Targeted Content Standard(s): Student Friendly Learning ...ioer.ilsharedlearning.org/ContentDocs/bc2cc184-41bf-464b-a363-11a554da4126/303/G6_U1...6. Problem #5 is the first opportunity

Grade 6 Lesson Title: Interpreting and Computing Quotients of Fractions (6NS1) Unit 1: Fractions and Decimals (Lesson 2 of 3) Time Frame: 5-7 days Essential Question:

o How is division related to realistic situations and to other operations? o How can division be represented and interpreted?

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Targeted Content Standard(s): Student Friendly Learning Targets 6.NS.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.

For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length3/4 mi and area ½ square mi?

I can…

Divide fractions by fractions

Interpret quotients of fractions.

Solve word problems involving division of fractions by fractions by using visual fraction models and equations to represent the problem.

Targeted Mathematical Practice(s): 1 Make sense of problems and persevere in solving them 2 Reason abstractly and quantitatively 3 Construct viable arguments and critique the reasoning of others 4 Model with mathematics 5 Use appropriate tools strategically 6 Attend to precision 7 Look for and make use of structure. 8 Look for an express regularity in repeated reasoning

Supporting Content Standard(s): (optional)

Purpose of Lesson: Students will develop understanding of the reasoning behind the algorithm for dividing fractions by fractions and be able to represent and solve problems involving division of fractions.

Explanation (Expectation) of Rigor: (Fill in those that are appropriate.) Conceptual: Students use their understanding of fractions to develop models for dividing fractions by fractions and connect these models to real-world situations and equations. They develop an understanding of reciprocals and the role they play in the division algorithm.

Procedural: Students develop procedures for dividing whole numbers and fractions, mixed numbers and fractions, and fractions by fractions.

Application: Students read and interpret real-world situations by recording appropriate equations to represent the problem. They may use visuals to support the creation of equations.

Vocabulary: Reciprocal Compose Algorithm

Quotient Numerator Denominator

Decompose Inverse operation

Dividend Divisor



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Evidence of Learning (Assessment): Pre-Assessment: Division Models Pre-Assessment Formative Assessment(s): Fraction Division Routine (Segments 3, 4, 5 and 7), Context Stories (Segment 5), Reasoning with Reciprocals (Segment 6), Reflecting Upon the Fraction Division Algorithm (Segment 7), Fraction Division Tile Game (Segment 7), Optional: Matching Representations (Any time after Segment 3) Summative Assessment: Interpreting and Computing Quotients of Fractions (Segment 8) Self-Assessment: Interpreting and Computing Quotients of Fractions (Segment 8)

Lesson Segments: 1. Pre-assessment - Division and fractions 2. Division of whole numbers and fractions 3. Division of mixed numbers/fractions by fractions – number line models 4. Division of mixed numbers/fractions by fractions – area models 5. Patterns in division of fractions 6. Reciprocals and the division algorithm 7. Looking for structure in fraction division equations/fraction division practice (Tile Game) 8. Assessment



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Lesson Procedure: Segment 1

Approximate Time Frame: 20-25 minutes

Lesson Format: Whole Group Small Group Independent

Modeled Guided Collaborative Assessment

Resources: Division Models Pre-Assessment

Focus: Assessing prior knowledge of fraction operation models

Modalities Represented: Concrete/Manipulative Picture/Graph Table/Chart Symbolic Oral/Written Language Real-Life Situation

Math Practice Look For(s):

MP2: Students reason about the values of fractions used in the various models.

MP4: Students use visual models and equations to represent real-world contexts and determine appropriate contexts for given mathematical models.

Differentiation for Remediation: If students struggle on the pre-assessment, they may need instruction regarding the relationship between division and fractions or in partitioning. See Grade 5 Unit 3 resources for partitioning to develop this understanding.

Differentiation for English Language Learners: English Language Learners may need explicit vocabulary instruction regarding the terms “share,” “expression,” “repeated addition model,” or “pattern.”

Differentiation for Enrichment: Students who are successful on this pre-assessment may benefit from additional pre-assessment of current content to determine where curriculum may be compacted for acceleration purposes.

