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.

Fractions, part 1: modeling equivalents and beginning addition

Fractions can be very confusing to students who have no concrete experience with them.  One 6th grade remedial math class was learning to reduce fractions, and they learned the hard way that half of 1/4 is not 1/2.  We were halving a brownie recipe (countertop ovens are an essential in my math classroom) and as the brownies were baking, the room smelled deliciously chocolatey... the students could not wait to taste them.  To their dismay, the miscalculation of the salt fraction made the entire batch completely inedible.

So, before any operations can be done with fractions, students need hands-on experience with the concept.  Cooking is a great way to accomplish this, and even toddlers can help measure ingredients for a batch of cookies.  Click here for a lovely sugar cookie recipe full of fractions!  Measuring cups can also be played with in the sandbox, bathtub, or at the kitchen table with some dry rice or beans.

The basic purpose of playing with measuring cups is to note the name of the fraction and its size relative to the other measurements, and to compare the quantities.  Which is bigger, 1/2 or 1/4?  How many 1/4 cups does it take to make 1 cup?  Can 1/3 go into 1/2 evenly? How many cups can you make out of five 1/4 cups?  How many 1/3 cups can you get out of 2 cups?





The more exposure your child has to measuring, the more natural the concepts will be for him.  Once your child is comfortable with fractions in measuring, try pie circles.  These can be purchased or made at home from paper, fabric, craft foam, or anything else you might have.  Templates are available online.

 The first objective here is to get students familiar with what is meant by 1/2,  1/3, 1/4, etc.  Concepts to emphasize include:

1. The fraction's denominator (bottom number) refers to the size of the piece-- how many pieces it takes to complete the whole (in this case, the whole circle).

2. The greater the denominator, the smaller the fraction size:  1/2 is larger than 1/4, even though "4" is greater than "2". (This is very important when you're making brownies!)

3. The numerator (top number) refers to how many pieces you have in the fraction.  2/3  means two pieces that are 1/3 each.

4. When the numerator and denominator in a fraction are the same number, you have all the pieces you need to complete the circle.  So 2/2 = 3/3 = 4/4 = 5/5 = 6/6 = 1

5. Some fractions can be put together to be the same size as a different fraction.  1/2 = 2/4 = 3/6, and  2/6 = 1/3.  These are called equivalent fractions.  But you can't make 1/2 out of thirds or fifths, and you can't make 1/3 out of fourths or fifths.

6. Sometimes different fractions can be added together to make whole circles.  1/2 and 3/6 make a whole, as do 2/4 and 1/2, or 2/3 and 2/6.

Plenty of time should be allowed for these concepts to sink in.  A child who is first learning about fractions may want to play with them for a few minutes every day, for a week or two, before you even start labeling them as fractions or specifying numerators and denominators.  For preschoolers, just playing with the fractions is enough.




At this point, you may help your school-aged child start a list (or booklet) of Fraction Discoveries (Or Fraction Facts, Fraction Truths, etc).  This is where she records what she sees in "math code"-- also known as math sentences or equations.  She can illustrate each equation with pictures of the fraction circles (tracing jar lids or using a compass to make the circles).  Her discoveries may include:

1 = 2/2
3/3 = 1
4/4 = 1
1/2 = 2/4
1/3 = 2/6
1/ 3 + 1/3 + 1/3 = 1
1/2 + 1/2 = 1
2/4 + 1/2 = 1
2/3 + 2/6 = 1

When circle fractions are mastered, you may go over the same concepts using bar models or rods.  These are Cuisenaire rods,

but you can also use homemade fraction strips.  These can be printed out pre-colored or ready to color; having each size a different color makes them easier to tell apart.  Then all you have to do is cut them out.

Warning: with the Cuisenaire rods, the orange "one" that is equivalent to two yellow "halves" is a different size than the blue "one" that is the same as three green "thirds."  The black "one" can only be evenly divided into seven white "sevenths."  That is great for demonstrating prime and composite numbers, but if you think it might confuse your student, you probably would be safer with the fraction strips.

What new Fraction Discoveries might your child come up with now?