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Introducing Common Fractions Using Counting Boards

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Fractions are mostly used together with integers. So, we introduce boards in pairs. Examples                            In each case you create a fraction board by dividing all whole numbers on the integer board by their least common multiple. If you use only one counter per square, the total is represented as a sum of unit fractions. (This way of representing fractions was used in ancient Egypt.) But you may stack more tokens on one square to represent bigger numerators. Examples 1/6 + 1/3 = 1/2 1/6 + 1/6 + 1/5 + 1/15 + 1/15 + 1/30 = 7/10 1 + 1/4 + 1/6 + 1/6 + 1/6 + 1/10 + 1/10 + 1/15 + 1/15 + 1/30 = 127/60 The most important fact is that the rules of regrouping are the same on an integer board and its corresponding fraction board. So for example, if you want to compute 2/5 - 1/3, you may compute instead 24 - 20 = 4 and get 1/15 as the answer to the original problem. Using the fraction board... → → 1/5 + 1/5 = 1/3 + 1/15 1/3 - 1/3 = 0 Using the integer board... → → 12 + 12 = 20 + 4 20 - 20 = 0 After regrouping, the integer board holds 4. On the fraction board, this corresponds to the answer 1/15. Other Examples 2/15 + 1/30 Using the fraction board... → → 1/30 + 1/15 = 1/10 1/15 + 1/10 = 1/6 Using the integer board... → → 1 + 2 = 3 2 + 3 = 5 After regrouping, the integer board holds 5. On the fraction board, this corresponds to the answer 1/6. 1/6 - 1/10 Using the fraction board... → → 1/6 = 1/10 + 1/15 1/10 - 1/10 = 0 Using the integer board... → → 5 = 3 + 2 3 - 3 = 0 After regrouping, the integer board holds 2. On the fraction board, this corresponds to the answer 1/15. 1/2 + 1/15 - 4/10 Using the fraction board... → → 1/10 + 1/10 + 1/10 + 1/10 = 1/3 + 1/15 1/15 - 1/15 = 0 → → 1/2 = 1/6 + 1/3 1/3 - 1/3 = 0 Using the integer board... → → 6 + 6 + 6 + 6 = 20 + 4 4 - 4 = 0 → → 30 = 10 + 20 20 - 20 = 0 After regrouping, the integer board holds 10. On the fraction board, this corresponds to the answer 1/6. Webpage Maintained by Owen Ramsey Number Board index