# GRADE 6 MATH: SHARE MY CANDY

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1 GRADE 6 MATH: SHARE MY CANDY UNIT OVERVIEW The length of this unit is approximately 2-3 weeks. Students will develop an understanding of dividing fractions by fractions by building upon the conceptual knowledge gained in grade 5 of dividing fractions by whole numbers and whole numbers by unit fractions (5.NF7). Students use the meaning of fractions, the meaning of multiplication and division, and the relationship between multiplication and division to understand and explain why the procedures for dividing fractions make sense. Following this unit, students can use this knowledge to reason about ratio and rate problems. TASK DETAILS Task Name: Share My Candy Grade: 6 Subject: Math Depth of Knowledge: 3 Task Description: In the task Share My Candy, students show their ability to interpret and compute quotients of fractions in a real- world Students may choose to perform the division of the fractions using either visual models or numerical symbols. Standards: 6.NS.1: Interpret and compute quotients of fractions, and solve real word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to interpret the Standards for Mathematical Practice: MP.1. Make sense of problems and persevere in solving them. MP.2. Reason abstractly and quantitatively. MP.3. Construct viable arguments and critique the reasoning of others. MP.. Model with mathematics. MP.6. Attend to precision. 1

2 TABLE OF CONTENTS The task and instructional supports in the following pages are designed to help educators understand and implement Common Core aligned tasks that are embedded in a unit of instruction. We have learned through our pilot work that focusing instruction on units anchored in rigorous Common Core aligned assessments drives significant shifts in curriculum and pedagogy. Callout boxes and Universal Design for Learning (UDL) support are included to provide multiple entry points for diverse learners. PERFORMANCE TASK: SHARE MY CANDY...3 RUBRIC...6 ANNOTATED STUDENT WORK...8 INSTRUCTIONAL SUPPORTS...21 UNIT OUTLINE SOLUTION PATHS FOR TEACHERS...36 ADDITIONAL INSTRUCTIONAL TASKS: CHERRY PIES AND POUND CAKES..38 Acknowledgements: The unit outline and culminating performance task was developed by Common Core Math Fellow Laginne Walker, District 7, X22. Feedback and revisions of the final unit were also given by Common Core Math Fellows Jennifer Hernandez, Prema Vora, Alice Grgas, and Cheryl Schafer. 2

4 Name: Class: Date: Share My Candy Jason has 3 candy bars. He wants to share the candy bars with his friends. He gives as many of his friends as possible himself. of a candy bar. He keeps the rest for PART A: How many friends can he give of a candy bar to? Show your work and write your answer in a complete sentence. SHOW WORK: ANSWER SENTENCE: PART B: How much of the candy bar will Jason keep for himself? Show your work and write your answer in a complete sentence. SHOW WORK: ANSWER SENTENCE:

5 PART C: Explain your reasoning for parts A and B. PART D: Show how you can check if your responses to parts A and B are correct. 5

6 GRADE 6 MATH: SHARE MY CANDY RUBRIC Attached is a rubric to assess student mastery of the learning standards in the performance task Share My Candy. This rubric assesses five categories: presentation, implementation, precision, communication, and solving. All categories are based on a - point scale with a maximum possible of 20 total points. 6

8 GRADE 6 MATH: SHARE MY CANDY ANNOTATED STUDENT WORK This section contains annotated student work at a range of score points. The student work shows examples of understandings and misunderstandings of the task. 8

9 LEVEL SAMPLE RESPONSE WITH ANNOTATIONS 6.NS.1: Student uses equations to represent a solution to a word problem involving division of fractions. 6.NS.1: Student interprets the quotient of 3 ½ 3/ using a clear and correct explanation. Student properly explains the meaning of both the whole number and the fraction within the context of this 9

11 LEVEL 3 SAMPLE RESPONSE WITH ANNOTATIONS 6.NS.1, MP.1 The is improperly labeled how much his friends get when it should be labeled how many friends get the portion of the candy bar. MP.1 Answer sentence does not indicate the interpretation of the product. This could be interpreted as half of the candy bar or half of the total number of candy bars. MP.3, MP.6 The explanation is partially correct. Student properly identifies the number of friends who get candy, but does not adequately explain why it is and not. Student also should have identified of what in the response. 11

