Download - SFUSD Mathematics Core Curriculum Development Project · 1 SFUSD Mathematics Core Curriculum, Grade 6, Unit 6.2: Fractions and Decimals, 2014–2015 SFUSD Mathematics Core Curriculum

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SFUSD Mathematics Core Curriculum, Grade 6, Unit 6.2: Fractions and Decimals, 2014–2015

SFUSD Mathematics Core Curriculum Development Project

2014–2015

Creating meaningful transformation in mathematics education

Developing learners who are independent, assertive constructors of their own understanding

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Grade 6

Unit 6.2: Fractions and Decimals

Number of Days

Lesson Reproducibles Number of Copies

Materials

2 Entry Task CPM CCC1 Lesson 6.1.1 (2 pages) HW: CPM CCC1 Lesson 6.1.1 (2 pages)

1 per pair CPM eBook

Licorice, 2 per student (optional) Straws, more than 2 per student Scissors Poster paper and markers Glue sticks or tape

3 Lesson Series 1 CPM CCC1 Lesson 6.1.2 (3 pages) HW: CPM CCC1 Lesson 6.1.2 CPM CCC1 Lesson 6.1.3 (3 pages) HW: CPM CCC1 Lesson 6.1.3

1 per pair CPM eBook 1 per pair CPM eBook

Counters, 10 per team (optional)

1 Apprentice Task Rabbit Costumes Interpreting a Division Computation

1 per student 1 per student

3 Lesson Series 2 CPM CCC1 Lesson 6.1.4 (4 pages) Resource Page 6.1.4 HW: CPM CCC1 6.1.4 (3 pages) Setting Goals

1 per pair 1 per student CPM eBook 1 per pair

1 Expert Task How much Money Reasoning about Multiplication and Division and

Place Value

1 per student 1 per student

4 Lesson Series 3 CPM CCC1 Lesson 7.2.1 (3 pages) Resource Page 7.2.1 HW: CPM CCC1 Lesson 7.2.1 CPM CCC1 Lesson 7.2.2 (2 pages) Resource Page 7.2.2 (half-page) HW: CPM CCC1 Lesson 7.2.2 CPM CCC1 Lesson 7.2.3 (2 pages) HW: CPM CCC1 Lesson 7.2.3

1 per pair 1 per student CPM eBook 1 per pair 1 per pair CPM eBook 1 per pair CPM eBook

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CPM CCC1 Lesson 7.2.4 (2 pages) HW: CPM CCC1 Lesson 7.2.4

1 per pair CPM eBook

1 Milestone Task Ribbons and Bows (2 pages) Constructed Response

Provided by AAO

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Unit Overview

Big Idea

The inverse relationship between multiplication and division can be used to explain why the procedures for dividing fractions make sense.

Unit Objectives

● Students will be able to use operations with decimals and long division to apply to division of fractions by fractions. ● Students will be able to use visual models to represent and solve problems involving division of fractions. ● Students will develop fluency with long division of multi-digit numbers and decimal numbers.

Unit Description

This unit begins by capitalizing on students’ prior knowledge and experiences with division of whole numbers, decimal numbers, and fractions with common denominators. Lessons move through tasks that emphasize long division skills, operations with decimal numbers that include addition, subtraction, multiplication and division, and culminate in activities focused on dividing fractions by fractions with unlike denominators. Students explore many different visual models for representing the division of fractions, and also investigate several methods for dividing, including a common denominator method and using a “Super Giant One” to simplify complex fractions.

CCSS-M Content Standards

The Number System

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

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 length 3/4 mi and area 1/2 square mi?

Compute fluently with multi-digit numbers and find common factors and multiples.

6.NS.2 Fluently divide multi-digit numbers using the standard algorithm.

6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

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Progression of Mathematical Ideas

Prior Supporting Mathematics Current Essential Mathematics Future Mathematics

In previous grades, students add and subtract fractions with unlike denominators, understand that a fraction is a division problem, and multiply and divide fractions by whole numbers. Students understand the place value system and perform operations with multi-digit whole numbers and with decimals to hundredths.

Students will learn about the relationship between multiplication and division, as well as between fractions and decimals. Students will apply this knowledge through real-world problem-solving tasks. They will be able to demonstrate these skills using equal groups, arrays, area models, number lines, multiplication tables and word problems.

In seventh grade, students will apply and extend previous understanding of operations with fractions to add, subtract, multiply and divide rational numbers. They also represent and solve problems involving proportions.

