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 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 Example 3:

5. More partial products Now we are going to explore this technique of partial products with fractions. Draw a rectangle and label the sides with 2 and 4 ½ Can you make two regions in the rectangle and label the sides?

6. More partial products Now we are going to explore this technique of partial products with fractions. Draw a rectangle and label the sides with 2 and 4 ½ Can you make two regions in the rectangle and label the sides?

7. More partial products Now we are going to explore this technique of partial products with fractions. Draw a rectangle and label the sides with 2 and 4 ½ Can you make two regions in the rectangle and label the sides? So we have Yellow: 2 × 4 = 8 Pink: 2 × ½ = 1 So 2 × 4 ½ = 9 Were you expecting 9?

8. More partial products What if we needed to find 2 1 / 3 x 4 ½ Can you extend the rectangle underneath?

9. More partial products What if we needed to find 2 1 / 3 x 4 ½ Can you extend the rectangle underneath?

10. yellowpinkgreenblue 2 x 42 x 1 / 2 1 / 3 x 4 1 / 3 x 1 / 2 So now we have 4 partial products

11. yellowpinkgreenblue 2 x 42 x 1 / 2 1 / 3 x 4 1 / 3 x 1 / 2 81 4 / 3 = 1 1 / 3 1/61/6 So now we have 4 partial products So 2 1 / 3 x 4 1 / 2 Can be found by adding 8 + 1 + 1 1 / 3 + 1 / 6 = 10 ½

12. We can also consider lower and upper bounds to check our answers.

13. 2 1 / 3 x 4 ½

14. Upper bound 3x5 = 15

15. Lower bound 2 x 4= 8 Upper bound = 15

16. Lower bound = 8 Upper bound = 15 So we know that our answer (to 2 1 / 3 x 4 ½) lies between 8 and 15.