Potential Pitfall(s):

Students lack the understanding of the models (area and/or number line) for fraction operations.

Independent Practice (Homework):

Steps:

1. Give Division Models Pre-Assessment to determine student readiness for this lesson. The pre-assessment addresses students’ previous work with whole number division represented as fractions and division of whole numbers and unit fractions.

Teacher Notes/Reflection:

If students struggle on the pre-assessment, refer to Grade 5, Unit 3 Fractions for resources. Questions align to the lesson as follows:

Question #1 – Lesson 2, Segment #1 Questions #2, 3 and 4 – Lesson 2, Segment #2 Questions #5 and 8 – Lesson 2, Segment #3 Question #6 – Lesson 2, Segment #4 Question #7 – Lesson 2 Segment #5



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Segment 2

Approximate Time Frame:

45-60 minutes



Resources: Handout 1: Division of Fractions and Whole Numbers

Focus:

Division of fractions by whole numbers or whole numbers by fractions.



MP7 Look for and make use of structure: Students should be interpreting how the operation is modeled by the visuals and equations using the relationship between multiplication and division.

MP2 Reason abstractly and quantitatively: Students interpret the values of quotients based upon the models, with particular attention to the fractions that represent remainders.

Differentiation for Remediation: Students may need to partition by cutting apart paper or using tactile tools (fraction squares or fraction tiles).

Differentiation for English Language Learners: English Language Learners will benefit from a focus on the vocabulary terms that correspond to the fraction models.

Differentiation for Enrichment: Students may not need to be guided and can interpret the visual models by discussing them in small groups. It is important that these students interpret the models and do not merely solve the problems with the division algorithm.

Potential Pitfall(s): Students just DO the math and not DISCUSS the math. Students can make the models, but do not interpret them.


Students would benefit from fluency practice with multiplication of fractions and/or interpretation of area models involving multiplication of fractions.

Steps: 1. Give students 4 index cards per pair. Ask students to use them to

determine how many 2/5 portions are in 4. Before beginning to partition, have students predict if the quotient will be greater than, less than, or equal to 1. If students are not sure what to do, guide them to partition/fold each index card into fifths and then partition out groups of 2/5. Ask what operation they are doing (DIVISION) and have students justify their responses by explaining to their partners.

Record a division equation to represent the model. (4÷ 2

5= 10 )

Ask students if they predicted if the quotient would be a whole number and, if so, why.

Teacher Notes/Reflections:



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2. Give students the Division of Fractions and Whole Numbers handout to take notes. (This is a guided notes handout and is not intended for students to complete independently.) Remind students that multiplication and division are inverse operations, so the question can be thought of as a missing factor

problem, “What times 2

3 would equal 6?” and can be recorded with

a multiplication equation ? x 2

3= 6. (Students record this in the

Multiplication Equation box. Ask how they can partition the diagram to show 2/3 portions. Have students share their models that represent partitioning into thirds and showing the 2/3 portions. If no student suggests it, show how to partition each “whole” box into thirds and shade 2/3 portions. (See the examples in the teacher notes portion of the document.) Have students record the division equation to represent the problem.


3. Have students complete problem #2 by working collaboratively with their partners. If some students appear to be struggling, you may wish to pull a small group for more guidance. If partners seem to be grappling with the problem well, circulate and listen to their conversations, joining in when guidance or enrichment may be appropriate. Ask students to interpret the fraction 1/3 that represents the “remainder” or the 1 unshaded box in the teacher notes. Why is it 1/3 if you partitioned each box into fourths?

4. Guide students to interpret problem #3 using the number line model. First have them write the multiplication equation that represents the question. Then they can work on completing the model. 3/5 is already indicated on the number line with the first arrow. Have students continue to indicate 3/5 portions using arrows until they have the 10 arrows indicated. Finally, ask them to write the division equation with its solution.

