Multiplication and Division: The Inside Story

1. Multiplication and Division: The Inside Story A behind-the-scenes look at the most powerful operations

2. Three sessions • Today: Multiplication and Division • Dec. 3: Fractions and Decimals • Jan. 28: Geometric Shapes and Volume

3. Today How children learn • Multiplication and division problem-solving • Multiplication and division combinations • Multi-digit multiplication and division • Connections with area and perimeter

4. The first way we teach children to think about multiplication: Skip-counting of rows in an array. An example is 4 rows of 5 chairs lined up in a room.

5. Is 5 rows of 4 the same number? Make up 3 examples with jumps of 2-3-4.

6. 4 • 4 • 4 • 4 • 4 The second way we teach children to think about multiplication: Equal groups. This is a generalization of equal-size rows of objects in an array. An example is 5 bags with 4 cookies in each bag. Make up 3 more examples using 10-11-12.

7. The third way we teach children to think about multiplication: • My dog can run 5 times as fast as your rabbit. • Your rabbit can jump 3 times as far as my dog. • My dog eats 10 times more food than your rabbit. • Your rabbit is 1/4 the height of my dog (or my dog is 4 times taller than your rabbit). • Your rabbit is twice as old as my dog. • My dog can bark 100 times louder than your rabbit! Multiplicative comparison. Make up 3 more that involve everyday things.

8. Related problem types • Rate • Price • Combination • See the handout

9. Why is it important to recognize types of multiplication problems? The fixed costs of manufacturing basketballs in a factory are $1,400.00 per day. The variable costs are $5.25 per basketball. Which of the following expressions can be used to model the cost of manufacturing b basketballs in one day? A. $1,405.25b B. $5.25b − $1,400.00 C. $1,400.00b + $5.25 D. $1,400.00 − $5.25b E. $1,400.00 + $5.25b

10. Number Talk What number do you think will go in the blank to make the equation true? Try to solve this by reasoning, without doing the calculations. 4 x 9 = 12 x ___ How did you think about this?

11. The most powerful way of thinking about multiplication: This is powerful because it connects multiplication to the area of a rectangle. 8 x 7 = 56 8 in. x 7 in. = 56 sq. in.

12. The most powerful way of thinking about multiplication: Plus, it gives us insight in the process of multiplication, and new ways to compute: 8 x 7 = (8 x 5) + (8 x 2) This is the distributive property (3.MD.7)

13. The most powerful way of thinking about multiplication: Now you can multiply bigger numbers in your head. Try 56 x 5. Try 8 x 23.

14. Find a way to multiply 38 x 6 by representing 38 as a subtraction. Try 3,426 x 5 by decomposing into thousands, hundreds, tens and ones.

15. Number Talks book and DVD • Number Talks: Helping Children Build Mental Math and Computation Strategies, Grades K-5, by Sherry Parrish (DVD) Watch Array Discussion

16. How many rectangles…? How many different rectangles can you make from your bag of squares? Write a multiplication sentence to go with each rectangle. Watch Associative Property 12 x 15

17. Factors The word “factor” is an academic vocabulary term that is essential to understanding multiplication. 6 x 1 = 6 3 x 2 = 6 Which are the factors and which are the products in your rectangles? Watch 16 x 35

18. Rectangle multiplication What does this visual representation tell you about multiplication? (knees to knees, eyes to eyes) http://nlvm.usu.edu

19. The Factor Game • Common Core Collaboration Cards • With your team member, see if you can figure out a strategy for winning. • Also linked from our Elementary Math Resources wiki: Go to inghamisd.org, then click on Wiki Spaces.

20. How to help a child become fluent Acquisition – Fluency – Generalization Concepts, strategies, procedures Practice, practice, practice Extensions This learning progression is true for single digit “math facts” and for fluency with multi-digit procedures.

21. Math facts, if not already known Math fact strategy: • Only work on unknown combinations • Ensure knowledge of meaning of multiplication (acquisition) • Learn strategies through repeated problem-solving (acquisition) • Practice in game situations (fluency) • Use in division situations (generalization)

22. IISD Fluency Packet • Resources for helping those students who still need work on combinations.

23. The Product Game • Good practice for children who don’t have all their combinations from memory yet. • A combination game from PhET

24. Research Recommendation Interventions at all grade levels should devote about 10 minutes in each session to building fluent retrieval of basic arithmetic facts. • Provide about 10 minutes per session of instruction to build quick retrieval of basic arithmetic facts. Consider using technology, flash cards, and other materials for extensive practice to facilitate automatic retrieval. • For students in kindergarten through grade 2, explicitly teach strategies for efficient counting to improve the retrieval of mathematics facts. • Teach students in grades 2-8 how to use knowledge of properties, such as commutative, associative, and distributive laws, to derive facts in their heads.

25. Box and Books of Facts

26. Procedures… The C-R-A Concrete-Representational-Abstract Concrete: Multiply 16 x 12 using base 10 blocks.

27. Procedures… The C-R-A Concrete-Representational-Abstract Representational: National Library of Virtual Manipulatives nlvm.usu.edu

28. Procedures… The C-R-A Concrete-Representational-Abstract Abstract:

29. Learning Progression

33. Problem-solving with area and perimeter Table for 22: Real-World Geometry Problem

34. How is division tied to multiplication? List several ways the two are connected…

35. Two types of division Partitive (fair shares) We want to share 12 cookies equally among 4 kids. How many cookies does each kid get? How would you solve this with a picture? The number of groups is known; the number in each group is unknown.

36. Measurement (repeated subtraction) For our bake sale, we have 12 cookies and want to make bags with 2 cookies in each bag. How many bags can we make? How would you solve this with a picture? The number in each group is known; the number of groups is unknown.

37. Partial quotient method 6 )234 -120 20 114 -60 10 54 -30 5 24 -24 4 0 39 Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value. 4.NBT.6 This type of division is called repeated subtraction

38. You try it 24)8280 Now the standard algorithm Keep in mind that 8280 = 8000 + 200 + 80 + 0 or 8200 + 80 or 82 hundreds + 8 tens

39. The standard algorithm: How many equal groups of 24 can be made from 82? 3 groups, with 10 left over. 82 what? 10 what? Why do we put the 3 there? 3 24)8280 72 10 24 24 24 10

40. The standard algorithm: How many equal groups of 24 can be made from 82? 3 groups, with 10 left over. 82 what? Why do we put the 3 there? How many equal groups of 24 can be made from 108? 4 groups, with 12 left over.108 what?Why do we put the 4 there? 34 24)8280 72 1080 96 12 12 24 24 24 24

41. The standard algorithm: How many equal groups of 24 can be made from 82? 3 groups, with 10 left over. 82 what? 10 what?Why do we put the 3 there? How many equal groups of 24 can be made from 108? 4 groups, with 12 left over.108 what? 12 what?Why do we put the 4 there? How many equal groups of 24 can be made from 120? 5 groups, with 0 left over.120 what? Why do we put the 5 there? 345 24)8280 72 1080 96 120 120 0 24 24 24 24 24

42. 345 24)8280 72 1080 96 120 120 0 8280 = 8000 + 200 + 80 + 0 or = 7200 + 960 + 120 = 24x300 + 24x40 + 24x5

43. What about remainders?

44. Algorithms