Fraction problem with mental math

Someone asked me,
How many times does 3/4 fit into 15 6/8?

Here are two ways to solve this:

1) These numbers look awkward, but if I changed them to easier ones, for example
"How many times does 2 fit into 834", then we'd all soon realize that we need to use DIVISION.

So the original problem is solved by fraction division:

15 6/8 ÷ 3/4


Are you ready? Remember how to divide fractions?




2) But wait a minute! These numbers aren't so difficult after all... because 6/8 equals 3/4. Let's use this thinking cap of ours - mental math.

3/4 goes into 1 1/2 two times. Doubling that, we find 3/4 goes into 3 four times. And so to 15... fives times that: 3/4 goes into 15 4 × 5 or 20 times!

And, of course 3/4 goes into 6/8 exactly one time.

So all total 3/4 goes into 15 6/8 exactly 21 times. No leftovers. And that was easy!




On another note, Denise in Illinois has made up a mnemonic poem for kids to remember better the "invert and multiply" rule.

Comments

denise said…
A great example of why it's sometimes faster (not to mention easier) to just stop and think, than to plow ahead with a calculation. Of course, all students need to practice the calculation until they are very comfortable with it. But the students who can manipulate numbers in their heads like this are the ones who will have the easiest time with future math.
Lynx said…
As far as memorizing invert and multiply, there is a better way. One does not have to invert and multiply when dividing fractions. Just like adding and subtracting fractions, find a common denominator for the two fractions. Then divide across. When you divide the denominators you'll end up with 1, so the answer is the quotient of the numerators.

Example: 1/2 divided by 2/5 Find a common denominator of 10 and the problem becomes 5/10 divided by 4/10. 5 divided by 4 yields 4 1/5 which is the numerator. 10 divided by 10 yields one which is the denominator. That leaves 4 1/5 as the answer, which you will also get by solving with "invert and multiply."

I find this works better because it connects to something the kids already do.
denise said…
Your alternate method is valid, Lynx, but it is not better. It requires more steps (for any but the simplest fractions, common denominators are a nuisance), which means more places for the student to make a careless error. More important, it does not easily apply to algebra fractions. To make it through algebra, every student HAS to know the invert-and-multiply rule.

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