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.