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.