5. Have students complete problem #4 by working collaboratively with their partners. If some students appear to be struggling, you may wish to pull a small group for more guidance. If partners seem to be grappling with the problem well, circulate and listen to their conversations, joining in when guidance or enrichment may be appropriate. Some groups may need reminding of how many sections the number line should be partitioned into and why they are dividing each whole into eighths. Others may need reminding of how large each “leap” on the number line should be and why it is a leap of 3. Ask students to interpret the fraction 1/3 that represents the “remainder” or the 1 portion without an arrow in the teacher notes. Why is it 1/3 if you partitioned each box into eighths?



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6. Problem #5 is the first opportunity to divide a fraction by a whole number. Guide students to shade the fraction on the model by shading 5 of the 8 sections. Then ask how they could show that each section will be divided by 2. If no student suggests dividing each eighth section into 2 portions, show them how dividing all of the eighths portions into 2 portions creates sixteenths. Then they should find that 5 of the portions that are shaded are indicated. Therefore, the quotient is 5/16.


7. Students then work on problem #6 with their partners and share out responses and partitioning strategies.

8. Next guide students to create the number line model for dividing a fraction by a whole number in Problem #7. First ask students how number line should be partitioned to represent 5/6. Once a student responds that it should be divided into six sections to represent sixths, then have them label 5/6 on the number line. Next have students partition each (sixth) section into 4 partitions. Students will need to draw the 4 equal leaps on the number line to indicate 4 portions of the 5/6 length on the number line. The will see that each leap is 5/24 of the entire number line, which is the quotient.

9. Finally, have students work collaboratively with their partners on problem #8 and then share out their solutions and strategies.



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Segment 3


45-60 minutes



Resources: Handout 2: Division of Mixed Numbers and Fractions (Area Models) Fraction Division Routine

Focus: Division of mixed numbers/fractions by fractions – area models



MP7: Students look for patterns in area model diagrams and describe them to peers.

MP4: Look for students to describe real-world situations that correspond to

Differentiation for Remediation: Use the Matching Representations area models and division equations as a tool to have students match the models to the equations. You can also have them match the real-life situations.

Differentiation for English Language Learners:

Differentiation for Enrichment: Students create their own area models and trade with other students to interpret.


Students may struggle with interpreting the fractions in the remainder or with creating the division models. They may want to just use the algorithm without the models.


Students would benefit from practice dividing whole numbers and fractions with models to retain what was taught in Segment 2.

Steps: 1. Have students study the model for question #1 briefly, and turn

and talk with their partner to discuss what it shows. Then ask what strategy they could use to answer question #1. Students are provided the model so that they can interpret it. The goal is for students to see that the figure is already partitioned to show 5 sets of ½. They could count to find how many ½ portions are in the shaded portion, as the shaded portion shows 4 ½. Using their understanding that multiplication and division are related, they could think of this as a missing factor problem, as they did in segment 2. ? x ½ = 4 ½? Then they can

record the division equation, 41

2 ÷

1

2= 9. Students should

discuss a real world situation where they would divide 4 ½ by ½ and interpret what the quotient represents in that context.




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2. Have students work collaboratively on problems 2 and 3 while you circulate to observe and ask guiding questions or pull a small group who need more support.


3. Ask students what is different about problem #5. Have them

share out their solution strategies for addressing the problem with this model. Note: The process is the same, but students may not recognize the model since there are discrete portions instead of one long model that is connected.

4. For Problem #5, ask students to identify the dividend and how the model should be partitioned to represent the dividend. (15/16) If no students recognize that 15/16 is being divided by 3/8, help them to consider the missing factor multiplication expression and the division expression first before making the model. Then guide them to partition the bar to represent the 15/16. Have them then partition out portions of 3/8 to find how many total portions of 3/8 are in the figure. Some students may forget to convert between eighths and sixteenths. Allow students to work collaboratively in partners to address parts a and b based upon their model.

5. Continue guiding students to model problem #6. Again ask students to identify the dividend and how the model should be partitioned to represent the dividend. (1/8) Have them shade the 1/8 portion. Then use an arrow to indicate ½ of the figure. Ask students how much of the ½ portion is shaded. When students indicate 1 of the 4 sections, ask what fraction is used to represent that portion and record the equations to represent the model. Allow students to work collaboratively in partners to address parts a and b based upon their model.