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Unit Design All SFUSD Mathematics Core Curriculum Units are developed with a combination of rich tasks and lessons series. The tasks are both formative and summative assessments of student learning. The tasks are designed to address four central questions: Entry Task: What do you already know? Apprentice Task: What sense are you making of what you are learning? Expert Task: How can you apply what you have learned so far to a new situation? Milestone Task: Did you learn what was expected of you from this unit?

2 days 3 days 1 day 3 days 1 day 4 days 1 day Total Days: 15

Lesson Series 1

Lesson Series 2

Lesson Series 3

Entr

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Entry Task Dividing Licorice

Apprentice Task Rabbit Costumes

(Interpreting a Division Computation)

Expert Task How Much Money?

Multiplication, Division, and Place Value)

Milestone Task Ribbon and Bows; Constructed

Response Questions

CCSS-M Standards

6.NS.1

6.NS.1, (6.NS.2, 6.NS.3) 6.NS.1, 6.NS.2 6.NS.1, 6.NS.2

Brief Description of task

Students work together to devise ways to share whole numbers of licorice equally among different numbers of people.

Make sense of and solve a real-world problem involving multiplication and division of fractions. If time permits, a second short task involves reasoning about the algorithm for long division.

The first part of the task challenges students to use their understanding of multiplication, division, and place value to solve problems without calculating by hand. The second part of the task asks students to use and explain a visual model that represents dividing fractions.

Students solve real-world and mathematical problems involving the division of fractions or decimals.

Source CPM Core Connections Course 1 Section 6.1.1, Problems 6-1 to 6-4

Part 1: MARS Task 2003 Part 2: Illustrative Mathematics

Part 1: MARS Task 2005 Part 2: Illustrative Mathematics

SVMI 2013; CPM Core Connections 1 Assessment Bank

Lesson Series 1

Lesson Series 2

Lesson Series 3

CCSS-M Standards

6.NS.1, 6.NS.2

6.NS.1

6.NS.1, 6.NS.2, 6.NS.3

Brief Description of Lessons

This series builds on the context of dividing pieces of licorice evenly among several people to help students make sense of division. Students construct visual models and make connections among models to build their understanding of division and fractions. They also explore and make sense of the algorithm for long division. As they solve problems involving division, they must consider the meaning of the “whole” in different contexts and models. They reinforce their understanding by using multiplication to check their results.

Students extend their understanding to include division of fractions by fractions and continue to represent division using visual diagrams, including a new area model, number sentences, and verbally. The third day of this lesson series gives time for you to review. Have students do presentations, or give additional practice as needed.

Students deepen their understanding of division and move from visual models to more efficient methods for dividing fractions. Then students look at how division is like a ratio.

Sources

CPM Core Connections Course 1, Chapter 6 CPM Core Connections Course 1, Chapter 6 Illustrative Mathematics

CPM Core Connections Course 1, Chapter 7

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Entry Task

Dividing Licorice

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: ● Students will identify problems that can be solved using division. ● Students represent division of fractions using diagrams.

CCSS-M Standards Addressed: 6.NS.1 This task will take 2 days. Materials: licorice, about 2 strands per student (optional) straws, more than 2 per student scissors poster paper and markers glue sticks or tape Homework: Day 1: 6-5, 6-6, 6-9 Day 2: 6-10, 6-11, 6-12, 6-13, 6-14 Read the Teacher Guide pp. 522–525 to prepare for this task and Lesson Series 1.

Launch: Use the opening questions for Lesson 6.1.1 to introduce the theme of sharing equally. Tell students that you will be sharing licorice with them, but they need to figure out how to divide it fairly. Have students read problem 6-1, first individually, before sharing with a partner and if needed, the class. During: Divide the class into groups of 3, 5, and 6, and name them W, X, and Y to correspond to the scenarios in Problem 6-1. The teams will solve each problem using diagrams and models (straws). Encourage groups to find multiple ways to represent their solutions. Ask questions to prompt student thinking and remind students to keep track of the “whole” as they divide and subdivide the original straws or licorice pieces. Use questions to assess students understanding of fractions. In problem 6-2, each group will prepare a poster and a presentation. Ask group members to plan who will present each part; establish the norm that every group member should be responsible for presenting something mathematical in the presentation. Remind students of the requirements for their presentation: diagrams, a division number sentence, words explaining the solution, and an explanation of how students know that they are sharing fairly. Presentations may start on the first day if time permits. After all groups have presented, ask whether all students in the class would receive a fair share if you gave out 5 pieces to groups of 3, 9 pieces to groups of 5, and 10 pieces to groups of 6. Teams should work together to answer problem 6-3, and then share out part c. Closure: Have teams or pairs work on 6-4. Use this problem to assess students’ proficiency. with addition of fractions.

