Multiplication and Division

1. Multiplication and Division Math Content – Part 3 March 4, 2013

2. Afternoon Meeting Welcome Outstanding Coaches, Please greet at least three other people in the room and share why you what you want to know about multiplication and division. Activity: Form a circle. “I Have… Who Has….. From, Erny

4. The Devil’s Library

5. Learning • We are learning to… • Understand how the CCSSM views the development of multiplicative thinking. • Apply strategies that promoting fluency with single-digit multiplication and division. • We will be successful when… • We can help students apply properties of operations as strategies to multiply.

6. What is Multiplication? Respond in writing in your notebook… What is multiplication? Check with a neighbor to see: • How is your thinking similar? • How is your choice of language similar or different?

7. Repeated Addition

8. Repeated Addition Begin multiplication instruction with problems involving repeated addition and introduce symbolic multiplication as a shortcut for representing repeated addition problems. Children can be encouraged to first represent repeated addition as an addition expression and then devise a shorthand for such symbolic expressions. - Baroody, 1998

9. Strategies- Creatures

10. Standards… • Read and Reflect on Standards 3.OA: 1 – 6 • Share with your shoulder partner a few ideas that struck you as critical to developing a sound understanding of multiplication and division.

11. From Counting by Ones to Thinking In Groups • Place a large amount of counters in the middle of the table. • On the word “go” grab as many groups of two that you can before we say stop. Don’t count the total number of sets, just concentrate on making groups of two.

12. Return the counters to the middle of the table. • On the word “go” grab as many sets of ___3____ that you can before we say stop.

13. Making groups: What did you notice? • What kind of thinking were you doing as you were making groups? • What would this activity tell you about students’ thinking? • You were just unitizing!(To make or transform into a single unit)

14. Dot Images • How many dots so you see? • How do you see It? • Draw what you see in your notebook.

15. Image 1

16. Image 2

17. Image 3

18. Grounding thinking in CCSSM 3.OA.5: Apply properties of operations as strategies to multiply and divide. • Commutative • Associative • Distributive With your shoulder partner, use an example, remind each other how these “rules of numbers work.”

19. Reflect back on 3.OA.5. Think about how the images were described. Where do the properties show up in the reasoning?

20. Individually: • Quickly glance at the dot image and determine the number of dots. • Jot down using “language” how you saw it. • Write an equation that matches your image and your description. • Identify the property or properties you used. Turn and share.

21. Image 4

22. Image 5

23. Image 6

24. Revisit Standard 3.OA.5 and 3.OA.7 Take turns with a Shoulder Partner to summarize: Reflecting on our “dot image” work, what are the main messages of these standards?

25. Pose a word problem for: The total number of objects in 4 groups of 7 objects each. 4 x 7 = ☐ Group Size Number of Groups Examine your word problem: What does the 4 mean? What does the 7 mean?

26. Pose a word problem for: 28 ÷ 4 = ☐ Tell a different story for 28 ÷ 4 = ☐. Examine your word problem: What is unknown in your problem: • Number of groups? • Group size?

27. 28 ÷ 4 = ☐ Number of shares or Number of objectsin each share Total Division means to partition to find the number of shares or find the amount in each share.

28. 4OA1Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. 4OA2 Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

29. Students in earlier grades learned to compare quantities additively…. “This one is 32 feet higher than that one” 40 feet high, the other 8 feet high Students in Grade 4 learn to compare these quantities multiplicatively….. “This one is 5 times as high as that one.” What’s the new thinking being developed in 40A1 & 2?

30. Experiences with Compare Word Problems When you have a draft of each, turn and check-in with a partner. Individually pose compare word problems for: 6 x 9 = ☐ 54 ÷ 6 = ☐ with Group Size Unknown 54 ÷ 6 = ☐ with Number of Groups Unknown

31. 7 x 8 = ☐ Total Unknown 56 ÷ 8 = ☐ Group Size Unknown 56 ÷ 8 = ☐ Number of Groups Unknown Unknown: Group Size Connie has 8 times as much money as Melissa. Connie has $56. How much money does Melissa have?

33. Building Blocks of Algebra • Understand problem situations • Represent the situation with objects or diagrams • Represent quantitative relationships with equations • Use properties of operations as the basis for strategies p.13 OA Progressions

34. Expanding View of Multiplication • Grade 3: Equal groups --discrete objects, arrays…. • Grade 4: Comparison situations --continuous quantities • Grade 5: Stretches or Shrinks (scale factor) --Context for reasoning multiplicatively with continuous quantities

35. Oldest Multiplication Chart Napier Bones

37. Lattice Multiplication The lattice algorithm for multiplication has been traced to India, where it was in use before A.D.1100. Many students find this particular multiplication algorithm to be one of their favorites. It helps them keep track of all the partial products without having to write extra zeros – and it helps them practice their multiplication facts

38. Lattice Method of Multiplication

39. 286 X 34

40. 1. Create a grid. Write one factor along the top, one digit per cell. Write the other factor along the outer right side, one digit per cell. 2 8 6 1 0 2. Draw diagonals across the cells. 2 0 3 6 3.Multiply each digit in the top factor by each digit in the side factor. Record each answer in its own cell, placing the tens digit in the upper half of the cell and the ones digit in the bottom half of the cell. 4 8 2 0 3 9 4 8 2 4 4. Add along each diagonal and record any regroupings in the next diagonal 1 7 1 2 4

41. Answer 2 8 6 1 1 1 0 2 0 3 6 4 8 2 0 3 9 4 8 2 4 7 2 4 286 X 34 = 9 7 2 4

42. 732 X 57

43. 7 3 2 1 1 3 5 4 5 5 0 4 1 2 7 1 1 9 1 4 1 7 2 4 732 X 57 = 1, 7 2 4 4

44. Partial Products Algorithm for Multiplication

45. + To find 67 x 53, think of 67 as 60 + 7 and 53 as 50 + 3. Then multiply each part of one sum by each part of the other, and add the results 6 7 X 5 3 3,000 Calculate 50 X 60 350 Calculate 50 X 7 180 Calculate 3 X 60 21 Calculate 3 X 7 3,551 Add the results

46. + Let’s try another one. 1 4 X 2 3 200 Calculate 10 X 20 80 Calculate 20 X 4 30 Calculate 3 X 10 12 Calculate 3 X 4 322 Add the results

47. + Do this one on your own. 3 8 Let’s see if you’re right. X 7 9 2, 100 Calculate 30 X 70 560 Calculate 70 X 8 270 Calculate 9 X 30 72 Calculate 9 X 8 3002 Add the results

48. Partial Quotients A Division Algorithm

49. 12 158 The Partial Quotients Algorithm uses a series of “at least, but less than” estimates of how many b’s in a. You might begin with multiples of 10 – they’re easiest. 13 R2 There are at least ten 12’s in 158 (10 x 12=120), but fewer than twenty. (20 x 12 = 240) - 120 10 – 1st guess Subtract 38 There are more than three (3 x 12 = 36), but fewer than four (4 x 12 = 48). Record 3 as the next guess 3 – 2nd guess - 36 Subtract 2 13 Sum of guesses Since 2 is less than 12, you can stop estimating. The final result is the sum of the guesses (10 + 3 = 13) plus what is left over (remainder of 2 )

50. 36 7,891 Let’s try another one 219 R7 - 3,600 100 – 1st guess Subtract 4,291 - 3,600 100 – 2nd guess Subtract 691 - 360 10 – 3rd guess 331 - 324 9 – 4th guess 7 219 R7 Sum of guesses