13 LEVEL 2 SAMPLE RESPONSE WITH ANNOTATIONS MP.3, MP.6 Although the work is correct, the answer sentence, is incorrect. Each of the friends get three fourths of a candy bar, rather than a whole candy bar as noted in the answer sentence. 6.NS.1 Although the check is correct and complete, part C lacks any clear explanation of the reasoning used in parts A and B. There also is not explanation of why each step was taken. 13

15 LEVEL 1 SAMPLE RESPONSE WITH ANNOTATIONS MP.3, MP.6 Although the student s work is correct, the answer sentence is incorrect. Each friend gets three quarters of a candy bar rather than a whole bar. MP.1, MP.3, MP.6 The work in B, C, and D is incorrect. The explanations show a limited conceptual understanding and difficulty using mathematical language. 15

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21 GRADE 6 MATH: SHARE MY CANDY INSTRUCTIONAL SUPPORTS The instructional supports on the following pages include a unit outline with formative assessments and suggested learning activities. Teachers may use this unit outline as it is described, integrate parts of it into a currently existing curriculum unit, or use it as a model or checklist for a currently existing unit on a different topic. 21

22 Unit Outline Grade 6 in Math: Share My Candy UNIT TOPIC AND LENGTH: The length of this unit is approximately 2-3 weeks. It will focus on developing an understanding of dividing fractions by fractions by building upon the students basic conceptual knowledge gained in Grade 5 of dividing fractions by whole numbers and whole numbers by unit fractions (5.NF.7). Students use the meaning of fractions, the meaning of multiplication and division, and the relationship between multiplication and division to understand and explain why the procedures for dividing fractions make sense. Subsequent to this unit, students will use this knowledge to reason about ratio and rate problems. COMMON CORE CONTENT STANDARDS: Ø 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 interpret the MATHEMATICAL PRACTICES Ø MP.1: Make sense of problems and persevere in solving them. Students are required to make conjectures about the form and meaning of the solution. Additionally, students are expected to be able to check their answers to problems. Ø MP.2: Reason abstractly and quantitatively. Students are required to make sense of quantities and their relationships in problem situations. Students are also asked to attend to the meaning of quantities, not just how to compute them. Ø MP.3: Construct viable arguments and critique the reasoning of others. Students are required to justify their conclusions and communicate them to others. Ø MP.: Model with mathematics. Students are required to apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. They can analyze relationships mathematically to draw conclusions. Ø MP.6: Attend to precision. Students are required to calculate accurately and efficiently and express numerical answers with a degree of precision appropriate for the problem context. 22

23 BIG IDEAS/ENDURING UNDERSTANDINGS: Ø There are several ways to divide fractions by fractions. Ø It does not always make sense to have an answer that is a mixed number. Ø Division means repeatedly making equivalent groups. Ø Division and multiplication are inverse operations and, therefore, can be used to check each other. ESSENTIAL QUESTIONS: Ø What real life situations can be modeled by dividing a fraction by a fraction? Ø Why does the division of fractions algorithm work (i.e., multiplying by the reciprocal)? Ø How can the fractional portion of a mixed number be interpreted? CONTENT: Prior Knowledge: Ø Fluency with multiplication and division of whole numbers to generate factors and multiples Ø Understanding of fractions and fraction equivalence Ø Ability to convert a mixed number to an improper fraction Ø Understand when to multiply a fraction by a mixed number Ø Ability to explain why the procedure used to multiply a fraction by a mixed number works SKILLS: Ø Ascertain the reasonableness of answers. Ø Write all answers in complete sentences with appropriate labels. Ø Adequately justify a response or explain the applied strategy Ø Procedurally demonstrate the ability to multiply a fraction and a mixed number. VOCABULARY/KEY TERMS: part, whole, product, quotient, partial number, remainder, solution, mixed number, common denominator, inverse operations, sum, difference, fraction, portion 23

