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?