1. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Multiplication of Fractions Part 3.5 March 5, 2013 Common Core Leadership in Mathematics2 (CCLM) This material was developed for use by participants in the Common Core Leadership in Mathematics (CCLM^2) project through the University of Wisconsin-Milwaukee. Use by school district personnel to support learning of its teachers and staff is permitted provided appropriate acknowledgement of its source. Use by others is prohibited except by prior written permission.

2. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Agenda Continue Understanding Multiplication of Fractions Word Problems MP 5 – Use appropriate tools strategically.

3. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 We are learning to …. Understand multiplication involving fractions using meaningful visual models and real-world contexts involving ‘parts-of’ and ‘groups-of’ problems. We will be successful when we can …. Represent, contextualize, and justify problems involving multiplication of fractions by fractions (4.NF.4, 5.NF.4, 5.NF.6) using tape diagrams and area models. Learning Intentions and Success Criteria

4. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Getting Up to Speed: Understanding the progression from 4 th to 5 th grade for word problems

5. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 One Meaning of Multiplication of Fractions: Groups of Problems 3 x ⅔  Tell a story  3 groups of ⅔  4.NF.4c 4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. (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.

6. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 ⅔ x 3  Tell a story  ⅔ part of 3  5.NF.4a Second Meaning of Multiplication of Fractions: Parts of Problems 5.NF.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…

7. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Multiplication of Fractions – Story Problems 4 th to 5 th Grade Progression 4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. (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. 5.NF.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. 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.  Standards Progression

8. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Delving a deeper into 5.NF.4

9. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 5.NF.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.) Study this standard. Then work in small groups to create a visual model and identify a context for (2/3) x 4 and for (2/3) x (4/5). Standard 5.NF.4a

10. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 I have 30 marbles. 2/5 of them are red. How many of the marbles are red? Example for Standard 5.NF.4a Partition 30 into fifths. Each fifth represents 6 marbles. We need 2 of the partitions (2/5) 30 ÷ 5 x 2 = 12 marbles. 5.NF.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. 6 marbles

11. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 5.NF.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. Task: Write (2/5) x 30 as a x q ÷ b (2/5) X 30 = (2 x 30) ÷ 5 60 ÷ 5 = 12 marbles I have 30 marbles. 2/5 of them are red. How many of the marbles are red? Example for Standard 5.NF.4a

12. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Where does “a sequence of operations a × q ÷ b” come from? How did we get here? (2/5) X 30 = (2 x 30) ÷ 5 (2/5) X 30 = ( 2 x 1/5) x 30 = 2 x (1/5 x 30) = 2 x (30 x 1/5) = (2 x 30) x 1/5 = (2 x 30) ÷ 5 I have 30 marbles. 2/5 of them are red. How many of the marbles are red? 6 marbles

13. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 TCM Article: Multiplying Fractions Count of by 6. Read your assigned problem. On your white board draw a strip diagram that you used to help you solve the problem. Present your solution to your table using your strip diagram and discuss how your problem and strip diagram connects back to 4.NF.4c, 5.NF.4a and/or 5.NF.6

14. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments & critique 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 and express regularity in repeated reasoning.

15. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 MP5 Use Appropriate Tools Strategically List 3 specific examples for: – Student Disposition: What did you do as students that illustrated this practice? – Teacher Actions: What experiences and opportunities did the teachers provide to foster the desired student dispositions?

16. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 Sense Making…. Share with your shoulder partner a few ideas that struck you as critical to developing a sound understanding of multiplication of fractions.

17. Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, 2012 - 2013 We are learning to …. Understand multiplication involving fractions using meaningful visual models and real-world contexts involving ‘parts-of’ and ‘groups-of’ problems. We will be successful when we can …. Represent, contextualize, and justify problems involving multiplication of fractions by fractions (4.NF.4, 5.NF.4, 5.NF.6) using tape diagrams and area models. Learning Intentions and Success Criteria