assignments.

Give students time to strategize how to equally divide the last fourth.

Once students have divided the remaining

14 , ask

what portion of the whole each section represents. Students should explain their thinking.

Students will need to determine that each of the 4

assignments requires

14+ 1

16 of an hour. Ask students approximately how much time this

represents.Page 5 of 17

Learning Experience


C. Formative Assessment:

Desmond has

12 of a candy bar that he wants to share

with 3 friends. Draw a model to show how the candy bar will be shared. How much will each person receive?

Activity 2






UDL Components: Principle I: Representation is present in the activity. Prior

knowledge is activated through the use of modeling fractions discussed and completed in earlier tasks.

Principle II: Expression is present in the activity. This task encourages students to think before choosing a correct answer for the remainder in division of fractions.

Principle III: Engagement is present in the activity. Students collaborate in small groups as they complete the task in order to discuss the processes they used to solve the problems, and to receive feedback from their peers.

Present the following scenarios.A. Carla purchased 6 pints of ice cream for a party. She will

serve

34 of a pint to each guest. How many guests can

she serve with 6 pints of ice cream? Use a model to explain how you solved the problem.

Before students begin to solve, ask students to estimate what might be a reasonable answer to the

Students should be able to identify quantities in a practical situation and use the appropriate diagram to represent the situation. (SMP # 4)



Students should be able to make a conjecture and build a logical progression of statements to

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Learning Experience


problem and why does this answer make sense.

Put students in groups of 2 – 3 and ask them to use the available materials to model a solution to the problem.

Monitor student work and ask questions to clarify and extend their thinking.

As a whole group, share a variety of problem solving strategies used by students. Also ask students to explain why their answer makes sense in context of the problem.

B. Sam has 2 1

4 gallons of gasoline for mowing lawns. Sam

uses

34 of a gallon to mow 1 lawn. How many lawns can

he mow?

Ask students to model a solution to the problem using a different manipulative.

Share student responses.

C. Formative Assessment:

Model5 1

3÷2

3 . Explain how you arrived at your solution.

explore the truth of their conjecture. (SMP #3)

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Learning Experience


Activity 3






UDL Components: Principle I: Representation is present in the activity. Prior




A. A local landscaping contractor has

34 ton of gravel.

She will use

38 ton in each landscaping job. How

many jobs can she complete with the gravel?

Kenisha drew the following model to show her solution to the problem.

Ask students to explain Kenisha’s drawing and why the drawing shows the solution.

How many jobs can the landscaper complete?




Students should be able to make a conjecture and build a logical progression of statements to explore the truth of their conjecture. (SMP #3)

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Learning Experience


Focus on Kenisha’s partitioning of

34 into

68 in order

to solve the problem. Show

68÷3

8=2

.Note to teacher:

Create several problems for the fraction division cards, such as: 1

4 ÷ 1

8

56

÷ 212

34

÷ 316

25

÷ 210

23

÷ 16

Fraction division cards (index cards) Put students in groups of 2-3. Distribute 4-5 fraction

division cards to each group. Explain that students will use models similar to Kenisha’s to solve each of the problems on their cards. Students should be prepared to show their solutions and defend their

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Learning Experience


answers. A student response to

14÷1

8 may look like the following:

Once students have completed all 5 problems, present the problems with common denominators and quotients.

28

÷18

=2

1012

÷212

=5

1216

÷316

=4

410

÷210

=2

46

÷16

=4

Ask students to examine the 5 problems and discuss any patterns that they see. Record student observations. Students should recognize that each fraction in a problem has a common denominator and there is a relation between the

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Learning Experience


quotient and the numerators. Students may discover that the quotient is the result of dividing the numerator by numerator. The denominators are also divided with the quotient always being one.





A. Lisa has

78 yd of material left over from a quilting

project. She will use the material to make potholders

for her aunts. Each potholder requires

38 yd of

material. Draw a model to represent this problem. Be prepared

to explain your drawing. The student model might look like the following:





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Learning Experience


Model and discuss what the remainder represents. Anticipate that students will think that the answer is

2 18 . Ask students to think about the work that they

did yesterday and the relationship between the quotient and the numerators of the fraction division

problems. Write the problem 3|7 on the board. Ask students to express the remainder as a fraction. Ask students to explain where the numbers in the fraction remainder come from and how they relate to the problem. Then ask students to look back at their

model and explain why

13 is an appropriate remainder

for

78 ÷

38 . Make sure students understand that the

remainder represents one of the three parts that defined a group.

