Division of Fractions Core Mathematics Partnership Building Mathematical Knowledge and High-Leverage Instruction for Student Success Wednesday July 29, 2015

Understanding Division with Fractions Yes, it is possible!

Let’s Check Our Understanding What happens when you divide using fractions? Estimate Greater than 5? Equal to 5? Less than 5? Connection back to the Practices. Sense making… It’s more than 5 because I can visualize using measurement division It’s less than 10 because 5 divided by ½ is 10

Apply and extend previous understanding of multiplication and division with whole numbers to multiplication and division with fractions. This is a capstone standard. This completes the extension of operations to fractions. Turn and talk. Identify 4-5 previous understandings from 3rd-5th grade around multiplication and division that they would be expected to apply. Gather and chart. Examples could include use of representations, understanding of unit fractions, understanding of fractions as numbers (quantity), the meaning of operations (multiplication and division), fraction times a whole number, application story problems, practice standards

Pose a word problem for 48 ÷ 4 = ? Which representation best matches your problem situation? Measurement (quotative) division Partitive division Did you pose an “equal sharing” or “fair shares” dealing out problem situation? Did your problem situation involve repeatedly measuring out or packaging a group of a specific size? How would you describe the difference between these two types of division situations?

48 ÷ 4 = 12 48 ÷ 4 = 12 Partitive Division 48 ÷ 4 = 12 48 ÷ 4 = 12 total amount number of shares (groups) size of each share (group) total amount size of each share (group) number of shares (groups) Partitive Division Measurement (quotative) Division “know the number of partitions” “know the size to measure out”

Standards 3.OA.1 & 3.OA.2 Cluster: Represent and solve problems involving multiplication and division. 3.OA.1. Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7. 3.OA.2. Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.

Standards 3.OA.1 & 3.OA.2 Cluster: Understand properties of multiplication and the relationship between multiplication and division. 3.OA.5. Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.) 3.OA.6. Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

Popcorn Party

Popcorn Party How many servings from: 8 cups of popcorn Serving Size: 2 cups How many servings from: 8 cups of popcorn 5 cups of popcorn

Summarize Popcorn Party #1: Serving Size 2 cups How many servings from 8 cups; 5 cups? Is this a measurement or partitive context? Write the equation for each situation (8 cups; 5 cups). Identify the meaning of each number in the equations, both contextual meaning and underlying mathematical structure.

8 ÷ 2 = 4 4 × 2 = 8 cups of cups per full servings using up 8 ÷ 2 = 4 cups of cups per full servings using up popcorn serving all the popcorn (exhaustive) total amount size of each share (group) number of groups (shares) 4 × 2 = 8 full cups per cups of popcorn servings serving number of groups (shares) size of each share (group) total amount

5 ÷ 2 = 2 1 2 cups of cups per servings (2 full servings, serving What does the 2 represent? What does the ½ represent? Are you sure? 1 2 5 ÷ 2 = 2 cups of popcorn cups per serving servings (2 full servings, one-half of another serving) using up all the popcorn total amount size of each share (group) number of shares (groups)

5 ÷ 2 = 2 ½ 2 ½ × 2 = 5 cups of cups per servings popcorn serving 5 ÷ 2 = 2 ½ cups of cups per servings popcorn serving total amount size of each share (group) number of shares (groups) 2 ½ × 2 = 5 servings cups per cups of popcorn serving number of shares (groups) size of each share (group) total amount

Popcorn Parties #1 & #2 Facilitator poses one problem at a time. Each individual silently solves it using fraction strips (envision, fold, cut apart as needed). On facilitator’s cue, state the answer. One person explains verbally while another person demonstrates step-by-step with paper strips how that person solved the problem. Whiteboard: Write an equation which describes the context and the structure (as a group). Let’s us know when you want to move on to Popcorn Party #2 with a new facilitator for your table group.

Popcorn Party #1 Serving Size: cup of popcorn How many servings can be made from: 1 cup of popcorn 2 cups of popcorn 3 cups of popcorn

Serving Size: cup of popcorn How many servings can be made from: Popcorn Party #2 Serving Size: cup of popcorn How many servings can be made from: of a cup of popcorn 6 cups of popcorn 2 cups of popcorn 4 cups of popcorn 1 serving; 8 servings; 3 servings; 6 servings all full or complete servings.

Think about the “size of a serving.” Just as with whole numbers, it is important to understand the meaning of the answer and how to interpret and relate ways of reporting “remainders.” Think about the “size of a serving.” Think about the “leftover or extra” amount. Think about the “number of servings” determining both how many full servings and what portion or part of a serving. .

3 4 How many servings from 2 ½ cups of popcorn? Serving Size: cup 3 4 How many servings from 2 ½ cups of popcorn? Work individually using your paper strips to solve the problem. THEN compare results.

