Students often complain that the visual models we use to teach fraction operations are more confusing than the rules they memorized. This indicates a deeper misunderstanding of the underlying concepts, and the solution is not to get rid of all the visuals. We need to find ways to make them more accessible and tangible for our students.
When students learn multiplication in second grade, they learn about groups of items. With fraction multiplication, I like to start by drawing a connection to the idea of groups using visuals like this:
4 groups of 2/3
4 groups of ¾
When both factors are proper fractions, though, this can be confusing because how can you have half of a group of 2/3?
I love to let students ponder this question as a means to discovering why multiplying a number by a proper fraction results in a product smaller than the number you started with.
For fraction x fraction problems, area models are a great way to demonstrate fraction multiplication, but students often find them confusing at first. To help them understand, I would give my students each a sticky note and ask them to cut it in half. Then I would have them take that half and cut off one third and ask them what fraction of the whole sticky note they just cut off. Then we draw lines so that all pieces are the same size and determine the fraction of the whole. Physically seeing and cutting off the pieces really helps them to make the jump to the more abstract area model.
I even use sticky notes to teach mixed number multiplication. I ask students to make a column of 2 and ¾ sticky notes, for example. And then I ask them to lay out 3 and ½ columns (groups) of 2 and ¾.
Eventually, with some nudging, they would come up with something like this.
Then they would figure out the value of each piece in order to find the total product.
Going through this process with physical sticky notes makes it much easier for them to understand the area model with mixed numbers.
It also is a great way to convince students that multiplying the two whole numbers and then the two fractions doesn’t work.
Side Note: I don’t tell my students to ALWAYS convert mixed numbers to improper fractions before multiplying. I believe in helping them develop the procedural fluency and understanding to know when that’s a good idea and when it’s not. I do make sure that if they are going to apply the distributive property, they need to make sure they multiply each part of one factor by each part of another factor.
We have discussions about situations like 100 ¼ x 2 ½ and whether converting to improper fractions makes things easier or harder.
Sometimes students do find the visual models we use more confusing than memorizing a rule. But let’s take their confusion as an impetus to make things even more accessible for them with sticky notes they can physically fold, cut, and write on.
All digital images were generated at my website, visualfractionlibrary.com.