Fractions, Ratio, and Proportion Sequences
Make use of tens
Halve and doubles
Get rid of the fraction
Use all the factors in pretty ways.
9 * 30 (used factors) 9 * 3 * 10 = 27 * 10 = 270
15 * 18 = 15 * 2 * 9 = 30 * 9 = 270
10 * 18 + 5 * 18, (think of 5 * 18 as 1/2 of 180) = (180 + 90) = 270
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 and multiplication of fractions
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
Developing strategies for computation with decimals
All the strategies for whole numbers can work with decimals.
Friendly numbers
71.87 + 28.2 = 72.07 + 28 (compensation subtract .2 and add .2)
71.87 + 28.2 = 71.07 + 29 (compensation subtract .8 and add .8)
71.87 + 28.2 = 72 + 28 + .07
Using money
.20 * 9 = (5 * .20 = 1.00 and 4 * .20 = .80) 1.80
.20 * 9 = (.20 * 10 - .20) 1.80
.25 * 9
16 * .25 = 4
How many students would know that a 4x4 array of quarters is 4 dollars?
Using fractions and decimals interchangeably
75 * 80 = 3/4 * 80 * 100 ( because 75 was treated as 75/100, divide 100 need to multiply 100)
1/4 * 80
.25 * 80
25 * 80
1/2 * 60
.5 * 60
.50 * 60
.50 * .60