Rectangles and Multiplication: Understanding Partial Products

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3. Rectangles and Multiplication Here is a rectangle with sides 3 and 7. The total number of squares can be found by multiplying 3 and 7. 7 3

4. Rectangles and Multiplication Here is a rectangle with sides 3 and 7. The total number of squares can be found by multiplying 3 and 7. Note that if we colour the squares we can work out the number of blue squares and the number of yellow squares separately and add. 7 3

5. Rectangles and Multiplication Here is a rectangle with sides 3 and 7. The total number of squares can be found by multiplying 3 and 7. Note that if we colour the squares we can work out the number of blue squares and the number of yellow squares separately and add. 7 3 Blue 3 × 5 = 15 Yellow 3 × 2 = 6 Total 15 + 6 = 21

6. This technique is useful for larger rectangles.

7. 6 Here is a rectangle with sides 15 and 6, so the total number of squares can be found from: 15 × 6. Again, if we colour the squares we can work out the number of blue squares and the number of yellow squares separately and add. 15

8. 6 Here is a rectangle with sides 15 and 6, so the total number of squares can be found from: 15 × 6. Again, if we colour the squares we can work out the number of blue squares and the number of yellow squares separately and add. Blue: 10 × 6 = 60 Yellow: 5 × 6 = 30 Total 60 + 30 = 90 10 5

9. Now consider even larger rectangles

10. Here is a rectangle with sides 54 and 23. 54 23

11. Here is a rectangle with sides 54 and 23. The total number of squares can be found from 54 × 23. 54 23

12. Here is a rectangle with sides 54 and 23. The total number of squares can be found from 54 × 23. Again, we can divide the rectangle into regions. What regions will you choose? 54 23

13. Here is a rectangle with sides 54 and 23. The total number of squares can be found from 54 × 23. Again, we can divide the rectangle into regions. What regions will you choose? Did you choose these 4 regions? 50 4 20 3

14. It would be easier if we drew the rectangle on grid paper.

15. Here is a rectangle with sides 54 and 23. 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 (as shown)

16. Here is a rectangle with sides 54 and 23. 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 (as shown) Orange: 50 x 20 = 1000 Yellow: 4 x 20 = 80 White: 50 x 3 = 150 Blue: 4 x 3 = 12 Total: 1242

17. Here is a rectangle with sides 54 and 23. 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 (as shown) Orange: 50 x 20 = 1000 Yellow: 4 x 20 = 80 White: 50 x 3 = 150 Blue: 4 x 3 = 12 Total: 1242 These are sometimes called ‘partial products’

18. Now your turn: Sketch a rectangle and label the sides with 25 and 75. What regions will you choose to divide it into?

19. 70 5 20 5 Did you choose these four regions? No matter what regions you choose, if you work out the partial products and then add, you will still get the same answer (25 ×75 = 1875)

20. 70 5 20 x 70 =1400 20 x 5 =100 20 5 5 x 70 =350 5 x 5 =25 Here are the 4 partial products for the 4 regions that were chosen.

21. 70 5 20 x 70 =1400 20 x 5 =100 20 5 5 x 70 =350 5 x 5 =25 So the result is found by adding the 4 partial products: 1400 + 100 + 250 + 25 = 1875