1. More partial products Recall that we can use a drawing of a rectangle to help us with calculating products. The rectangle is divided into regions and we determine “partial products” which are then added to find the total.

2. More partial products Recall that we can use a drawing of a rectangle to help us with calculating products. The rectangle is divided into regions and we determine “partial products” which are then added to find the total. 3 7 Blue 3 × 5 = 15 Yellow 3 × 2 = 6 Total = 21 Example 1:

3. More partial products Example 2: The total number of squares can be found from 54 × 23. One of the ways to calculate 54 × 23 is to divide the rectangle into 4 regions Orange: 50 x 20 = 1000 Yellow: 4 x 20 = 80 White: 50 x 3 = 150 Blue: 4 x 3 = 12 Total: 1242 So 54 × 23 = 1242

4. More partial products Now we are going to extend this technique of partial products with decimal numbers. Draw a rectangle and label the sides with 2 and 4.

5. More partial products Now we are going to extend this technique of partial products with decimal numbers. Draw a rectangle and label the sides with 2 and 4. We know 2 × 4 = 8 2 4

6. More partial products Now we are going to extend this technique of partial products with decimal numbers. Draw a rectangle and label the sides with 2 and 4. We know 2 × 4 = 8 2 4 What if we need to find 2 × 4.7? Can we draw another small region on the right?

7. More partial products Now we are going to extend this technique of partial products with decimal numbers. Draw a rectangle and label the sides with 2 and 4. We know 2 × 4 = 8 2 4 How wide will it be?

8. More partial products Now we are going to extend this technique of partial products with decimal numbers. Draw a rectangle and label the sides with 2 and 4. We know 2 × 4 = 8 2 4 How wide will it be? (0.7) 0.7

9. More partial products So, to find 2 × 4.7 we can use partial products: Pink: 2 × 4 = 8 Yellow: 2 × 0.7 = 1.4 2 4 0.7

10. More partial products So, to find 2 × 4.7 we can use partial products: Pink: 2 × 4 = 8 Yellow: 2 × 0.7 = 1.4 2 4 0.7 So 2 × 4.7 = 9.4

11. More partial products This is much easier to see if we use grid paper Pink: 2 × 4 = 8 Yellow: 2 × 0.7 = 1.4, So 2 × 4.7 = 9.4

12. More partial products What about 2.1 × 4.7? Can we draw an extra region underneath? What size will it be? Should we leave it as one region or make two regions?

13. More partial products What about 2.1 × 4.7? Here there are two more regions coloured green and orange. Green: 0.1 × 4 = 0.4 Orange: 0.1 × 0.7 = 0.14

14. More partial products pinkyellowgreenorange 2 x 42 x 0.70.1 x 40.1 x 0.7 81.40.40.07 So 2.1 x 4.7 = 8+1.4+0.4+0.07 = 9.87

15. Another example Here is one for you to try: Step 1: Sketch a rectangle and label the sides with 1.5 and 3.6 Step 2: Draw lines to make 4 regions. Step 3: What are the lengths of the sides of these regions?

16. Partial products: 1.5 x 3.6 30.6 0.5 1 Does your drawing have these 4 regions? Did you have these lengths? Now find the 4 partial products.

17. Partial products: 1.5 x 3.6 30.6 0.5 1 1 x 3 = 3

18. Partial products: 1.5 x 3.6 30.6 0.5 1 1 x 3 = 3 0.5 x 3 = 1.5 Think: One half of 3 is 1.5, or 3 groups of 5 tenths is 15 tenths

19. Partial products: 1.5 x 3.6 30.6 0.5 1 1 x 3 = 3 0.5 x 3 = 1.5 1 x 0.6 = 0.6

20. Partial products: 1.5 x 3.6 30.6 0.5 1 1 x 3 = 3 0.5 x 3 = 1.5 1 x 0.6 = 0.6 0.5 x 0.6 = 0.3 Think: One half of 6 tenths is 3 tenths, or 5 / 10 × 6 / 10 = 30 / 100 = 3 / 10

21. Partial products: 1.5 x 3.6 30.6 0.5 1 1 x 3 = 3 0.5 x 3 = 1.5 1 x 0.6 = 0.6 0.5 x 0.6 = 0.3 So, the answer to 1.5 × 3.6 is the total of the 4 partial products: Add them up to find your total.