24 LEARNING PLAN & ACTIVITIES: Understanding of Division of Fractions: Understanding division with fractions presents considerable difficulties for elementary, middle, and junior high school students. Most are unable to provide any justification for the standard algorithm, invert and multiply. Consequently, many students learn the algorithm and apply it mechanically without understanding the rich mathematical ideas behind it. Understanding the relationships among fractional quantities requires multiplicative reasoning which is the building block to proportional reasoning. Throughout this unit, students should consider why, when dividing fractions, do we invert and multiply? In order to promote optimal conceptual understanding, it is essential to connect prior understanding of unit fractions obtained in Grade 5 to the learning in this unit. Students will begin to move away from a strict memorization of algorithms and procedures and move toward deeper conceptual understanding of fractions and their applications. Class Discussions: Students should know that the rule of invert and multiply applies to division in general. It is a general principle, for example: 20 5, I can invert and multiply 20 x 1/5 =. With whole numbers, division can be thought of as making equal parts. When you divide something by 5, you are dividing it into 5 parts. In 5 th grade, students learn that dividing by 5 is the same as multiplying by 1/5. We can also think of fractional division as how many times does the divisor fit into the dividend? Students can also use this question to consider the reasonableness of their answers. In the example , clearly 1/2 can fit into more than two times. Or consider 2/8 11/12. Here the divisor is greater than the dividend. It will not even fit once. Students can begin to see the reasonableness of their answers before even performing any calculations. An alternative method for fraction division is to first change both the divisor and dividend into equivalent denominators, and then divide the numerators: = = 5 1 = 9 5 = 1 5 The quotient makes sense because 1/3 can go into 3/5 almost two times. This helps to make sense of the procedure. So instead of a student asking how many times does 5/15 go into 9/15, by making the denominators the same it becomes easier to see how many times 5 goes into 9. Students sometimes have an easier time understanding the idea of having 9 unit fractions and making groups of 5 unit fractions. Inverting and multiplying as an algorithm can be used as a shortcut, but only after students begin to understand the rationale behind it. Shortcuts are okay, but we also want to build on the conceptual knowledge. In addition, students should be taught how to check if division equations are true by multiplication. In the above example, students should know that if = 1 then it must be true that 1 = 2

25 INITIAL ASSESSMENT Application problems for adding, subtracting, multiplying fractions FORMATIVE ASSESSMENTS 1. Merry Models: Creating visual models from numerical models 2. Minding My Dividing: Dividing fractions by whole numbers and interpreting quotients and remainders 3. Penny s Pieces: Dividing fractions by fractions and interpreting remainders ADDITIONAL INSTRUCTIONAL TASKS 1. Cherry Pies 2. Pound Cakes FINAL PERFORMANCE TASK Share My Candy: Students may choose to perform the division of the fractions using either visual models or numerical symbols. Not specifying the method in which students should solve the problem makes the task more accessible to different types of learners. 6.NS.1 UNIT MAP DAY PURPOSE STANDARD SUPPORTS 1 Initial Assessment various Review using visual models to Review adding, subtracting, and add, subtract, and multiply multiplying two fractions. factions. 2 Realistic Remainders Students explore when it makes sense to have a partial number and when does it not. 3-5 Modeling fraction division Area model Number line Fraction strips Grouping tallies 6.NS.1 6.NS.1 Various types of models support learners of different learning modalities. Have students justify any numerical expression with a visual model to help build understanding of the link between the two representations 25

26 6 Formative Assessment 1 Assess whether students can model division of fractions visually. 6.NS.1 DAY PURPOSE STANDARD SUPPORTS 7-8 Whole numbers divided by a 5.NF.7 Use visual models and fraction and fractions divided by manipulatives. whole numbers Building upon what students learned in Grade 5 (dividing only by unit fractions), students now learn how to divide by any fraction, including ones where the numerator is not 1 and improper fractions. Also included in these lessons is a review of converting mixed numbers to improper fractions. Scaffold by starting with common denominators, progressing to denominators where one is a multiple of the other, then moving to relatively prime denominators Solving and creating word problems that involve 1 fraction and 1 whole number Focus on key words that indicate division. Also review modelling with BOTH numbers and pictures and use one method to check the other. 11 Formative Assessment 2 Assess division involving fractions, whole numbers, and mixed numbers Word problems with all types of positive rational numbers Students explore problems that include fractions, mixed numbers, and whole numbers. 15 Formative Assessment 3 Assess division involving fractions, whole numbers, and mixed numbers. 16 Unit Review Students can create word problems and create answer keys. Students can then swap problems with one another and explore various methods of solving the same 17 Culminating Task: Share My Candy 5.NF.7 6.NS.1 Various 6.NS.1 5.NF.7 Use visual models and manipulatives. Scaffold by starting with common denominators, progressing to denominators where one is a multiple of the other, then moving to relatively prime denominators. Provide some carefully selected, scaffolded problems. Students should also be encouraged to create and complete problems of their own. 6.NS.1 Problems should be scaffolded from easier to more complex. Word problems should always start by finding key words that indicate the function. 6.NS.1 26

