Using Manipulatives to Reduce Fractions

Instructor: James Anderson

James Anderson is a PhD Candidate in Agricultural Sciences specializing in Plant Breeding. He has taught several classes during this time, including Plant Genetics, Plant Breeding, Field Plot Techniques, Precision Agriculture, and Soil Fertlility Evaluation.

Using manipulatives is a fancy way of trying to find a way to make a fraction as small as we possibly can. Why do we want to do that? Because we tend to like fractions that are smaller so we can get a grasp of them. Do you know what 32/96 is? No? Well, what about 1/3? Those two numbers are the same, and we use the manipulatives to make those numbers smaller.

Take a fresh apple pie. It comes out of the oven and is cut into eight pieces. If you eat four of those eight pieces you can see that it was half of the pie, but how do you figure that out based on what you ate? The four pieces that you ate are placed in the numerator or top part of the fraction. The total number of pieces, eight, is placed in the denominator or bottom of the fraction. This shows that out of the eight pieces of pie we had, four of them were eaten, or 4/8.

Reducing Fractions

What we do to reduce fractions is to figure out how to make the number smaller. Looking at the number before, 4/8, we look to see how we can make it smaller. The easiest way to do this is to look for numbers that will divide into both the numerator and denominator. To do this we look at what numbers are can divide evenly into both. 4 can be divided by 1 and 4 as well as 2 and 2 (with 1 times 4 equaling 4 and 2 times 2 equaling 4). 8 can be divided by 1 and 8 and 2 and 4 (with 1 times 8 equaling 8 and 2 times 4 being 8. So, we look at the divisors that both sets have. They both have the numbers 1, 2, and 4 that can divide into both, called a common divisor. At this point you can start simplifying the fraction. You take the numbers that both sets have and dividing the numerator and denominator by the common divisor.

Greatest Common Denominator

Dividing 4 by 1 gives 4 and 8 by 8 gives 8, leaving the original fraction of 4/8, which does not help. Dividing by the second common divisor gives us slightly different results. Dividing 4 by 2 gives a 2 and 8 by 2 gives a 4, reducing the fraction to 2/4. This is closer, but not quite right as we can see that both the 2 in the numerator and the 4 in the denominator are divisible by 2. So, this can be simplified even more. At this point we look at the third common divisor, 4, which is called the greatest common divisor (GCD). The GCD is the largest number that can divide into two sets of numbers. We divide 4 by 4, giving a result of 1. We divide 8 by 4, giving an answer of 2. This leaves a fraction of 1/2, which cannot be reduced any more.

So, there you go. We were able to identify the common divisor for the fraction 4/8, figure out which one was the greatest common divisor, and reduce the fraction to its smallest form. All while not thinking about eating an apple pie at all no sir!

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