How Would You Visualize a Fraction Divided by a Fraction

I loathe “Keep, Flip, Change.” When we teach students tricks instead of number sense the result is often that students fail to understand what they are doing. In Mathematical Mindsets by Jo Boaler she states that math is “Creative and Visual.” Instead of teaching tricks, consider having students visualize and explain the fraction. So how would you visualize a fraction divided by a fraction?

What Does Divide Mean?

Instead of going straight to the rules of dividing fractions… which many people do not understand… lets take some time to think about division.

How Many Ways Can You Describe It?

It can help a lot to have students share out different ways to express what the division means. What does 20 ÷ 4 mean?

Divide 20 by 4. Create 20 circles and break into 4 groups. Circle each group of 5.
  • Divide is to break into groups
  • How many are in each group
  • 20 ÷ 4 is to break into four groups. How many are in ONE group?
  • What is ONE of the four groups?
  • Create 4 groups. Evenly divide the 20 pieces into all the groups
  • How many will each group have when all 20 pieces are evenly split
  • What are other ways you can say this?

By not always presenting it or saying it in the same way students help to understand the concept of division.

20 times one fourth is the same as 20 divided by 4

Why is 20 times one fourth the same as 20 ÷ 4 ?

When you divide by 4 you express how many are in ONE of the groups.

What Would Change if You Divided by One Fourth?

What if instead of 20 times one fourth, you had 20 divided by one fourth?

20 divided by one fourth

Compare and Contrast

How is dividing by 1/4 different than dividing by 4?

Each piece is divided into a fourth. 1/4 of each piece is a small piece.

Think of taking a bar of chocolate (that has segments, like a Hershey™ bar) and breaking it up into each chunk. You went from 1 piece (bar) to 12 pieces.

You have 20 pieces and you divide each piece into quarters (1/4th) … then each big piece becomes 4 little pieces … for a total of 80 pieces.

What if you had 20 hours of yard duty for the semester. No one wants to do yard duty so it is agreed to divide it up into 15 minute (quarter of an hour) spots. How many people are needed to cover the 20 hours of duty? One person is only doing a fraction of an hour. So if there are 20 total hours of yard duty in the semester it will take more than 20 people to cover this. Each hour has 4 quarters… so that is 4 people needed each hour. Four people each hour for 20 hours is … 80 people. Or 80 duty slots to be covered.

Fraction Divided by a Whole Number

So when we had 20 divided by one fourth we ended up with 80 small pieces. But what if we started with a fraction and wanted to divide it up. I chose divide by 2 because most of us intuitively know that means 1/2. YOU KNOW THAT ÷2 equals 1/2

one fourth divided by 2. Visualize as 4 pieces and you want ONE of the four pieces. Now cut that in half.

Cut each of those 1/4 pieces in half. You want ONE out of the TWO pieces that are created by cutting the piece up.

So first you take 1/4, which means you cut the whole thing into 4 pieces. Then you take 1 out of the 4 pieces (1/4) and divide that into two pieces. You want ONE out of the TWO smaller pieces. Breaking it up into smaller pieces means you have more pieces. So the whole thing would have a total of 8 pieces but you only have 1 out of the 8 smaller pieces.

You started with one piece.
Broke into 4 pieces
and Broke that into 8 pieces.
And you have one out of the 8 pieces
This is one eighth

You want half of the one fourth piece.

Fraction Divided by a Fraction

Let’s compare that to dividing by one half.

This is not the same math problem. I am NOT dividing each piece into 2 pieces. I am dividing each piece into half a piece.

Remember how 20 pieces divided into 1/4 size pieces ended up with 80 smaller pieces.

20 divided into 1/4 sizes is 80. (notice how I keep rewording it! So important to keep rethinking different ways to say what it means. Sense making is mathematical practice #1) How many quarter cups of flour are in 20 cups of flour?

Of all the 20 pieces, each was cut into 4 smaller pieces.
Of all the 1/4 pieces, each was cut into 2 smaller pieces

Obviously 2 of these newly created smaller pieces pushed back together would make the 1/4 piece. There are 4 of the newly created 1/8 pieces.
Visually, push all the triangles together and you end up with 4 out of the 8 pieces… or half the whole thing.

