1. Exploration 3.13 You try Lattice Multiplication for 27 13 1313 2 7 1313 2 1 6 72 3 5 1

2. Link the pictorial models Suppose I want to multiply 4 3.

3. Link the pictorial models Suppose I want to multiply 40 3. This is much harder to draw, but it can be done.

4. Link the pictorial models As the numbers get bigger, it is harder to draw in all the little dots. But the area model will work well: 40 3: 3 40

5. Area Model Let’s look a little closer: Let this be a unit square--that is, a square that measures 1 unit on each side. Then, this is a representation for 7 15. 7 15

6. Area Model Look at this more closely: This is the same as 7 10 + 7 5 7 10 and 7 5 are partial products.

7. Area Model This idea works for more: 32 14 10 + 10 + 10 + 2 10 + 4

8. 32 14 Do you see 4 2? 4 30? 10 2? 10 30? Add the partial products.

9. Can you explain what is going on here? 15 24 =

10. Area Model You try: 46 23 Draw a picture. Now, can you explain where these products are in the diagram? 46 23 = (46 20) + (46 3) or = (23 40) + (23 6)

11. Compare models Can you explain how this is related to the lattice multiplication model you did for Exploration 3.13? Can you explain how this area model is related to the standard multiplication algorithm? Can you explain how this area model is related to the four students’ models?

12. Children’s Thinking Videos In general, watch the videos, and try to notice: a) what is the child doing? b) mathematically, what is going on? c) mathematically, why is it correct or incorrect.

13. Children’s Thinking Can you explain how the area model is related to the four students’ models?

14. Ryshawn and Nicholas 20 4+ 9 4

15. Jemea 30 12 - 12

16. Thomas 17 36 = ((17 10) 3)+ (6 10) + (6 7)

17. Is there an error? Is there more than one? 46 46 X 37 X 37 42 42 280 46 46 X 37 X 37 2842 18 12 42 28 4 3 21

18. Explain why… Which property or properties make these statements true? Can you show or explain why? Give a reason? Draw a picture? Use base 10 blocks? 2 (3 14) = (2 3) 14? 2 (3 14) = (3 2) 14? 2 (3 14) = 3 (2 14) 2 (3 14) = 14 (2 3) 2 (3 + 14) = 3 14 + 3 14 2 (3 14) ≠ 2 3 2 14