Fraction Addition (And Subtraction): We’re not in kindergarten anymore
Addition and subtraction are only easy in elementary school. Once middle school starts, continuing throughout any Math class taken that point forward, addition and subtraction are much harder than multiplication and division. Why? The Common Denominator. To a kid who is not fluent in his multiplication facts, finding The Common Denominator is an exercise in torture.
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What is a common denominator? A common denominator is a multiple of both denominators in a fraction addition (or subtraction) problem. For example:
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In the above example, 6 is a common denominator of 2 and 3. But is it the only one? No. How many common denominators are there between two fractions? Infinite. For example:
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Why would we want to use 7830 as a common denominator? Why not? The point is that any number that both denominators divide into evenly can act as a common denominator. We are far less restricted than we thought.
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So if we’re virtually unrestricted in choosing a common denominator, why not pick the one that is the product (multiply) of the two denominators? For example:
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Just multiply the denominators to find a common denominator. This is easy.
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At this point in the traditional method of adding fractions, we’d begin to ask our questions: “How many 8’s go into 16?” Ok, 2. “2 times 3 is …?” Ok 6. So 3/8 = 6/16 . Though this process is easy to a person who is fluent in their multiplication and division, it will give reason for a non-fluent Math student to seize up.
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A great alternative way of adding fractions is the Crisscross Method of adding (and subtracting) fractions. In this method, we use the common denominator just once (this method will not create two equivalent fractions to the original two) and multiply “crisscross” to find two new numerators.
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In 3/8 + 5/2, we’ll first multiply the denominators to find our new, common denominator:
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Next, we’ll multiply 3 • 2 (always starting our crisscross in the top left corner) to find the first missing numerator:
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And then 8 • 5 to find the second missing numerator:
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But why are we allowed to do this? Let’s back up to see what really happened.-
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First, we found the common denominator 16 by multiplying the denominators (8 and 2) of both fractions. We’re guaranteed that our denominator is common if we created it by multiplying the two original denominators to get it. To get the first numerator 6, we multiplied the numerator of the first fraction (3) by the denominator of the second fraction (2).
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In the process, we multiplied both numerator and denominator by 2. In other words, we multiplied 3/8 by 2/2. Any number divided by itself is just a fancy 1, and multiplying any number by 1 does not change the number’s value. As a check to see if this process worked, 3/8 = 6/16 . The old and new fractions are equivalent.
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The same is true to get the second numerator 40:
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Both numerator and denominator were multiplied by 8. In other words, we multiplied 5/2 by 8/8, which is just a fancy 1. Multiplying by 1 does not change a number’s value. As a check, 5/2 = 40/16. The old and new fractions are equivalent.
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Now we simply add the numerators:
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The Crisscross method also works for fraction subtraction – we’d have a subtraction in the numerator. Why was this method not taught in school?
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Hurray for Fraction Addition (and Subtraction)!
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You can download a PDF ebook that uses pictures to explain fraction division, multiplication and addition on CurrClick at Fractions: A Picture Book!
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