Fractions, part 2: modeling mixed numbers, reducing, converting

If your student has not yet played with fraction manipulatives, please read Fractions, part 1.

Now that she familiar with the names of the fractions and what they represent, it is time for your child to start making those numbers do some work!

In this post, we will discuss 1) converting between mixed numbers and improper fractions, 2) adding fractions, and 3) subtracting fractions.  It is best to introduce the concepts in that order.

First, the definitions:  a mixed number is a number that includes a whole number part and a fraction part.  1 2/3,  3  4/5,  16 1/2  are all mixed numbers.  (Most children are already familiar with mixed numbers, because we tell their ages that way.  Preschoolers understand that someone who is three-and-a-half is older than a three, but younger than a four.)

An improper fraction is a fraction whose numerator (top number) is greater than the denominator (bottom number.)   5/3,  19/5, and 33/2 are improper fractions.

Once your child has learned that 3/3 = 1 and 5/5 = 5,  he can play with other equivalents.  Using fraction circles or strips, see how many he can match up exactly.  He should see that  1/2, 2/4 and 3/6 are the same length.  And 1/3 is the same as 2/6.  These are equivalents.  But 2/3, 2/4, and 2/5 are NOT equivalent.  Their denominators make them different sizes, and he can clearly see that two small things put together is smaller than two big things put together.

If your child is keeping track of fraction facts or fraction discoveries in a notebook, you can help him draw or glue in examples of equivalents and label them.

Older students can learn to reduce fractions.  Reducing means to write the fraction with the smallest accurate denominator.  In the photo here, 3/6 and 2/4 can both be reduced to 1/2.  2/6 can be reduced to 1/3.  But 2/3 and 2/5 cannot be reduced, because no fraction with a lower denominator fits.

Students who are familiar with multiplication facts may recognize that reducing a fraction on paper is a lot like dividing whole numbers.  The only difference is that she must divide the top and bottom by the same number.  It must be a number that goes into both the top and bottom evenly.  The best number to divide by is called the "Greatest Common Factor," or GCF.  The GCF is the biggest number that will go into the other two numbers evenly.

For example, the GCF of 10 and 15 is 5, because 5 is the biggest number that can go into both 10 and 15.  30 is the GCF of 30 and 90, because 30 x 1 = 30, and 30 x 3 = 90.  21 and 15 have a GCF of 3.  12 and 7, however, have a GCF of 1, because there's no other number that goes into both 7 and 12 evenly.

Dividing the top and bottom of a fraction by the GCF immediately reduces that fraction as far as it will go.

For example:

4     divided by 2    =    2
6     divided by 2          3

25   divided by 5    =   5
30   divided by 5         6

Of course, younger children who have not begun multiplication can just stick to using the strips and circles.

In the next step, you will need at least two sets of each fraction size: two circles or cut into halves, two into thirds, two into fourths, etc.  Or you can print out, color and cut a second set of fraction strips.

With the extra set(s) of fractions, the student can show that 3/2 (an improper fraction) is the same as 1 1/2 (a mixed number)  She can record her observations in "math code," writing 3/2 = 1 1/2.

Your student may note that while 7/5 is the same as 1 2/5,   8/6 can be reduced from 1 2/6 to 1 1/3.

For a bit of incentive, you can challenge her to find as many improper fraction/mixed number sets as she can in five minutes.  She'll need to write them down as she finds them so she can re-use the fraction pieces without losing her fraction discoveries.

Next, see how easily she can convert a mixed number to an improper fraction with and without using the fraction pieces:  can she show you that 1 1/6 is the same as 7/6?  Or that 10/5 is the same as 2?   See if she can tell you what the improper fraction will be before she shows you.

When your student can comfortably convert from a mixed number to an improper fraction and back again, she is ready to try addition and subtraction of fractions.