27 Name: Class: Date: Initial Assessment For each of the following questions, show your work and write your answer in a complete sentence. PART A: On Monday, Paul walked 2 miles. On Tuesday, Paul walked 1 Wednesday, Paul walked miles. How many miles did Paul walk all together? miles. On Answer Sentence: PART B: If Jaime runs 2 kilometers per day, how many kilometers can Jaime run in 1 days? Answer Sentence: PART C: Sarah lives 6 blocks from school. If she walked further does she have to walk? blocks so far, how much Answer Sentence: 27

29 Initial Assessment Answer Key PART A: = 7 = 8 Paul walker 8 miles all together. PART B: 2 1 = " = "# = 31 PART C: 6 = 6 = 5 = 1 Jaime runs 31 kilometers. Sarah has 1 blocks left. PART D:

30 Name: Class: Date: Formative Assessment 1 Merry Models Draw a picture or model that represents each of the following problems. Then write your quotient on the line Quotient: Quotient: Quotient: Quotient: Quotient: Quotient: 7. Explain the model you created for problem #6. 30

31 Formative Assessment 1- ANSWER KEY Merry Models NOTE: There are various picture models that are complete and correct. This answer key is completed using grouping with tally marks. Any correct picture model is acceptable. Students are not required to use all types of models, but rather only those that make sense to them I I I I I I I I I I I I I I _ I I I I Quotient: Quotient: 1 Quotient: I I I I I I I I = = I I I I I I Quotient: 7 Quotient: Quotient: 6 7. Explain the model you created for problem #6. I got a common denominator so that the size of the unit piece in the total could be the same as the size of the unit piece in the divisor. Then I divided the total unit pieces that I started with into 6 equal pieces, as specified by the divisor. We can ignore the denominator now because both denominators are the same, which means the size of the unit pieces are the same. (Or similar response) 31

32 Name: Class: Date: Formative Assessment 2 Minding My Dividing PART A: Jose solved a division He got a quotient of 5. He cannot remember what the problem was about. Below are 5 choices that Jose can choose from. Which ones can he eliminate? Write possible next to the quotient that could be true or impossible next to the quotient that could not be true. For each impossible quotient, briefly explain why it is not possible. 5 people: 5 candy bars: 5 blocks: 5 chairs: 5 cups of water: PART B: Write a question that could be represented by the expression 3. PART C: Solve the question that you created in Part B. Show your work and write your answer in a complete sentence. Answer Sentence: 32

33 Formative Assessment 2 ANSWER KEY Minding My Dividing PART A: Jose solved a division He got a quotient of 5. He cannot remember what the problem was about. Below are 5 choices that Jose can choose from. Which ones can he eliminate? Write possible next to the quotient that could be true or impossible next to the quotient that could not be true. For each impossible quotient, briefly explain why it is not possible. 5 people: Impossible Cannot have two thirds of a person 5 candy bars: Possible. I can break up a candy bar into thirds 5 blocks: Possible. I can break up a block into thirds OR Impossible Can t break a child s wooden building block into thirds 5 chairs: Impossible I cannot have two thirds of a chair 5 cups of water: Possible. I can measure water in thirds of a cup PART B: Write a question that could be represented by the expression 3. Various correct responses PART C: Solve the question that you created in Part B. Show your work and write your answer in a complete sentence = = = = OR I I I I I I I I I I I I I I I I I I I I I I Answer Sentence: Varies depending on question created for PART C. 33

