Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Multiplication and Division Arrays and Strategies Common Core Leadership in Mathematics (CCLM^2) Tuesday, July 10, 2012 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.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Learning Intention & Success Criteria We are learning to… – Make connections between the distributive property, the use of arrays, and the area model for multiplication. – Understand how “applying properties of the operations” leads to fluency with single-digit multiplication and division. We will be successful when… – We can explain and give examples of CCSSM expectations for fluency with single-digit multiplication and division. – Extend the application of the distributive property to larger numbers.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute OA.5 & 3.OA.7 3.OA.5 Apply properties of operations to multiply and divide. 3.OA.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division or properties of the operations. What properties will students be asked to use and apply as they multiply and divide?

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Scaffolding to CCSSM Expectations for Multiplication and Division Facts Jigsaw “Levels in problem representation and solution” p OA Progressions Level 1, Level 2, Level 3 Be prepared to teach your partners: Summarize key ideas of your Level and use 3 x 8 to demonstrate the thinking.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Thinking back to Monday… Breaking Apart a 3x4 Array What are the ways a 3 x 4 array can be broken apart? –Split rows or columns, not individual tiles (3x3) + (3x1) (3x2) + (3x2) (2x4) + (1x4)

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 (3x3) + (3x1) (3x2) + (3x2)

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Level 3: Distributive Property 6x8 Pretend you don’t know the answer to 6x8 Sketch a 6x8 array on grid paper. Partition the array into no more than 2 easier problems. Keep track of the way you partitioned it. –Use numbers and highlighters to keep track of how you partitioned your arrays.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute x 8

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute OA.5 & 3.OA.7 3.OA.5 Apply properties of operations to multiply and divide. 3.OA.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division or properties of the operations. In what ways do we engage students in 3.OA.5 and 3.OA.7 as we work with “breaking apart a hard fact” into smaller, easier to solve facts?

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 CCSSM Fluency Expectations: Single-Digit Multiplication and Division

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Third Grade Fluency 1 st viewing: Listen for the fluency demonstrated by the students. 2 nd viewing: Listen and note how the students use the distributive property to fluently get their answer. Finally, using the script as a reference, write an expression that corresponds to each student’s reasoning and that highlights the distributive property.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Fluency Expectations Read the 1 st paragraph on p. 27 of your OA Progressions. Share with your partner 1-2 ideas you have highlighted that contribute to your thinking around fluency with single-digit multiplication and division.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 True or False ?? 13  9 =  9 = 130 – 13 Individually Decide if the statement is true or false. Make notes to keep track of your thinking. Group Share your thinking.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute Select a facilitator. 2.Facilitator pulls out an equation strip and shares it with group. 3.Individually decide if the statement is true or false. Keep track of your thinking. 4.Facilitator asks each person to share their decision and the reason behind it.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Extending to larger numbers 3.NBT.3, 4.NBT.5, 5.NBT.6

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 The second graders at Clemens School are performing a play for their families. The chairs in the gym are arranged in 8 rows with 15 chairs in each row. How many chairs are there? Cut an array to represent 8  15. Partition the array into two smaller arrays. Label each partial product directly on your array. Write an equation that matches how you partitioned your array. 8  15 = _______________________

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute x 34 The chairs in the auditorium are arranged in 27 rows with 34 chairs in each row. How many chairs are in the auditorium. Sketch an array to represent 27  34. Partition into smaller arrays using benchmarks of 10. Label each partial product directly on your array. Write an equation that matches how you partitioned your array. 27  34 = _____________________

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute x 307 x 4 Write the partial product for each array and calculate the total. 600 = 20 x 30 (Step 1) 80 = 20 x 4 (Step 2) 210 = 7 x 30 (Step 3) 28 = 7 x 4 (Step 3) x x 4 27 x This is commonly call the Partial Product Algorithm. Why?

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Division Strategies

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Standard 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.) 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.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = = 56. (Distributive property.) What is the relationship between these two standards? Compare and contrast them.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Standard 3.OA.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 What Strategies Should Children Know for Division?

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 What Strategies Should Children Know for Division? Unknown Factor Problems ‘Think Multiplication’ when you know the multiplication fact. Use Strategies. 35 ÷ 5 = ? 5 x ? = 35 Answer is 7 Use ‘Count-bys’ 35÷ 5 = ? Count by 5, 10, 15, 20, 25, 30, 35 … 7 counts Use easier facts 35 ÷ 5 = ? 5 x ? = 35 Use easier facts 5 x 5 = 25 and 5 x 2 = = 35 so =7

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Revisiting Math Practices As a table group, revisit SMP 3: Construct Viable Argument and Critique the Reasoning of Others and SMP 7: Look for Structure Divide your slate in half. On one side, show how we were engaged in SMP3 and on the other side show SMP7.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Learning Intentions We are learning to… Understand two types of division situations. Learn division strategies that develop fluency in basic division facts. We will be successful when we can … Articulate the meaning of third grade standards that focus on division

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Review 3.MD.7c How does 3.MD.7c support this work? Using your slate, draw a model to explain this standard.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2012 Taking a look back: 2.OA.4 Divide your white board in half. On one side rephrase the standard in your own words. On the other side provide an example. In what way does 2.OA.4 prepare students for Level 2 solution strategies in 3 rd grade?