Ratios is a concept that doesn’t always go smoothly in the classroom, and I know many teachers who grumble about the fact that their students don’t understand it better. I’d suggest that, as often is the case, this is at least partly attributable to the fact that we focus on procedures before the students have an intuitive idea of what ratios really represent.

I was reminded of this when I was reading the submissions to recent Math Fundamentals Problem of the Week, Anthony’s Famous Butter Rolls. Students are asked to figure out how many rolls Anthony has to make based on the instructions his mother gave him. One instruction is, “His mom wants him to make enough so that there will be 3 rolls for every 2 adults and 4 rolls for every 3 children.”

A ratio, you say! We can use proportions to do something! Yup, we could. But the problem doesn’t mention ratios, and only one kid even used the word “ratio” in their submission. Nobody used “proportion”. Instead, they solved the problem from a completely conceptual perspective, using four different methods. Below are excerpts from nine student solution.

By far the most common method was what these three kids did:

Mitchell Z, Highlands Elementary School

Since there will be 12 adults and each 2 adults gets 3 rolls, you first divide 12 by 2 because the answer is the number of the groups of adults. The answer to that is 6. Then you multiply 3 rolls by 6 groups of 2 adults. That would be 18 so now you know that Anthony needs to make 18 rolls for the adults.

Elisabeth M, Fred W. Miller School

[T]o find out how many rolls the adults will need, 12 adults in all,3 rolls for every 2 adults, I divided 12 by 2 and got 6 groups of adults. Then, I multiplied 6 by 3 and got a total of 18 rolls for adults.

Evan S, Fred W. Miller School

I figured out how many groups of 2 adults there were in 12 people.
12 divided by 2 = 6. Now that I know there are 6 groups of 2 and each group gets 3 rolls, there must be 18 rolls for the adults. (6×3 = 18)

Slightly less popular was the idea of making a table:

Molly M, Highlands Elementary School

In the problem, it said that for every two adults, Anthony should make three rolls. I wrote down numbers to twelve counting by two’s to start my graph, because there is twelve adults and for every two of them, Anthony needed to make three rolls. Here is my adult chart with my work.

Adult Rolls
Adults Rolls
2 3
4 6
6 9
8 12
10 15
12 18

I knew now that the adults needed 18 rolls.

Andrei C, Lorne Park Public School

Since there were 3 rolls for every 2 adults and there were 12 adults:

3 rolls= 2 adults
3 rolls= 2 adults
3 rolls= 2 adults
3 rolls= 2 adults
3 rolls= 2 adults
3 rolls= 2 adults
18 rolls 12 adults

Another method that is really close to making a table is to draw a picture. We didn’t get any actual pictures from submitters (other than some yummy-looking pictures of rolls!), but some students did a nice job of explaining their picture. Here are two examples:

Adabelle W, Steele Elementary

I made a chart with 27 circles. There is 27 because 12 adults and 15 children = 27 people all together. I put an “A” in 12 circles for Adult and an “C” in the other 15 for children. Then I put a big circle around 2 of the adults and wrote a 3 with them for how many rolls they needed. I did that until there were no more adults left to group. Then I did the same thing with the children except instead of 3 rolls for every 2 adults I grouped 4 rolls for every 3 children. Anthony needed to make 18 rolls for the adults. 3*6=18. He needed to make 20 for the children.

Annabel L, Steele Elementary

I first of all, drew a picture of 12 adults and 15 children. I then split the adults into groups of 2 and put a 3 beside them because they’re 12 adults and they’re 3 rolls for every 2 adults. I then did the same thing to the children, only I split them into groups of 3 and put a 4 beside them because they’re 15 children and they’re 4 rolls for every 3 children. I then, added all of the rolls up and got 38.

The final method we saw wasn’t used by very many submitters, but it works just fine! These two students figured out how many rolls needed to be made for each individual adult (and child).

Nick L, Highlands Elementary School
I knew that for every 2 adults there should be 3 rolls so I divided 3 by 2 to get 1.5 rolls per adult. Next, I knew that there was 12 adults so I multiplied 12 by 1.5 to get 18 rolls for all the adults.

Elise B, Elgin Street School
3 rolls every 2 adults. Each adult will get 1 1/2 rolls
One more roll for each person: [that's part of the directions]
An adult will get 2 1/2
A child will get 2 1/3
For 12 adults:
24+6=30 rolls for all adults

These students aren’t solving this problem using a procedure that someone has taught them. They’re solving it by doing something that makes sense to them. They understand the concept of what it means for there to be three rolls for every two adults, and they’ve come up with methods that will find the correct answer every time. Two of these methods (the second and third) aren’t something you would want to use with a big number of people, because it would take a while, but the students who used those methods are ready for a conversation about what it might look like if you had 120 people. The other two methods (the first and the fourth) could be used for any number of people. All four methods were used in the context of the problem.

What sorts of methods would your students use to solve a problem like this? How would you move students from this sort of conceptual understanding to developing the efficient procedures that they’ll eventually be expected to execute? We’d like to know!

Some Anthony’s Famous Butter Rolls links in case you are interested: