1. Fraction Multiplication

2. There are 3 approaches for modeling fraction multiplication u A Fraction of a Fraction  Length X Width = Area u Cross Shading We will now examine each of these 3 approaches.

3. Area of a Rectangle What is the area of this rectangle? To find the area of a rectangle we can multiply the length by the width. Area = Length X Width = 4 x 3 = 12 units 2

4. To find the answer to, we will use the model to find of. 3 5 We use a fraction square to represent the fraction. 3 5 1 2 3 5 1 2 3 5 X

5. Then, we shade of. We can see that it is the same as. 3 5 3 5 1 2 of 1 2 3 10 3 = 3 5 X 1 2 So,

6. In this example, of has been shaded 3 4 1 2 of 1212 3434 What is the answer to ? 1212 3434 X

7. Modeling multiplication of fractions using the fraction of a fraction approach requires that the children understand the relationship of multiplication to the word “of.” We can establish this understanding showing whole-number examples like: 6 threes is the same as 6 X 3.

8. We will think of multiplying fractions as finding a fraction of another fraction. 3 4 We use a fraction square to represent the fraction. 3 4

9. Then, we shade of. We can see that it is the same as. 3 4 3 4 2 3 of 2 3 6 12 = 3 4 X 2 3 6 But, of is the same as. 3 4 2 3 3 4 X 2 3 So,

10. In the second method, we will think of multiplying fractions as multiplying a length times a width to get an area. 3 4 This length is

11. In the second method, we will think of multiplying fractions as multiplying a length times a width to get an area. 2 3 This length is 3 4

12. We think of the rectangle having those sides. Its area is the product of those sides. 2 3 3 4 This area is X 3 4 2 3

13. We can find another name for that area by seeing what part of the square is shaded. 2 3 3 4 This area is X 3 4 2 3 It is also 6 12

14. We have two names for the same area. They must be equal. 2 3 3 4 This area is X 3 4 2 3 It is also 6 12 3 4 2 3 X = 6

15. Length X Width = Area This area is X 3 4 1 2 3 4 1 2 It is also 3 8 3 4 1 2 X = 3 8

16. What is the answer to X ? 4 5 1 4 1414 4545

17. Modeling multiplication of fractions using the length times width equals area approach requires that the children understand how to find the area of a rectangle. A great advantage to this approach is that the area model is consistently used for multiplication of whole numbers and decimals. Its use for fractions, then is merely an extension of previous experience.

18. In the third method, we will represent both fractions on the same square. 3 4 is 1 2

19. The product of the two fractions is the part of the square that is shaded both directions. 3 4 is 1 2 3 4 1 2 X = 3 8

20. We will look at another example using cross shading. We shade one direction. 4 5 4 5

21. 4 5 2 3 The answer to X is the part that is shaded both directions. 4 5 2 3 4 5 2 3 X = 8 15 Then we shade the other direction. 2 3

22. The End