34 Name: Class: Date: Formative Assessment 3 Penny s Pieces Penny had a piece of string that was 3 yards long. She wanted to cut the string into strips that were 1 yards long. Use this information to answer the following questions. PART A: Show how you can use a picture or diagram to find how many pieces Penny can cut. PART B: Write a number sentence that represents this situation. PART C: How many full strips can Penny cut? Show your work and write your answer in a sentence. Answer Sentence: PART D: How many yards of string does Penny have left after she cuts as many strips as possible? Show how you know, and write your answer in an answer sentence. Answer Sentence: 3

36 POSSIBLE SOLUTION PATHS FOR TEACHERS Share My Candy Jason has 3 candy bars. He wants to share the candy bars with his friends. He gives as many of his friends as possible of a candy bar. He keeps the rest for himself. PART A: How many friends can he give of a candy bar to? Show your work and write your answer in a complete sentence. WORK: SOLUTION PATH 1: 3 = = = " = have some left.) = He can give ¾ of the candy bar to friends (and he ll SOLUTION PATH 2: (Let F represent Friend so F1 means Friend 1) F1 F1 F1 F2 F2 F2 F3 F3 F3 F F F ANSWER SENTENCE: He can give friends ¾ of a candy bar. PART B: How much of the candy bar will he keep for himself? Show your work and write your answer in a complete sentence. WORK: SOLUTION PATH 1: = of 3 = = 2 or 1 2 SOLUTION PATH 2: From the picture above, he has 2 or 1 2 left. He can t share it with his friends because he wants each of his friends to get 3, which is greater than 2 ANSWER SENTENCE: He has or candy bar left for himself. PART C: Explain your reasoning for parts A and B. SOLUTION PATH 1: Jason wants to share his 3 ½ candy bars with his friends. So we can use division to find out how many friends he can give ¾ of a candy bar to. We can use the division algorithm to find the solution by finding out how many ¾ are in 3 ½, which is the same as 3 SOLUTION PATH 2: Jason wants to share 3 ½ candy bars with his friends. I can draw 3 ½ candy bars and then split each whole candy bar into ths because he only wants to give ¾ to each friend. 36

37 PART D: Show how you can check if your responses to parts A and B are correct. SOLUTION PATH 1: 3 = multiplication over addition. so to check I can use multiplication and distributive property of = + = + ( ) = 3 + = 3 SOLUTION PATH 2: Each candy bar is split into fourths. To give ¾ of each candy bar to a friend, I need to give 3 parts of the equal parts from each candy bar. (This solution may be easier for some kids because they can visualize by drawing pictures.) 37

38 Cherry Pies 1. Heather bought 3 cherry pies. She decides she wants to serve each of her guests of a pie. How many guests can she serve? Number Line (Guests) 0 Tally Model (Pies) I I I I I I I I I I I I I I I I I I I I I I I I 9 Guests I I I I I I I I I I I I III Algorithm 3 = " = " " = " = " = 9 " " " " 9 Guests 2. What fraction of the pie will she have left? Number Line Three bars remain of a total of twelve possible bars so that would give a remainder of " = pie. Tally Model Three tallies remain of a total of twelve possible tallies so that would give a remainder of " = pie. 38

39 Algorithm of a pie piece remain and each pie piece was cut into, so: of = = = of a pie remaining. " 39

40 Pound Cakes 1. Jack has 1 pound cakes. He wants to cut the cake in equal pieces. How many pieces can he make? Picture Model 1 cake + ½ cake = 6/6 + 3/6 2 3 = 2 X 2 3 X 2 cakes = 6 There are 2 groups of 6 or 2 3 in the pound Number Line 1 2 (Pieces) 0 1 (Pound Cakes) Tally Model I I I I I I I I I 2 Pieces Algorithm 1 = = = = = 2 2 Guests 2. What fraction of the pound cake will he have remaining? Picture Model One rectangle representing remains, so there is pound cake remaining. Number Line One bar remains of a total of six possible bars so that would give a remainder of pound cake. 0

41 Tally Model One tally remains of a total of six possible tallies so that would give a remainder of pie. Algorithm of a cake piece remains, and each cake piece was cut into, so: of = = = of a pound cake remaining. " 1

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