April 22, 2011

How to help a student with a fraction word problem

Today, my daughter had to tackle this word problem in her 5th grade Math Mammoth (lesson Multiply Mixed Numbers):

The square on the right measures 10 in. × 10 in. and
the rectangle inside it measures 6 7/8 in. × 3 1/2 in.
How many square inches is the colored area?

This requires the student to
  • multiply mixed numbers, 
  • subtract mixed numbers, 
  • understand about area, and 
  • understand how to find the "colored" area by subtraction of areas. 

So it is a multi-step word problem.

She didn't understand it, she said. My first "help" was this:
"Let's say we change those fractions to whole numbers 6 and 3. Can you mark those in the image? Would you be able to solve the problem now?"

The strategy I used is:
If you can't solve the problem at hand, change it and make it easier. Then try to solve the easier problem.

She was able to mark 6 and 3 on the sides of the rectangle (that is inside the square). But she said she couldn't solve it. She said it's not possible to find the area of the colored area!

Then I asked her, "Is there anything you CAN find? Is there anything you CAN solve using this information?"

And this is another great strategy in solving any problem (whether math or not): If you can't find the answer to the question in the problem, solve what you CAN solve. That might lead you to find the answer to your question somewhere along the way!

I said, "Well, we CAN find the area of the square. It is 100 square inches. We CAN find the area of the rectangle inside it."

THEN immediately after I said that, she saw it: "OHH! SUBTRACT!" And on she went to multiply the mixed numbers in the problem. So the story had a happy ending!

Here's the complete solution:

First multiply the mixed numbers 6 7/8 and 3 1/2 to find the area of the rectangle. Keep in mind they need to be changed into fractions before multiplying.

6 7/8 × 3 1/2 = 55/8 × 7/2
= 385 / 16 square inches

Change this into a mixed number. Here one needs to use long division to divide 385 ÷ 16 = 24 R1. This tells us the whole number part is 24. The remainder,1, tells us how many 16th parts are "left over."

So, 385 / 16 = 24 1/16 square inches. So this is the area of the rectangle.

Now, the area of the surrounding square is simply 10 in x 10 in. = 100 square inches.

And lastly, the area of the colored area is found by subtracting 100 − 24 1/16 = 75 15/16 square inches.
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