1 cup of popcorn 1 cup of popcorn 1/2 cup of popcorn Represents 1 cup of popcorn 1 cup of popcorn 1 cup of popcorn 1/2 cup of popcorn

1 cup of popcorn 1 cup of popcorn 1/2 cup of popcorn Represents 1 cup of popcorn 1 cup of popcorn 1 cup of popcorn 1/2 cup of popcorn

What portion of a serving is this amount? Represents 1 cup of popcorn 1 serving 1 serving 1 serving What portion of a serving is this amount?

3 4 How many servings from 5 cups of popcorn? Serving Size: cup 3 4 How many servings from 5 cups of popcorn? Work individually using your paper strips to solve the problem. THEN compare results.

Represents 1 cup of popcorn I made 6 complete servings, each with ¾ of a cup of popcorn, but what portion of a serving is this amount? 1 serving

Represents 1 cup of popcorn Well, one serving is 3 parts of size ¼, and this is 2 parts of size ¼. So, this amount is 2/3 of a serving of popcorn. 1 serving

I think I’m starting to understand!!

Why or Why Not? Work in pairs. Place all the cards face down on the table in a 4 × 6 grid One player turns over two cards, and states a reason why the cards do or do not match before turning them back face down (or taking the cards if they match) Alternate turning over cards 2 1/3 servings; 4 4/5 servings

Why or Why Not? How did this game connect the language of fraction division to visual models? 2 1/3 servings; 4 4/5 servings

What has been a major “aha” or new insight? Discussion What has been a major “aha” or new insight?

CCSSM Standards

Study Standards 5.NF.7 & 6.NS.1 Which aspects of these standards relate to our work with division and fractions?

Standard 5.NF.7 Cluster: Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 5.NF.7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. c. Solve real world problems involving division of unit fractions by non- zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins.

Standard 6.NS.1 Cluster: 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?

Connections to an Algorithm

Which, if any, of these questions yields a different answer? Same or different? Which, if any, of these questions yields a different answer? • How many 3s are there in 6? • How many groups of 3 tens are there in 6 tens? • How many groups of 3 fives are there in 6 fives? • How many groups of 3 tenths are there in 6 tenths? • How many groups of 3 @s are there in 6 @s? • How many groups of 3 anythings are there in 6 anythings (as long as both anythings refer to the same unit)? http://www.learner.org/courses/learningmath/number/session9/part_a common_denominator.html

Estimate: Is the answer more or fewer than 6? Different denominators How many ¾ pound packages of fudge are needed to make up 6 pounds of fudge? Estimate: Is the answer more or fewer than 6? Write a number sentence for this problem. Use your observations from the last slide to solve the problem.

Package size: ¾ pound of fudge. Want 6 pounds of fudge. How many packages? Common Denominator Strategy: Rename 6 pounds as 24 parts of size ¼ of a pound.

Common Denominator Method As a group, make sense of the common denominator method for dividing fractions. Try it with these problems: Use visual models to confirm your answers.

PRR: Gregg and Gregg Read Gregg and Gregg Measurement and Fair Share Models for Division of Fractions p. 491 to top of p. 494 The Common-Denominator Algorithm Sequence How did the examples and discussion in this reading help you better understand measurement division and the common denominator method?

More Practice Will this procedure always work?

Three Key Learning Targets for Your Students “Measurement Division with Fractions”

Measurement Division Three Key Learning Targets 1. Meaning of the Numbers and Operation: Context & Structure—Measuring out equal groups. 2. Effect of the Operation: Estimate whether the answer is... more or less than 3? 3. Understanding the Solution: How many “complete” groups of size 4/5 of this stuff can I make? Do I have any remaining stuff to also make “part of” a group? more or less than 6? I understand! Chart equation with context and structure.

Measurement Division Common Denominator Method Explain what is happening with this sequence of equations by contextualizing the numbers and operations (SMP 2) and by considering the underlying mathematical structure of the operations (SMP 7).

Measurement Division Common Denominator Method Put 4 pieces of size 1/5 into each group. Partition each whole thing into 5 fifths. I have a total of 15 pieces of size 1/5. I can make 3 complete groups and ¾ of another group.

Apply and extend previous understanding of multiplication and division with whole numbers to multiplication and division with fractions. This is a capstone standard. This completes the extension of operations to fractions. Turn and talk. Identify 4-5 previous understandings from 3rd-5th grade around multiplication and division that they would be expected to apply. Gather and chart. Examples could include use of representations, understanding of unit fractions, understanding of fractions as numbers (quantity), the meaning of operations (multiplication and division), fraction times a whole number, application story problems, practice standards

Disclaimer Core Mathematics Partnership Project University of Wisconsin-Milwaukee, 2013-2016   This material was developed for the Core Mathematics Partnership project through the University of Wisconsin-Milwaukee, Center for Mathematics and Science Education Research (CMSER). This material may be used by schools to support learning of teachers and staff provided appropriate attribution and acknowledgement of its source. Other use of this work without prior written permission is prohibited—including reproduction, modification, distribution, or re-publication and use by non-profit organizations and commercial vendors. This project was supported through a grant from the Wisconsin ESEA Title II, Part B, Mathematics and Science Partnerships.