The other key multiplicative situation middle-primary students encounter is multiplication or division as comparison. This can also be referred to as proportional reasoning or scaling. In this situation, we do not count the number of objects in equal groups. We choose one factor as the base unit or set, and we compare this with the product or quotient.
Let’s use 5 x 2=10 and look at the two multiplicative situations that can be represented by this statement, side by side. On first glance they look very similar.
Equal groups structure
They differ because we are performing different functions. In the equal group structure we are counting or adding. In the comparative structure we are comparing and considering proportions. For Middle Primary students these situations can usually be distinguished by the use of comparative language in the problem.
They can also be distinguished by how we talk about the model and the solution.
Dr. Marian Small, our guest professor and author, explains more about the structure in this video.
Thinking comparatively about multiplication and division is more difficult for children than thinking about multiplication as equal groups (repeated addition) but research suggests that it’s important to start developing such thinking in the middle primary years. Once the idea that ‘multiplication is repeated addition’ is locked in, it can be difficult to transition to comparative/proportional reasoning.
Here is an activity written by Dr. Small that you might like to use to kick start comparative thinking with your students. It’s called ‘Four times as big’ and can be found in the Year 2 section of the eBooks.
This is the task.
This is the interactive.
These are the teacher notes and printable student page.