Finding the fraction of a number (multiplying a whole number by a fraction)

When students are beginning to work with fractions,  one concept they need to understand is that a fraction involves both division and multiplication at the same time.  It's not as easy to see when using fraction bars or circles, because you're always starting with a fraction of a single thing-- a whole circle or a whole bar.  It is not until you are trying to find the fraction of a whole number that you can see what's really going on.

In this demonstration, I am using pennies and a set of fraction squares that I made from clear plastic page protectors drawn on with permanent markers.  I traced over a piece of graph paper to get my sections even (24 squares x 24 squares).  The same effect could be obtained from squares of paper or drawn a whiteboard, as long as the student has a selection of fractions to choose from.

Now we use our model.

A) What is 1/2 of 12?
B) What is 2/3 of 12?
C) What is 3/4 of 12?

We start with 1/2 of 12, because the student will likely already know the answer, but can also show it with the model.

1. Choose the fraction block that matches the denominator in the problem.  In this case, we are asked about 1/2, so we choose the square that is divided into 2 equal sections (halves).

2. Count out the correct number of pennies (or blocks or beads or whatever you're using) -- 12 in this problem-- and divide them evenly into the sections of the fraction square.  In this case, six pennies go into each of the two sections.

3. Select the number of sections indicated by the numerator-- in this case, 1.  Count the number of pennies in that/those section(s).  So we see that 1/2 of 12 = 6.

To find two thirds of 12, we: 1. Choose the square divided into three equal sections (thirds).  2. Count out 12 pennies and place an equal number of them into each of the three sections of the square (4).  3. Select two of the sections and count the pennies in the sections.  2/3 of 12 = 8.

To find three fourths of 12, we: 1. Choose the square divided into four equal sections (fourths).  2. Count out 12 pennies and place an equal number of them into each of the four sections of the square (3).  3. Select three of the sections and count the pennies in the sections.  3/4 of 12 = 9.

Next, try different whole numbers with various fractions.  Just be sure the whole number is always evenly divisible by the denominator in the fraction.  For example, 3/5 of 15 would be a good practice problem, but we're not ready for 3/5 of 27.

After enough practice, the student should recognize that what she is doing is dividing the whole number by the denominator and then multiplying that product by the numerator.   At this point, she can try using mental math to solve similar problems with larger whole numbers..

2/5 of 100 (40)
3/8 of 64 (24)
4/7 of 56 (32)
9/10 of 120 (108)

Finally, the student can be told that in math, "of" is generally translated "x" (multiplied by).  So 2/5 of 100 is actually 2/5 x 100, and so on, for the rest of the problems.

Multiplying vs dividing by fractions: Let's Make Brownies!

Multiplying or dividing fractions by a whole number is pretty easy to demonstrate.  Multiplying and dividing fractions BY fractions, however, can be a confusing business!  The algorithms aren't so bad-- to multiply one fraction by another, simply multiply across the top (numerators) and then across the bottom (denominators), reducing the end result when possible:

Division works the same way, except that the second fraction is "flipped" (inverted) before multiplying:

The typical textbook fraction multiplication model involves shading a rectangle with one fraction horizontally and shading the other fraction vertically.  The intersection is the solution.  That's fine.

1/2 x 2/3 =1/3

Here we have the vertical fraction, 1/2, shaded in pink, and the horizontal fraction, 2/3, shaded in black.  The place where they overlap is the solution-- 2/6 of the rectangle, or 1/3.

Division modeling gets a little trickier.  To model 1/2 ÷ 2/3, you have to show how many 2/3's are in 1/2-- which is not even one, but another fraction.

This lovely box diagram shows 1/2 of the box shaded in pink, and then 2/3 of the box shaded in black.  This time, both fractions are shaded in the same direction (here, vertically).  Again, the overlapping shading gives us the solution-- the difference here is that the overlapped part is measured in terms of the black shaded section-- the overlapped part covers 3/4 of the black shaded area. 

The difference in division modeling is also obvious when the dividend fraction is divided by a smaller divisor fraction.

In this case, the shaded 1/2 section is being divided by the outlined 1/6.  How many 1/6 are in 1/2?  There are three.

Not surprisingly, many students still don't understand what it is they're actually doing when they're multiplying or dividing fractions, so translating it into word problems can be frustrating.  To make the concepts clear, all you need is a fresh-baked batch of brownies, some measuring cups, and maybe some paper and scissors.

First, bake the brownies.  Then, consider this problem:

John made a pan of brownies and shared them with his friends.   They ate half of the brownies in the pan.  After they left, Dad ate two thirds of the remaining brownies.  How much of the original batch did Dad eat?

Is this a multiplication or a division problem?

To help your child understand what's going on, she needs to know two things:

1) The multiplication sign can generally be translated "of" in English.  2x100 means two (sets) of 100, which is 200.  1/2 x 100 means one half of 100, which is 50.

2) Division answers the question, "How many (sets of) _the second number_  are in _the first_?"  100 ÷ 50 is asking, How many sets of 50 are in 100? = 2.  50 ÷ 1/2  asks,  "How many sets of 1/2 are in 50? = 100.

So, back to the brownies-- if we have half a pan of brownies and Dad eats 2/3 OF those,  we can figure out how much Dad ate by multiplying.  1/2 x 2/3 = 1/3.  Dad ate a third of the original batch!

Obviously, to model this with brownies, you'll have to cut the brownies into six even pieces, eat half of them, and then put two thirds of the remainder onto a plate, to see that what's going on the plate is 1/3 of the original pan.

Here is half of the pan of brownies. (Of course, these could also be cookie bars,  rice cereal bars, even gelatin "jigglers," depending on what your students are allowed to eat.)

Here is the 2/3 of the remaining half.  Note that it can also be described as 2/6, but since 2/6 = 1/3, our final answer is 1/3.








So, what would a division word problem look like?  Try this:

John has a recipe for brownies that calls for 1/3 cup of cocoa per batch.  There is 3/4 cup of cocoa in the box.  How many batches of brownies can he make with the cocoa he has?

In this case, we want to know how many sets of 1/3 are in 3/4?  So we divide:

So, John has enough cocoa to make two full batches of brownies and 1/4 of another batch.

A less high-calorie method of modeling fractions is to use measuring cups with water or dry rice. For example, you can measure out 3/4 cup ofto your heart's content. rice, and see how many times you can fill 1/3 cup with that amount.  You will show that 3/4 cup can be divided by 1/3 cup twice, with one fourth of the 1/3 cup left over-- that is, there are 2 1/4 sets of 1/3 cup in 3/4 cup.   Similarly, you can measure out 2/3 cup 1/3 cup of rice, divide it in half and show that the half amount fills a single cup: 1/2 x 2/3 = 1/3.

Another way to model the fractions is by using paper fraction strips.  Print out as many as you need, and then you can model "half of 2/3" (1/2 x 2/3 = 1/3) or "how many 1/4's in 6/8" (6/8 ÷ 1/4 = 3). More on that on a future post.

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?