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Dividing Licorice

Students will work with a team to make sense of diagrams

Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them 3. Construct viable arguments and critique the reasoning of others.

Structures for Student Learning: Academic Language Support: The Universal Access section of the CPM Teacher Edition describes vocabulary strategies such as personal dictionaries or a class word wall.

Vocabulary: division; quotient; ratio; part; whole; fair share; portion Sentence frames: This looks like___ whole pieces. This looks like _____ whole units divided into_____ parts.

Differentiation Strategies: Have students create problems of their own, using dividing 2 pizzas among 11 children. Participation Structures (group, partners, individual, other): Teams of three, five, and six students. Also pairs for think-pair-share.

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Lesson Series #1

Lesson Series Overview: This series builds on the context of dividing pieces of licorice evenly among several people to help students make sense of division. Students construct visual models and make connections among models to build their understanding of division and fractions. They also explore and make sense of the algorithm for long division. As they solve problems involving division, they must consider the meaning of the “whole” in different contexts and models. They reinforce their understanding by using multiplication to check their results. CCSS-M Standards Addressed: 6.NS.1, 6.NS.2 Time: 3 days

Lesson Overview – Day 1 Resources

Description of Lesson: Students make sense of hypothetical diagrams and explain how licorice is being divided among groups. Time: 1 day Notes: Read the CPM Teacher Notes for Lesson 6.1.2. Some are excerpted below. Problem 6-15 revisits the licorice division scenario, in which students are asked to think about the connections between the amount of licorice each student receives and the operation of division. This could be done as a Think-Pair-Share. Before pairs or groups begin to work, post and introduce the focus questions for the lesson:

1. What does this represent? 2. How does this connect to what we already know how to do? 3. How can we apply this strategy?

As teams discuss each of the diagrams and then try to apply the ideas, listen for statements such as, “This looks like 1/3 four times”, “This looks like one whole piece was given to each person and then another 1/3,” and “This looks like four whole units divided into three parts.” As teams finish problem 6-15, bring the class together for a discussion. Ask students to describe the differences among the diagrams that represent dividing licorice. Students may

CPM CCC1 Lesson 6.1.2 Problems 6-15 to 6-19, 6-20 (optional) Homework: 6-21, 6-22, 6-24

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notice that in some diagrams, the amount that each person receives is a sum of small parts. You may wish to lead problem 6-18 as a full-class activity because of the complexity of the instructions. It is an important problem for tying together the three standards for this unit. For problem 6-19, make sure that students use long division and not calculators to find the decimal representation. This is part of developing fluency. 6-20 is a Learning Log or journal prompt. This is optional.


Description of Lesson: In this lesson the context shifts from licorice to pies. Students identify problems that can be solved with division, interpret different meanings of division, and make connections between division and multiplication. They use multiplication to check division, and they continue to use diagrams to model and solve problems. Very importantly, students move among diagrams and number sentences involving division and multiplication. Notes: Read the CPM Teacher Notes for Lesson 6.1.3. Some are excerpted below. The notes include very good suggestions for team strategies and a deep discussion of the mathematics of the lesson. You may wish to introduce and post the focus questions and remind students to use these questions to help them get “unstuck”:

1. How can we represent this with a diagram? 2. Is there another way to see it? 3. What is it “part” of? What is the whole?

For problem 6-26, make sure that groups come up with the number sentence 3 ÷ (1/8) = 24, which relates to problem 6-27. You may ask groups to share the number sentences and prompt groups to write a number sentence using division.

CPM CCC1 Lesson 6.1.3 Problems 6-26 to 6-29 Homework: 6-33, 6-34, 6-36

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Description of Lesson: This day gives students additional practice in solving problems using division and checking with multiplication. Notes: Optional materials, 10 counters per team Make counters available for students to model concretely.