B. Put students in pairs and present

916

÷38 . Ask

students to solve in two different ways: draw a model and use common denominators. Students should be prepared to explain the whole number and fractional part of the quotient.


Students should be able to reasoning and create a coherent representation of the problem.

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Learning Experience





Present the following questions:

A. Will 5÷ 14 be greater than or less than 5?

After the students have tried to explain the answer to the question, present this scenario.

While shopping at the mall, you spot a toy crane vending machine. You have $5.00 to spend. You want to try to win a toy. Each attempt at the machine costs a quarter. How many tries will you get with your $5?

Lead a discussion that allows the students to understand that division breaks quantities into groups.

What was being broken into groups in 5÷ 14

? How

many groups were there?

If necessary, show the students the repeated

subtraction for 5 ÷ 14 . Remind the students that

(SMP #2)



Students will calculate their problems and notice the repeated pattern as they work.

(SMP #8)

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Learning Experience


multiplication is repeated addition. Division is repeated subtraction because it is the inverse of multiplication.

B. Present 6÷ 2 and 6÷ 12 . What is the difference in

these problems? Do you think the answers will be the same?

Have the students use what they’ve learned in the previous activities to find both quotients. Debrief by interpreting each problem as how many groups of 2 are in 6 and how many groups of one-half are in 6. Model using 6 graham crackers if the students need a concrete visual.

C. Present 8 ÷ 4 and 8 ÷ 14 and discuss the differences in

the quotients.

Make sure that the students understand why 8÷ 14 has

a larger quotient than 8 ÷ 2. The size of the divisor determines the size of the quotient (group).

Present the following series of problems and have the students use prior knowledge, including the common denominator algorithm, to find the quotients.

8 ÷ 8 =

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Learning Experience


8 ÷ 4 =

8 ÷ 2 =

8 ÷ 1 =

8 ÷ 12 =

8 ÷ 14 =

Have the students discuss the pattern of answers that is established and why they think its happening. [The size of the divisor is decreasing so it is possible to make more equal groups (quotient).]

D. Ask the students to give the value of 18 of 8.

Continue by asking for 14 of 8. Finish with

12 of 8.

Write these problems next to the division problems with equivalent solutions.

What are the similarities and differences between the associated problems?

Ask the students to make a conjecture about what related problem could be written next to

8 ÷ 1, 8 ÷ 12 , and 8÷ 1

4 . Use the conjectures to

solve the related multiplication problems to determine whether the answers are equivalent.

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Learning Experience


Use a student produced conjecture that works and have the students develop a rule, they think would work to divide fractions without finding common denominators. (The student produced conjecture should be comparable to the standard algorithm.)

Closure Present examples used in previous activities and have the students try their new rule.

Supporting InformationInterventions/Enrichments

Students with Disabilities/Struggling Learners

ELL Gifted and Talented

Students who are struggling may need to use a lot of manipulatives to understand how each problem is computed. They may need extra practice with many other problems before the concept is understood.

ELL students may need to have vocabulary words explained to them so they can understand the processes that are needed to solve each problem.

GT students can use complex fractions as they work through the problems.Materials Activity 1 Materials: rulers, fraction strips, clock face, centimeter cubes, cubes and rods, number lines

Activity 2 Materials: rulers, fraction strips, clock face, centimeter cubes, cubes and rods, number lines, whiteboards

Activity 3 Materials: rulers, fraction strips, clock face, centimeter cubes, cubes and rods, number lines, fraction, Division Cards

Activity 4 Materials: rulers, fraction strips, clock face, centimeter cubes, cubes and rods, number lines

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Activity 5 Materials: Graham Crackers

Technology Smart BoardDigital CameraPromethean Board

Resources Teaching Student-Centered Mathematics Grades 3-5, John A. Van de Walle, Pearson 2006.Teaching Student-Centered Mathematics Grades 6-8, John A. Van de Walle, Pearson 2006.

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mdk12.msde.maryland.govmdk12.msde.maryland.gov/share/ccsc/uos/resources/... · Web viewWhole number operations. Modeling fraction addition, subtraction, multiplication. Fraction