6. It is up to the teacher’s discretion to determine if students can work collaboratively on questions 7 and 8, or if they will need to be guided with questions from the teacher.

7. Have students complete the Area Model and Create a Real-World Problem portions of the Fraction Division Routine. (They may choose their own, or you may give them all the same expression.)



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Segment 4


45- 60 minutes



Resources: Handout 3 Division of Mixed Numbers and Fractions (Number Line Models) Segment 4 PowerPoint (optional) Fraction Division Routine

Focus: Division of mixed numbers/fractions by fractions – number line models


Math Practice Look For(s): MP 7: Look for and make use of structure. Students look for the patterns in the number line models.

Differentiation for Remediation: Use the Matching Representations measurement models and division equations as a tool to have students match the models to the equations. You can also have them match the real-life situations. Differentiation for English Language Learners: Differentiation for Enrichment: Students create their own measurement models and trade with other students to interpret.



1. Write a number line like this on the board (or show ppt. slide #2):

0 Ask students where 2/3 would be. Then record the following on the number line:

0 3 Ask students where 2/3 would be now. Have students turn and talk about why the location of points might change on number lines.




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2. Distribute Handout 3 Division of Mixed Numbers and Fractions (Number Line Models) to students for their guided notes. Ask students to study the number line model in problem number 1 for 30 seconds. (THINK) Then have them discuss what information is already provided in the number line with a peer. (PAIR) Have some pairs share what they discussed. (SHARE)


3. Ask students how they might restate the question as a

multiplication equation (? X 3

5=

33

10)

4. Ask students what the arrow indicates on the diagram. When

someone says it is a portion of 3

5, ask how they would determine

how many 3

5 portions are in

33

10. When a student suggests

continuing to record the arrows of 3

5 portions, have the students

do so. They will see that there are 5 complete portions of 3

5. Then ask them what to do with the remaining 3 sections on

the number line. If no suggestions are reasonable, ask how many sections on the number line are needed for a new leap of 3

5. Students will need to determine that the 3 sections out of

are represented by the remaining portion on this number line.

They should record the division equation as 33

10÷

3

5= 5

3

6 or 5

1

2 .

(You may wish to have students discuss that both of these responses are correct because they are equivalent values.)

5. Ask students, “Is it likely for a real-world situation to be presented as an improper fraction? Or is it more likely to have it presented as a mixed number?” Have students consider the

following context: “Manny has 3 2

5 hours set aside to exercise

each week. If his exercise sessions last 3

5 of an hour each time,

how many sessions can he do each week?” Does this context match the diagram and equations?

6. Have students compare the number line in question #2 to question #1. What is similar and what is different? How might the question be restated as a multiplication

equation? (? X 3

8= 1

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4) Ask how the number line can be

partitioned to model the problem. (See the teacher notes.) Students partition the number line and record the division equation that matches by working collaboratively with their partners. Have students consider possible contexts for this problem (THINK) and then discuss their context with a partner (PAIR). Then solicit contexts to share with the whole class. (SHARE). Students record a sensible situation to correspond with the diagram and equations.



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7. Before working on #3 with the students, have read the following context: Sam still needs to water 5/6 of the field. He is able to water 2/3 of the field in an hour. How many hours will it take him to finish watering? Have students record this context or share other ideas for when 5/6 can be divided by 2/3. Then proceed with recording the multiplication equation and division equation (without the solution) before partitioning the model. (See teacher notes.) Have student interpret the remainder in the context of the situation they recorded.


8. Problem #4 involves division of a smaller fraction by a larger fraction. Provide the following context: 2/3 of a box of cookies was placed on a platter to serve at a party. If each guest only ate 5/6 of his or her portion of cookies, how much of the box was actually eaten? You may have students brainstorm similar contexts or record this one on their notes page. Then proceed with recording the multiplication equation and division equation (without the solution) before partitioning the model. (See teacher notes.) Have student interpret the remainder in the context of the situation they recorded.

