1. Multiplication and division with a comprehensive K-5 curriculum from © E. Paul Goldenberg 2008 This presentation may be shown for professional development purposes only, and may not be sold, distributed, or altered.

2. Foundations for multiplication The logic of addition: any-order any-grouping The logic of addition: any-order any-grouping The image of multiplication: intersections, arrays, and area The image of multiplication: intersections, arrays, and area The logic of multiplication: any-order any-grouping The logic of multiplication: any-order any-grouping

3. Sorting in Kindergarten Picture a young child with a small pile of buttons. Natural to sort. We help children refine and extend what is already natural.

4. 6 4 7310 Back to the very beginnings Children can also summarize. “Data” from the buttons. bluegray large small

5. large small bluegray If we substitute numbers for the original objects… Abstraction 6 4 7310 6 4 73 42 31 GR 1 (GR K – exposure)

6. A Cross Number Puzzle 5 Don’t always start with the question! 21 8 13 9 12 76 3 GR 1

7. Building the addition algorithm Only multiples of 10 in yellow. Only less than 10 in blue. 63 38 25 13 50 20 5 8 30 50 + 13 = 63 25 + 38 63 GR 2

8. Relating addition and subtraction 6 4 7310 42 31 6 4 73 42 31 Ultimately, building the addition and subtraction algorithms GR 2

9. Foundations for multiplication The logic of addition: any-order any-grouping The logic of addition: any-order any-grouping The image of multiplication: intersections, arrays, and area The image of multiplication: intersections, arrays, and area The logic of multiplication: any-order any-grouping The logic of multiplication: any-order any-grouping

10. Naming intersections, first grade Put a red house at the intersection of A street and N avenue. Where is the green house? How do we go from the green house to the school?

11. Combinatorics, 1 st week 2 nd grade How many two-letter words can you make, starting with a red letter and ending with a purple letter? How many two-letter words can you make, starting with a red letter and ending with a purple letter? aisnt

12. Multiplication, coordinates, phonics? aisnt asas inin atat GR 2

13. Multiplication, coordinates, phonics? wsillitink bp stickacking brtr GR 2

14. How will they learn the facts? Doubling and halving (also for fractions!) Doubling and halving (also for fractions!) Multiplying and dividing by 10 (decimals!) Multiplying and dividing by 10 (decimals!) Arrays, arrays, arrays Arrays, arrays, arrays Begins in Kindergaten Begins in GR 1 Throughout TM!

15. Intersections, arrays, area 3 4 = 4 3 GR 2

16. Similar questions, similar image Four skirts and three shirts: how many outfits? Five flavors of ice cream and four toppings: how many sundaes? (one scoop, one topping) How many 2-block towers can you make from four differently-colored Lego blocks? GR 2

17. Foundations for multiplication The logic of addition: any-order any-grouping The logic of addition: any-order any-grouping The image of multiplication: intersections, arrays, and area The image of multiplication: intersections, arrays, and area The logic of multiplication: any-order any-grouping The logic of multiplication: any-order any-grouping

18. Representing 22 × 17 22 17 GR 3

19. Representing the algorithm 20 10 2 7 GR 4

20. Representing the algorithm 20 10 2 7 200 140 20 14 22 17 200 140 20 x 14 374 Recording the process using all four partial products GR 4

21. Building the common algorithm 20 10 2 7 200 140 20 14 220 154 374 34 340 GR 4 Summarizing across Summarizing down Product

22. Building the common algorithm 10 7 200 140 20 14 220 154 22 17 154 220 374 × 1 20 2 GR 4 374 34 340

23. 374 34 340 Building the common algorithm 20 10 2 7 200 140 20 14 220 154 17 22 34 340 374 × 1 GR 4

24. Getting ready for algebra 20 10 2 7 200 140 20 14 GR 4 22 17 200 140 20 x 14 374 We record the process using all four partial products We don’t summarize!

25. In algebra (d + 2) (r + 7) = d r 2 7 dr 7d7d 2r2r 14 Middle School dr + 7d + 2r + 14 We record the process using all four partial products We don’t summarize!

26. Multiplication, zillions of dots 22 17 374 22 × 17 = 374 GR 3

27. Multiplication, area 22 17 374 22 × 17 = 374 GR 3

28. Representing division (not yet the algorithm) “Oh! Division is just unmultipli- cation!” “Oh! Division is just unmultipli- cation!” 22 17 374 374 ÷ 17 = 22 22 17 374 GR 3

29. The division algorithm If the multiplication algorithm is really clear, division comes almost for free. If the multiplication algorithm is really clear, division comes almost for free.

30. 17 170 34 374 10 2 170 10 22 34 204 -170 -34 0 374 ? × 17 = 374 Representing the algorithm GR 4

31. Building the standard algorithm Easy once you understand the underlying math Easy once you understand the underlying math Practice old skills to build a table of multiples Practice old skills to build a table of multiples GR 5 (GR 4 exposure)

32. 123456789 37 74111148185222259296333 37 Total = 999 37 999 Left tensones Summary: 999 ÷ 37 = 20 740 259 7 7 2 5 92 5 9 0 27 GR 5