22. Partial products: 1.5 x 3.6 30.6 0.5 1 1 x 3 = 3 0.5 x 3 = 1.5 1 x 0.6 = 0.6 0.5 x 0.6 = 0.3 So, the answer to 1.5 × 3.6 is the total of the 4 partial products: 3 + 1.5 + 0.6 + 0.3 = 5.4

23. Partial products: 1.5 x 3.6 30.6 0.5 1 1 x 3 = 3 0.5 x 3 = 1.5 1 x 0.6 = 0.6 0.5 x 0.6 = 0.3 This is an alternative method to use to multiply decimals. Even if you don’t use this diagram and the four partial products it has other uses.

24. Upper and lower bounds We can use the diagram to help us with estimation: a lower bound and an upper bound

25. 1.5 3.6

26. 1 3 The purple rectangle is sitting on top of the green rectangle, and is definitely smaller, so 3 (from 1 x 3) is a lower bound for the size of the green rectangle.

27. 2 4 The orange rectangle is on top of the green rectangle and it is definitely larger, so 8 (from 2 x 4) is an upper bound for the size of the green rectangle.

28. 1.5 3.6 So we are sure that the area of the green rectangle is more than 3 and less than 8. So, we can reject any answer which is not between 3 and 8. This is a good strategy for mental estimation, before we do an exact calculation.

29. Practice Consider 42 x 28 Without actually finding the answer, can you give: a lower bound for the answer? an upper bound for the answer?

30. Practice Consider 42 x 28 Without actually finding the answer, can you give: a lower bound for the answer? -for example, 800 (40 x 20) an upper bound for the answer? for example, 1500 (50 x 30) So if we calculate the answer and it is not between 800 and 1500, then we know it is wrong.

31. Can you pick Sam’s error? Sam wrote 2.5 × 6.7 12.35

32. Can you pick the error? Sam wrote 2.5 × 6.7 12.35 Using our method of upper and lower bounds, we could predict the answer is between 12 and 21 (i.e. between 2 × 6 and 3 × 7). As 12.35 does lie between 12 and 21, we cannot reject it for this reason. Can you find the correct answer? Can you see what Sam has done incorrectly?

33. Partial products: 2.5 x 6.7 60.7 0.5 2 2 x 6 = 12 0.5 x 6 = 3 2 x 0.7 = 1.4 0.5x 0.7= 0.35 There should be 4 partial products which need to be added (12 + 3 + 1.4 + 0.35 = 16.75)

34. Partial products: 2.5 x 6.7 60.7 0.5 2 2 x 6 = 12 0.5 x 6 = 3 2 x 0.7 = 1.4 0.5x 0.7= 0.35 There should be 4 partial products which need to be added (12 + 3 + 1.4 + 0.35 = 16.75) Did Sam find these 4 partial products?

35. Partial products: 2.5 x 6.7 60.7 0.5 2 2 x 6 = 12 0.5 x 6 = 3 2 x 0.7 = 1.4 0.5x 0.7= 0.35 60.7 0.5 2 2 x 6 = 12 0.5 x 0.7=0.35 Sam only found 2 of these

36. Partial products: 2.5 x 6.7 60.7 0.5 2 2 x 6 = 12 0.5 x 6 = 3 2 x 0.7 = 1.4 0.5x 0.7= 0.35 So, even if you do not use the rectangular region and the partial products to actually do the calculation (you might prefer another method), it is a helpful diagram to make sure that you don’t use Sam’s method!