9. Have students compare their work for problems 4 and 5. Ask what they notice about the 2 problems and their solutions. Have them reflect with their peers about what they see in the 4 problems that were addressed in the number line models.

10. Give students the formative assessment: Context Stories. Students should complete this task independently. They will represent each of the 4 situations in different forms: multiplication and division expressions, area model, number line diagram and solution. (You may choose to have students make either an area model or a number line diagram, but they should record at least 1 visual to support their strategic thinking.)

11. Have students complete the Measurement Model section for a fraction division expression. You may also choose to have students do all 3 parts (measurement, area and real-world) for a different expression.



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Segment 5


40-50 minutes



Resources:

Segment 5 Division Patterns PowerPoint

Handout 4: Division Patterns

Fraction Division Routine

Focus: Students develop understanding of the algorithm for dividing fractions by fractions by examining patterns



MP7: Look for and make use of structure. Students look for patterns in diagrams and tables, relating these to the division operation.

Differentiation for Remediation: Students who struggle may not have much to discuss with peers. Pull a small group to facilitate the conversation while the other students are discussing in partners.


Differentiation for Enrichment: Students work through the Handout and then generate their own patterns in a table and in visual models.


Students do not share ideas with peers.


Steps: 1. Show the PowerPoint slide 2 and have students discuss with a

partner. Students take notes on Handout 4: Division Models by recording their thinking in #1. Then show PowerPoint slide 3 and have students compare what they discussed to the model. Students record their notes on Handout 4.


2. Show PowerPoint slide 4 and have students discuss with a partner. Students take notes on Handout 4: Division Models by recording their thinking in #2.

Then show PowerPoint slide 5 and have students compare what they discussed to the model. Students record their notes on Handout 4.



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3. Show PowerPoint slide 6 and have students discuss with a partner. Students take notes on Handout 4: Division Models by recording their thinking in #3. Then show PowerPoint slide 7 and have students compare what they discussed to the model. Students record their notes on Handout 4.




6. Show PowerPoint slide 12 and have students discuss what the problem is asking. Show slide 13 and ask students to interpret the visual model and then record it on #6. Then show slide 14 and have students answer the questions by referring to the model.

7. Show slide 15 and have students interpret the table. Students record their thinking on Handout 4 in number 7. Students describe the pattern in the table for #8.

8. Have students work on the table in #9 on Handout 4 collaboratively or independently. Have students share out their tables and the pattern they see in #10.

9. Have students complete the Measurement, Area, and Real-World portions for another fraction division expression on the Fraction Division Routine.



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Segment 6


80-100 minutes



Resources:

Handout 5: Rewriting Reciprocals

Reasoning with Reciprocals Assessment

Focus: Reciprocals and the division algorithm



MP7: Look for students to interpret the pattern regarding the use of reciprocals for division of fractions by fractions.

Differentiation for Remediation: Students will need to be guided through the exercise step-by-step.

Differentiation for English Language Learners: Pre-teach the term “reciprocals” prior to this segment. They will need to be guided to interpret the visual models throughout the exercise. Create visual (area/measurement) models for each of the 5 expressions.

Differentiation for Enrichment: Have students explain in writing how they used reciprocals in the activity. Explain why changing mixed numbers to improper fractions is a reasonable step.



Steps:

1. Students work in small groups to answer questions 1 and 2, creating diagrams to justify their reasoning. Have some partners share out their diagrams. (See Teacher notes on Handout 5.)


2. Guide students to interpret the diagram in problem 3. Ask what portion of the diagram shows 2/3? What does the white portion of the diagram show? How does the diagram show the “whole”? (See Teacher notes on Handout 5.)

3. Guide students to interpret the number line diagram in problem #4. Ask what the arrow indicates. Ask what number would be written below the end of the second arrow. Ask how many more sections would be needed to have a 3rd whole portion of 3/8. Ask why they think the mixed number is to be written as an improper fraction. (See Teacher notes on Handout 5.)



