Grappling with a problem is productive struggle when/if students can "make sense" of what they are working on. As a teacher, part of my role is to support their thinking by introducing visual models for thinking. This one demonstrates common misconceptions children may display during this lesson.
The intent of this lesson is to promote thinking, not to correct them. So, rather than tell a child who has unequal pieces that they are wrong, ask them, "Are those pieces equal?"
This is one of the first lessons I teach on division. It can also be used as a prelude to fractions. It works for either!
To begin, I ask students to silently think of at least 2 things they know about division and write them down. If they say they don't know anything, I prompt them to even write down something as simple as, "I know I will learn it in 3rd grade," or "I know what the symbol looks like."
Next, students share with a buddy and then 4 students share what their partner said with the entire class. I challenge students to be prepared to reflect on how their thinking has changed or evolved by the end of the lesson.
Then the fun part. I bring out a fruit that is NOT in a quantity conducive to equal sharing. For example, this year I brought 7 bananas. I have 30 students. (I love having a class number that works so well as a divisor or factor!) The students can very clearly see that 7 is not enough for each child to have one, and many of them quickly add that cutting them in half also won't be enough, but they still suggest that as the next step.
At this point, of course, several children will say they don't like bananas or whatever the fruit is and the teacher tells them to play along, for math's sake.
I follow their suggestions from there on. If they suggest uneven pieces (say cutting one of the halves in half but leaving the other intact), I do what they suggest but then guide them to assess if these are fair shares.
I follow the students' lead, so the results are different every time I teach this lesson. But there's always a way to use their misconceptions to build on how I guide the discussion. This year, when the bananas were cut into 1/2, 1/4, 1/4, we then had 21 uneven pieces. A child did see that if we cut the remaining 1/2 into 2 equal pieces, all the pieces (quarters) would be the same size.
This still did not solve our problem, as we have 30 students, two of whom were grudgingly pretending they ate bananas. Little did they know apples were arriving on the scene later, to alleviate their banana disappointment.
A student suggested I split two of the fourths (they were not using that term) in 1/2 again, so for two of the bananas I had 1/8 + 1/8 + 1/4 + 1/4 + 1/4. I made sure the children realized that the 1/8 were not equal pieces, and then let them eat their mini snack anyways.
The next task for students is to draw a simple diagram of how 7 bananas COULD be divided into 30 equal pieces.
7 divided by 30 is not a 3rd grade problem. It involves numbers less than one whole and it has a remainder (.2333333).
I, in no way, expect my 3rd grade students to arrive at this answer. I do cherish the discussions they have with each other about what is and isn't fair, and how to strategize about the problem, even down to the details of what kind of banana shapes they draw and the reality that the banana isn't the same size all the way through anyhow.
During their discussion and diagram drawing, I start to give them the words dividend and divisor to use in their conversations.
The entire class gathers again and share out their possible solutions by drawing them on the whiteboard or using the document camera to project the ideas crafted on their papers.
Again, there is guided partner discussion centered around specific questions related to the model. Some teacher questions might be:
What is a part of this process that worked/was successful?
What is part of this process that is something similar to what you did? How is it the same?
What is a part of this process that is incorrect/missing a step/doesn't have the banana divided into equal pieces. Why is it incorrect/missing a step/uneven? How would you change it?
Is there something ______'s model made you think of, that you hadn't thought of before?
This activity can run much longer than ten minutes, it really depends on the discussion. If the banana discussion moves quickly, or if you have time to extend, this whole process can be repeated with a second fruit such as an apple.
It can also be another number of objects that can't be equally divided OR you could give them a number of items that can be split into equal pieces so that there is one for each child.
This is the sort of lesson that can get your students deeply engaged, and I like to take advantage of that whenever I can so I have on hand the support materials that would allow me to go in either direction.