Adding and Subtracting Fractions Using Counting Boards



Supplies

The two 3 by 4 boards shown above, and 2-color tokens (red and white)

A white token on a number in a square has the value of the number, and the value of a red token on a square is the negative of the number written on the square. Notice that dividing each number on the whole number board by 60 yields the corresponding fraction on the fraction board.

The procedure for adding and subtracting fractions using the two boards is as follows: We place tokens on the fraction board to represent the problem. We also place tokens on the whole number board that correspond to the fractional numbers. We regroup the tokens on the whole number board, typically until all tokens are on the same whole number (or possibly on the number 60 as well). We then place tokens on the fraction board to match those on the whole number board, and read the answer from the fraction board.

1. Adding Fractions

Example 1. Let's add 1/3 + 1/4 + 1/5 + 1/6 + 1/20 We place white tokens on 1/3, 1/4, 1/5, 1/6, and 1/20. We also place white tokens on the corresponding numbers on the whole number board:



Now we regroup the tokens on the whole number board: 15 + 20 + 10 + 12 + 3 = 60. So we "regroup" on the fraction board, or simply copy from the whole number board; 60 on the whole number board corresponds to 1 on the fraction board. So 1/3 + 1/4 + 1/5 + 1/6 + 1/20 = 1.

Example 2. 2/3 + 1/2 We place two tokens on 1/3 and one token on 1/2 on the fraction board, and correspondingly, 2 tokens on 20 and 1 token on 30 on the whole number board:


We regroup on the whole number board and copy our answer on the fraction board:


So 2/3 + 1/2 = 1 + 1/6.

Example 3. 3/4 + 2/5 We put 3 tokens on 1/4 and 2 tokens on 1/5 on the fraction board, and 3 tokens on 15 and 2 tokens on 12 on the whole number board:


We regroup on the whole number board and copy it on the fraction board:


So 3/4 + 2/5 = 1 + 1/10 + 1/20. But we want to write the sum as a single fraction. So we regroup the 6 on the whole number board to two threes, and we do the same on the fraction board, regrouping 1/10 to 2/20.


3/4 + 2/5 = 1 + 3/20
Notice that the board handles fractions with denominators 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. And when the total on the board is not a single fraction, we may regroup on the whole number board so that all tokens are on the same whole number square, and then copy to the fraction board.



2. Comparing two fractions

Example 1. Which is bigger, 5/12 or 7/15? We can treat this as a subtraction problem, 7/15 - 5/12. 1/15 corresponds to 4, and 1/12 corresponds to 5 on the whole number board. So we put 7 white tokens on the 4 on the whole number board, and 5 red tokens on 5. If we know multiplication, 28 - 25 = 3, which corresponds to 1/20 on the fraction board, so 7/15 - 5/12 = 1/20. 7/15 is bigger.



3. Subtraction

Example 1. 3/4 - 3/10 We use white tokens for 3/4 and red tokens for 3/10:



On the whole number board, 45 - 18 is regrouped; 45 - 18 = 27:


and regrouped again (nine tokens on 3):


So we put nine tokens on 1/20 on the fraction board. 3/4 - 3/10 = 9/20.

Example 2. 1 5/12 - 3 2/3 The two boards look as follows:


We regroup on the whole number board, and then copy onto the fraction board:


So 1 5/12 - 3 2/3 = -(2 + 1/4)

Some problems for you to try: answers:
a.) 1/4 + 1/12 =           d.) 1/5 - 1/6 =           a.) 1/3           d.) 1/30
b.) 1/2 + 1/5 =           e.) 3/4 - 1/6 =           b.) 7/10           e.) 7/12
c.) 2/3 + 3/4 =           f.) 3 1/3 - 1 4/5 =           c.) 1 5/12           f.) 1 8/15

Webpage Maintained by Owen Ramsey
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