Multiplication of Fractions

1. WMC Annual Conference May 2012 Astrid Fossum, Milwaukee Public Schools Paige Richards, School District of South Milwaukee Multiplication of Fractions Isn’t this everything I need to know?

2. Current Connections to Fractions Thinking about your current grade level of instruction…talk to a shoulder partner… • Concepts you currently teach around fractions • Roadblocks that you encounter when you try to work with fractions with your students.

3. Learning Intentions We are learning to • Develop strategies related to multiplying fractions. • Understand how estimation should be an integral part of fraction computation development. • Read and interpret the cluster of CCSS standards related to multiplication of fractions. We will know we are successful when we can • Justify our thinking when multiplying fractions using concrete models and estimation strategies. • Clearly explain and provide examples for specific CCSS standards

5. Domain-Cluster-Standards Domain: Number and Operations: Fractions Cluster: Apply and extend previous understandings of multiplication and division to multiply and divide fractions. Standard: 5.NF.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

6. Big Ideas for Multiplication and Division of Fractions • Multiplication and division of fractions are among the most complicated fraction concepts that elementary students encounter. • Instructional opportunities that students encounter should include the meaning of multiplication and division on a range of situations and build procedural fluency with understanding.

7. Launch: Building on What We Know What do you know about multiplication and division of whole numbers? As a table group, make a list of what students have learned as they interact with multiplication and division problems. Are they all true for fractions?

8. The Importance of Models Researchers indicate that teachers need knowledge of concrete models to help students transition from multiplication of whole numbers to multiplication of fractions. (Fendel, 2000)

9. Thinking in Whole Numbers 3 X 4 = 12 Write a word problem for this equation. What does each number mean?

10. Multiplication of Fraction Standards • 4.NF.4a, 4.NF.4b, 4.NF.4c, 5.NF.4, 5.NF.5, 5.NF.6 • On your Standards Interpretation Sheet we will rephrase these standards and provide examples.

11. Some Stories Solve these problems. Use a visual model to record and then discuss your thinking with the group. Then write an equation. • There are 15 cars in Derek’s toy car collection. Two-thirds of the cars are red. How many red cars does Derek have? • Tia has 11 cookies. She wants to share them with her three friends. How many cookies will Tia and each of her friends get, if they share them equally? • Jason filled 5 glasses with 1/3 liter of soda in each glass. How much soda did Jason use?

12. Unpacking 4.NF.4 4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.) b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.) c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?

13. More Stories Solve these problems. Use a model to record and then discuss your thinking with your group. Write an equation for each problem. • You have of a pizza left. If you give of the left-over pizza to your brother, how much of a whole pizza will your brother get? • Frankie had 2/3 of the lawn left to cut. After lunch, she cut 3/ 4 of the lawn she had left. How much of the whole lawn did Frankie cut after lunch?

14. Unpacking 5.NF.4a 4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)

15. Multiplying with Mixed Numbers Determine the solution to these problems. Do not use a computational algorithm. 6 ½ x 4 2 ½ x 5 ¾

16. Unpacking 5.NF.6 6. Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

17. Revisiting Word Problems Examine the five problems on the handout. Which are story problems for 2/3 x 1/ 4 and which are not? Why?

19. Standards Interpretation:Summing It All Up As a teacher of mathematics… • What message are you walking away with regarding the CCSSM domain of fractions? As a leader of mathematics… • What message will you be taking back to teachers in your school/district?

20. Thank you! Astrid Fossum, Milwaukee Public Schools fossumag@milwaukee.k12.wi.us Paige Richards, School District of South Milwaukee prichards@sdsm.k12.wi.us