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4. Ask students to interpret the number line model in #5. If students do not see the response to the question, guide them to count the leaps to the 1 on the number line and the total number of leaps it would take to get from 0 to 7/4. (See Teacher notes on Handout 5.)


5. Have students reflect upon what they notice in #6. Then have them share out what they noticed. (See Teacher notes on Handout 5.)

6. Students work collaboratively on problems 7-11 and then share out responses. (See Teacher notes on Handout 5.)

7. Direct students to the 3rd page of Handout 5. Ask students why they are being asked to think “? x 5/6 = 2/8” (Because division can be thought of s a missing factor problem) Complete #1 as a whole group by asking students, “What times 5/6 = 1? Let’s fill that in as the fraction on both sides of the equal sign. “We will learn more about this in unit 5, but know that the two sides of the equal sign are equivalent as long as you do the same action on both sides. So, we will multiply both sides by the same fraction.” Ask students what the new equation would be on the last line?

8. If students seem to notice the patterns, have them work on problems 2-5 collaboratively. If not, guide students in either a whole group or small group to address their misunderstandings.

9. Have students reflect about what they notice happens in the last step of each of these division problems.

10. Students complete Reasoning with Reciprocals either alone or in partners to demonstrate their understanding of the use of reciprocals.



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Segment 7




Resources:

Fraction Division Routine

Fraction Tile Game

Reflecting Upon the Fraction Division Algorithm

Focus: Look for structure in fraction division equations/fraction division practice (Tile Game)



MP2: Students reason abstractly and quantitatively when they predict the greatest quotient.

MP3: Look for students to explain their predictions and their answers to each other, especially when they disagree.

MP6: Students use precision in calculation by using the reciprocal appropriately.

MP7: Students look at the structure of the equations to make their predictions—noticing patterns that indicate greater quotients. Students reflect upon the structure by writing about their ideas and providing appropriate examples.

MP8: Some students may make generalizations about the structure of the equation leading to greater quotients.

Differentiation for Remediation: If students do not complete the routine with ease, pull them into a small group to discuss the algorithm and models prior to playing the game.


Differentiation for Enrichment: Have students circulate the room to find examples of situations that where the predictions for largest quotient might be too close to predict and explain why.



Steps:

1. Give students the Fraction Division Routine with a new expression as an entry ticket. Determine which students struggle and which students can model division independently.




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2. Teach students to play the Fraction Division Tile Game by playing the game against a student in front of the class. Have students play the game with partners and circulate while they work.

Directions: This game requires two players. Cut out the fraction tiles below. All fraction tiles should be turned over face down. Each player draws one tile. Both players record the numbers in the appropriate squares on their own record sheets and predict whether Problem A or Problem B will have the greater quotient. Players then compute both quotients and determine if their predictions were accurate.


3. Give students the Reflecting Upon the Fraction Division Algorithm Formative Assessment



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Segment 8


40-60 minutes



Resources:

Interpreting and Computing Quotients of Fractions

Focus: Summative Assessment



MP1: Students are making sense of the real-world and mathematical situations by examining models and the work of others.

MP2: Students reason about the values of fractions when considering if the calculations are reasonable according to the situations.

MP3: Students critique the work of others and explain their own thinking.

MP4: Students use models to solve real-world problems and interpret models.

MP6: Students attend to precision by calculating precisely and by closely reading the real-world problems.

MP7: Students use their understanding of the fraction division algorithm to divide fractions.

Differentiation for Remediation: You may wish to read problems with students who struggle.

Differentiation for English Language Learners: You may wish to read problems with English Language Learners to make sure they interpret the vocabulary appropriately.

Differentiation for Enrichment:



Steps:

1. Give students the Interpreting and Computing Quotients of Fractions Assessment. Students complete the problems independently. Circulate to observe the models students are using to make sense of the problems.


Documents

Targeted Content Standard(s): Student Friendly Learning ...ioer.ilsharedlearning.org/ContentDocs/bc2cc184-41bf-464b-a363-11a554da4126/303/G6_U1...6. Problem #5 is the first opportunity