CPM CCC1 Lesson 6.1.3 Problems 6-30, 6-31, 6-32 (optional, on homework page) Homework: 6-38 to 6-42

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Apprentice Task Rabbit Costumes; Interpreting a Division Computation



Math Objectives: ● Students demonstrate understanding of multiplication and division of

fractions. ● Students solve real-world problems involving multiplication and

division of fractions. ● Students use their understanding of place value, multiplication, and

division to make sense of the algorithm for long division. CCSS-M Standards Addressed: 6.NS.1 (for part 2, 6.NS.2, 6.NS.3)

Potential Misconceptions: Students may have trouble working with mixed numbers. They may not check every constraint in the problem. They may use the wrong operation, such as multiplication instead of division.

Launch: Introduce the scenario and its vocabulary to the class: Gail is making costumes for a play but she has a limited amount of fabric. You may relate the making of a costume to following a recipe. For each rabbit costume, she has to use 1½ yd of white fur fabric, 1 yd of blue striped fabric and ¼ yd of pink felt. During: You may have students work individually on this task and then later work with a partner, depending on the progress students make. The explanation for the second question is important; let students know that they need to convince you of their reasoning. Closure/Extension: If time permits, have students do the second task, Interpreting a Division Computation. This task has students make sense of the long division algorithm to find products.

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Rabbit Costumes; Interpreting a Division Computation

How will students do this?

Focus Standards for Mathematical Practice: 2. Reason abstractly and quantitatively. 7. Look for and make use of structure.

Structures for Student Learning: Academic Language Support: Make sure students understand the contextual vocabulary: costumes, fabric, felt, yard. Also, students may have trouble with the fractions spelled out: “a quarter of a yard,” “one and three quarter yards.”

Vocabulary: division, representation, fraction, decimal Sentence frames:

Differentiation Strategies: Participation Structures (group, partners, individual, other): Think-pair-share

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Lesson Series #2

Lesson Series Overview: Students extend their understanding to include division of fractions by fractions and continue to represent division using visual diagrams, including a new area model, number sentences, and verbally. The third day of this lesson series gives time for you to review. Have students do presentations, or give additional practice as needed. CCSS-M Standards Addressed: 6.NS.1 Time: 3 days


Description of Lesson: Students extend their understanding to include division of fractions by fractions and continue to represent division using visual diagrams, number sentences, and verbally. Notes: Read the CPM Teacher Notes for Lesson 6.1.4. Some are excerpted below. The notes include very good suggestions for team strategies and a deep discussion of the mathematics of the lesson. You may wish to introduce and post the focus questions and remind students to use these questions to help them get “unstuck”:

1. How can we represent this problem with a diagram? 2. Can we represent it in more than one way?

Bring the class together to share and discuss their findings for 6-43 and 6-44. If you hear a group having a productive discussion, ask them to share with the class. Problems 6-45 and 6-46 use the context of constructing a dollhouse, which lends itself to linear models. If students don’t finish, they can continue working on these problems on the next day.

CPM CCC1 Lesson 6.1.4 Problems 6-43 to 6-46 Resource Page 6.1.4 Homework: 6-52 to 6-55

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Description of lesson: This lesson introduces the area model for division and challenges students to write their own questions to correspond to division problems. Notes: It may be very challenging for students to write questions to fit the expressions in Problem 6-48. You might encourage them to look back at the problems and contexts they have seen in this unit (sharing licorice, pie, music notes, and construction).

CPM CCC1 Lesson 6.1.4 Problems 6-47 to 6-50 (6-50 is on the homework page) Homework: 6-57 to 6-60


Description of Lesson: This is a day to have students synthesize their learning or do more focused practice or review. Notes: You might have students make a poster for Problem 6-50. Ask them to provide examples that justify their answer for each question. They might also share and discuss the questions they wrote for problem 6-48. You can also select from the problems listed at right.

For dividing decimals, use Illustrative Mathematics Setting Goals.

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Expert Task

How Much Money?; Reasoning about Multiplication and Division and Place Value



Math Objectives: ● Students will demonstrate understanding of place value to find and

explain products and quotients of multi-digit numbers and decimals. ● Students will use a fraction model to solve and explain a real-world

problem involving the division of fractions. CCSS-M Standards Addressed: 6.NS.1, 6.NS.2, 6.NS.3

Potential Misconceptions: Students may not realize that they can use knowledge of place value to quickly solve the multiplication and division problems. Help students explain their thinking verbally and with number sentences. If 13 times 17 equals 221, then 13 times 1.7 will be 22.1 because it is one-tenth the original product. 13 x 1.7 equals 13 x 17 x 0.1, and 13 x 17 x 0.1 = 221 x 0.1.