Three Out of Four

How many 1/4’s are there in 20?

How many three fourths are in 20?
You have 20 cups of flour and you use a 3/4 cup measurer. How many 3/4’ths cups are there?

Now remember you have ALL 20 cups of flour. You are just making smaller baggies of flour that only have 3/4 of a cup of flour in them. How many baggies will you small bags of flour will you have? 20 + 6 + two out of 3

If you wanted to take the 26 bags and put them into third size bags so you have a common denominator (improper fraction) then each of those 26 bags in thirds would be a total of 78 third sized bags.

78 third sized bags + 2 third sized bag = 80 half sized bags.

26 baggies and 2/3 of a baggie.

Now With Fractions

How about 1/4 divided by 3/4?

This is NOT three fourths of the 1/4. This is One Fourth divided up into 3/4 sized pieces. You should end up with a larger number of pieces.

I end up not with 3 candy bars.. .but rather THREE one fourth sized chunks of a candy bar.

The answer is THREE but the size changed. Let’s think about it as 3 fun sized candy bars!

How About 3/5 Divide 1/4

I have 3/5 of a cup of flour. I want to divide this into 1/4 (not of a cup) sized baggies. How many 1/4ths are in 3/5ths?

I have 2 and 2/5 groups

Another Way

Even after you’ve figured it out… what is another way to express it? The more ways you have to express a problem the more flexible you are with numbers in different situations.

Three fifths is three … 1/5ths. Or three groups of 1/5. Being flexible to break up fractions makes many math problems a lot easier!

Thinking of 3/5 as THREE 1/5’s allows me to regroup the original question. Can you break numbers apart? Regroup? Use the Associative and Commutative properties to rethink how numbers can interact?

Using the Commutative Property I swapped the 1/5 and 3.

When I am breaking down numbers I often will switch the numbers completely so I can see how other numbers interact and then come back to the original set of numbers and apply the pattern I discovered. This is mathematical practice #7 and mathematical practice #8. Unsure what I can do with this regrouping I am going to look at some more familiar numbers:

12 Divided by 3 Times 4

Let’s take a look at the jerk math problem I would always give my high school students. WHY would I give them 12 divided by 3 times 4? Because I knew they would get it wrong. MY SOLE purpose for putting this on an assessment was to … take points off? Prove to them they are bad at math? Complain later that kids can’t do simple order of operations?

What it proved was… students do not have number sense. NOT that they are bad at math.

I do NOT have to go left to right. The Commutative Property says that a•b•c = c•a•b. SO if I have multiplication I can swap up the order. However, division is the multiplication of a fraction. Start reading the divide symbol as fraction. This will not only help you (and your students) be better at fractions, it opens up a whole new possibility for how to simplify expressions.

12 divide 3 times four is 12 fraction 3 times four or 12 times 1/3 times 4

Flatly, it is not 3 times 4 at all. The divide clearly puts the 3 in the denominator. Let’s have that conversation. WHAT is being divided. Instead of a rule that says “Left to Right”… BUT WHY?

The truth is, most people have no idea WHY. The response I get when I ask that is overwhelmingly “because that is what my teacher told me.”

Math should NOT be boiled down to a set of rules you memorized because your teacher told you. @alicekeeler

Be in the habit of looking at problems different ways. Compare and contrast. Why is this solution different than another (similar) problem.

I don’t know about you but I gain some insights by flexibly switching between the division symbol and a fraction. Feeling comfortable with equivalent expressions is having a better number sense.

Back to 3/5 Divided by 1/4

Divide by 5 means: “How many GROUPS of size 5 can you create?” So we first break up the three into smaller sized pieces. Dividing by 1/4 says to break each one into fourths. This creates 12 pieces. We now want to make groups of size 5. I can create 2 full groups with 2 out of 5 extra. So 12 pieces divided by 5 (12/5) or 2 and 2/5.

Google Jamboard

To view the Google Jamboard I made to explore the visualization of the fractions:

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    How Would You Visualize a Fraction Divided by a Fraction

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