Students are often introduced to the idea of division as taking a bunch of items and splitting them into n groups. This concept can be applied to fraction division as well but it’s a lot trickier to conceptualize. This is because the divisor is often a fraction or a mixed number.
How can I divide 10 items into 1/3 groups????
Instead I like to help students recall another way to conceptualize division, where the divisor represents the size of each group. This is so much easier to visualize because I can differentiate each “group” with a different color or label so kids can easily make the connection between the division they learned in 3rd grade.
8 ÷ 2
How many groups of 2?
8 ÷ 2/3
How many groups of 2/3?
This does get messy when the numbers don’t divide evenly and you end up with a partial last group. Kids have a hard time figuring out what fraction of a group that last piece should be because they are thinking fraction of a whole instead of fraction of a group (one divisor).
Here we can see 2 1/3 divided by 2/3 is 3 whole groups but what fraction of a group is that last whole piece?
Students using visuals to divide 2 and 1/3 divided by 2/3 will often find the quotient to be 3 and 1/3. I think this is a great starting point! Much better than the students who accidentally use keep change flip the wrong way and get an answer that is logically WAY too small. We just have to help them understand that since in this case, a whole group is 2/3 of the circle, if we only have 1/3, that is half of a group. Since we already have 3 whole groups, the quotient is 3 and 1/2.
This is a worthwhile conversation to have to deepen their understanding but it also highlights the difficulties with ONLY using visuals to solve fraction division problems. Of course, this is time consuming and I would want students to eventually move on to more efficient strategies. But it’s great to have a conceptual understanding of what’s going on under the hood in case they ever forget or get stuck.
I also strongly recommend using a variety of real world contexts including food, pieces of string, pieces of paper or sticky notes and units of time that can be divided in different ways to help students develop a deeper understanding that extends beyond any one representation.