Instructional Sequence for Multiplication and Division of Fractions

Concepts

Fractional numbers must be multiplied with consideration of their part values
Fractional numbers must be divided with consideration of their part values.
Models can be used to solve problems

Prior knowledge

Background information

Sequence ideas for Multiplication of fractions

4 1/2 * 50 = 9 * 25 = 225

Or (4 * 50) + (1/2 * 50) = 200 + 25

Or (5 * 50) - (1/2 * 50) = 250 - 25 = 225

2 1/4 * 100

2 1/4 * 120

4 1/2 * 60

15 1/2 * 36 = (10 * 36) + (5 * 36) + (1/2 * 36) = 360 + (1/2 * 360) + 18 = 360 + 180 + 18

15 1/2 * 4 1/2 =

Open arrays
Are arrays too easily constructed procedurally so that students do not see the conceptual? A person really has to think to see the fractional relationship (an array within an array).

1/3 * 1/4 (array 3 x 4)

2/3 * 1/4 (use same array, is it twice the other? Why?)

2/3 * 3/4 (How does this compare with the two preceding?)

4/3 * 3/2 Make a 3 x 4 array 2 x 3 would be the whole…

Might try 1/2 of 4/3 first, then 2/2 * 4/3, then 3/2 * 4/3

Then do these and transition to swapping numerators and denominators

1/5 * 1/7

3/5 * 4/7

Swapping numerators and denominators

4/5 * 3/7

Or 4 * 3 * 1/5 * 1/7

If only an array is used students will notice that the inside array and the outside array are the same, only rotated 90 degrees, and see that multiplication is commutative (doesn't matter if multiply 1/2 * 2/3 or 2/2 * 2/3) but not understand multiplication of fractions conceptually.

3/8 * 4/9

5/6 * 3/5

4/5 * 5/8

Use to find when swapping is a useful strategy.

Getting rid of the fraction

3 1/2 * 18 ((double to get rid of the fraction)) 7 * 9 = 63

3 1/4 * 28 = 13 * 7 = 70 + 21 = 91

3 1/5 * 50

Division

3 1/3 / 1/3

This idea is related to the invert and multiply algorithm or multiply by the reciprocal.

1/3 * 1/4

2/3 * 1/4

2/3 * 3/4

1/5 * 1/7

3/5 * 4/7

4/5 * 3/7

3/8 * 4/9

5/6 * 3/5

4/5 * 5/8

6 * 10

12 * 5

24 * 2 1/2

8 * 30

16 * 15

32 * 7 1/2

64 * 3 3/4

18 * 5 1/2

9 * 11

4 1/2 * 22

___ * ___

14 * 3 1/2

4 / 1/2

8 / 1

16 / 1/4

32 / 1/2

64 / 1

5 1/2 / 1/3

16 1/2 / 1

2 1/2 / 1/5