CHAPTER 13 Developing Strategies for Multiplication and Division Computation

1. CHAPTER 13Developing Strategies for Multiplication and Division Computation Elementary and Middle School Mathematics Teaching Developmentally Ninth Edition Van de Walle, Karp and Bay-Williams Developed by E. Todd Brown /Professor Emeritus University of Louisville

2. Big Ideas • Flexible methods of computation in multiplication and division involve decomposing and composing numbers in a wide variety of ways. • Flexible methods for multiplication and division require a strong understanding of the commutative property, associative property, and distributive property of multiplication over addition. • Invented strategies provide flexible methods of computing that vary with the numbers and the situation. • The standard algorithms are clever strategies for computing that have been developed over time. • Nearly all computational estimations involve easier-to-handle parts of numbers or substituting difficult-to-handle numbers so that the resulting computations can be done mentally.

3. Invented Strategies for Multiplication Complete Number 63 x 5 Useful Representations 6 x 34

4. Invented Strategies for Multiplication cont. Partitioning Strategies reflect understanding of place value. Try each of the four strategies and teach someone else the methods.

5. Invented Strategies for Multiplication cont. Compensation strategies Manipulate numbers to make calculations easier Adjustment or Compensation Half-then-double strategy used when 5 or 50 involved Close compatible number

6. Multiplication of Multidigit Numbers • Multiplication of multidigit numbers supports the importance of place value and emphasis on the number rather than separate digits. Cluster Problems – using facts and combinations already known in order to figure out more complex computations. Describe the strategies you see in these examples.

7. Standard Algorithms for Multiplication • Begin with models • Area Model Figuring out the size of each subrectangle and combining to find the whole Notice the base-ten language Each section is a partial product

8. Standard Algorithms for Multiplication cont. Open Array Semi-concrete representation of the area model • Blank rectangle (not to scale) • Mark of the subdivisions based on the digits in the factors 3. Record partial products inside each subdivision. 4. Find to sum of partial products

9. Develop the Written Record for Multiplication of Multidigit Numbers

10. Invented Strategies for Division • Partitioning/Fair sharing

11. Invented Strategies for Division cont. • What number times 6 will be close to 164 with less than 6 remaining? • Alternatively- think of half of 164 and ask What number time 6 would be close to 82? Missing-Factor

12. Invented Strategies for Division cont. Cluster Problems Cluster for 381 ÷72 10 x 72 5 x 70 2 x 72 4 x 72 5 x 72 • Cluster for 527 ÷ 4 • 100 x 4 • 500 ÷ 4 • 25 x 4 • 6 x 4 Provide students with a sense that problems can be solved different ways with different starting points.

13. Standard Algorithm for Division Thinking process There are enough hundreds for each set to get 1 hundred. That leaves 1 hundred I can’t share. Trade that 100 for 10 tens for a total of 18 tens. Give each set 4 tens with 2 left over. Trade 2 tens for 20 ones for a total of 23 ones Give each set 5 ones with a remainder of 3. Gave each group 1 hundred, 4 tens and 5 ones with 3 left over. Partition or Fair Share • 583 ÷ 4 • Instead of 4 goes into 5 we want students to think 5 hundreds, 8 tens and 3 ones • Put in a context-we have 5 cartons, 4 boxes and 3 pieces of candy to share with 4 schools.

14. Standard Algorithm for Division cont. Standard Bring Down and Explicit Trade Methods • Share and record the number of pieces • Record the number of pieces shares. Multiply to find this number • Record the number of pieces remaining. Subtract to find this number. • Trade for smaller pieces, and combine with any of the same-sized pieces that are there already. • Record number in next column.

15. Standard Algorithm of Division The Common Core State Standards (2010) suggest that one-digit algorithm in 4th grade should provide the extension to fifth‐grade students so they should be able to “whole‐number quotients of whole numbers with up to four‐digit dividends and two digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division.” The CCSSO goes on to state that the student should “Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models” (p. 35).

16. Computational Estimation Strategies • Rounding in multiplication Compatible numbers in division

17. Try this oneActivity 13.4 Double, Double- No Toil and Trouble! • Materials- method to post a division problem and model a side bar chart • Directions- What is 3842 ÷ 4 • Missing factor approach and • repeated subtraction • Does this help with the • estimation approach?

18. Try this oneActivity 13.8

19. Try this oneActivity 13.9 Hit the Target • Materials- calculators • Directions-can be used with all four operations • Pick a start number and an operation • Students enter the start number (X + - ÷) = to make the result land in the stated target.