Launch: Let students know that they will be solving one problem today that uses a model for fraction division, and another problem (with multiple parts) about multiplying and dividing decimals. The important thing for both problems is that students show and explain their thinking completely. They should imagine that they are explaining their process to another student. During: Have students work alone on the tasks without calculators. For the Reasoning about Multiplication and Division and Place Value problem, students must either show all of their computations or explain how they got the answer. If they have to work some or all of the problems out by hand, it may take them quite a while to finish. Closure/Extension: Problem 2 from “How much Money,” can be used as an extension. If all students finish, you might discuss the problems as a class.

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How Much Money?; Reasoning about Multiplication and Division and Place Value


Focus Standards for Mathematical Practice: 4. Reason abstractly and quantitatively. 7. Look for and make use of structure.

Structures for Student Learning: Academic Language Support:

Vocabulary: part, whole, pieces, quotient, one-tenth, one-hundredth Sentence frames:

Differentiation Strategies: Some students may approach the decimals task by working out each problem by hand instead of using place value to find the products and quotients. Guide those students to look for a relationship between their calculated answer and the original number sentence. Participation Structures (group, partners, individual, other): Individual, pairs (optional)

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Lesson Series #3

Lesson Series Overview: Students deepen their understanding of division and move from visual models to more efficient methods for dividing fractions. CCSS-M Standards addressed: 6.NS.1, 6.NS.2, 6.NS.3 Time: 4 days


Description of Lesson: Students work with linear models for dividing fractions, building on their work from Lesson Series 2. Real-world contexts including sewing and comparing fractional distances. The method of “common denominator” division is introduced and verified with diagrams and reasoning about fractions and division. Notes: Read the CPM Teacher Notes for Lesson 7.2.1. You may want to do part of 7-35 as a full-class discussion. After groups or pairs finish 7-35, bring the class back together to share their strategies. Problem 7-38 introduces a common denominator algorithm for dividing fractions; this method is used in the problems 7-39, 7-40, and 7-41. Students will need problem 7-40 for the next day, so make sure they keep their work. Materials: Resource Page “Making Bows,” one per student

CPM CCC1 Lesson 7.2.1 Problems 7-35 to 7-43 Resource Page 7.2.1 Homework: 7-44, 7-45, 7-48 (7-46 and 7-47 are optional)


Description of Lesson: Students investigate strategies for dividing fractions, including the “standard” algorithm of invert and multiply. The lesson uses painting a fence as the context and begins with students estimating the amount of paint needed to cover a fence. Notes: Read the CPM Teacher Notes for Lesson 7.2.2.

CPM CCC1 Lesson 7.2.2 Problems 7-49 to 7-51 Resource Page 7.2.2 Homework: 7-52, 7-53, 7-55, 7-56

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Description of Lesson: Students will divide fractions using a Giant One strategy. They will also connect fraction division to decimal division. Notes: Read the CPM Teacher Notes for Lesson 7.2.3.

CPM CCC1 Lesson 7.2.3 Problems 7-57 to 7-63 Homework: 7-64 to 7-68


Description of Lesson: Students will connect ratios with fraction division. Notes: Read the CPM Teacher Notes for Lesson 7.2.4.

CPM CCC1 Lesson 7.2.4 Problems 7-69 to 7-73 Homework: 7-74 to 7-78

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Milestone Task

Ribbons and Bows (Performance Assessment); Constructed Response Questions



Math Objectives: ● Students solve real-world problems involving the division of fractions. ● Students solve real-world problems involving the division of decimals. ● Students create visual models to represent fractions and fraction

problems. ● Students use understanding of place value, multiplication, and division

to estimate reasonable answers for products and quotients of decimals.

● Students compute fluently with decimals. CCSS-M Standards Addressed: 6.NS.1, 6.NS.2, 6.NS.3 Potential Misconceptions:

This is an individual assessment. Students will complete the assessment without the use of calculators, but you may assist with vocabulary clarify instructions as needed.

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Ribbons and Bows (Performance Assessment); Constructed Response Questions


Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 3. Construct viable arguments and critique the reasoning of others.

Structures for Student Learning: Academic Language Support:

Vocabulary: estimate Sentence frames:

Differentiation Strategies: Participation Structures (group, partners, individual, other): Individual

Download - SFUSD Mathematics Core Curriculum Development Project · 1 SFUSD Mathematics Core Curriculum, Grade 6, Unit 6.2: Fractions and Decimals, 2014–2015 SFUSD Mathematics Core Curriculum

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