I had the great honour of helping run the TLDSB math olympics for the second time yesterday. We had teams from all over our board come to compete and enjoy the love of math! During the relay part of the day each team of four students tries to complete as many math tasks as they can in an allotted time. One task popped up that kept generating a common misconception with many groups. One of my colleagues called me over and asked me if I could help her look at it and determine why most of the students were coming up with this answer. I took it back to my table and solved it myself quickly to see why students were coming up with this solution. While I was doing it the idea of this blog popped into my head because I drew a visual model and it immediately made clear to me what the students were doing. Here is a pic with the prompt and the common solution we saw students handing in. Take a minute and solve it yourself before reading on. Think about this question: What do you think the misconception is?
After I read the task prompt I had a feeling that it had to do with the dimensions of the new cube but as soon as I sat at my table and drew some visual models I was like bingo! I took it back to the my colleague and showed her the picture and she was like yes, that makes sense. Here is a pic of the model I drew, with some added notes for this blog to explain what I was thinking.
The model on the top was the first one I drew to represent what the students had did by visually showing the 240 gram cube times by 3. Immediately you can see that only one dimension of the new shape is 6 cm and it is also not a cube. The model on the bottom I drew next to help visualize what the 6x6x6 cube would look like with the smaller 2x2x2 cube inside it.
How can this help students?
I think using this misconception could be very powerful for students to see if they take the time and use a visual model to represent their thinking it can help them see that their current thinking on how to solve the problem is not on the right track. They could also use the model to check their solution for reasonableness when they are finished. Maybe we wouldn’t have seen as many solutions with this misconception if it hadn’t been in the relay part of the day but I have often seen students make this type of mistake. Drawing a visual model is a great problem solving strategy and is sometimes overlooked similarly to using a concrete model with manipulatives as being for younger students. It is not and is always an effective way to make sense of the math!