Recently in a Gr 3, 4, 5 classroom we gave the students this question:
Students are learning about insects. They discover that an ant has 1 body, 2 antennae and 6 legs. They each make a model. How many bodies, antennae and legs will they need for 5, 10 and 25 ants?
The educators I was working with were wondering about how to move students from additive to multiplicative thinking. Multiplicative thinking is a key concept of proportional reasoning and something many students in the junior grades struggle with.
We introduced the problem by reading it together and the students promptly got to work. After about a half an hour, I asked for their attention so that I could begin the consolidation part of the lesson. Students did not have to be completely finished in order for me to do this. This is the time that I highlight the key mathematics from the lesson and where I do explicit teaching based on the student work and connected to my number relationships learning goal.
I told the students that when walking around I had noticed a lot of pictures of ants and thought a chart would be helpful in recording the information they had discovered. I started by asking for the information for 1 ant, then 2 ants. I then asked: How are 1 and 2 related? With this question I am looking for “2 is double 1”. (It took some probing but the doubling strategy is the one I want students to use to fill in the chart today.) Remember: Our goal is to move students from additive to multiplicative thinking.
I did not ask for 3 ants yet but instead asked: what is double 2? Once we found out for 4 ants we then did 5 ants? I asked if we needed to continue filling in the chart until we got to 25 ants. Students said no, that we could double the information for 5 ants to get 10 ants and double that information for 20.
To find 25 ants, students at this point could either continue the pattern or look at the relationship between 5 and 25.
After drawing 25 ants and counting all the legs, most students were very engaged when the patterns emerged on the chart. Tomorrow the students will be given a similar problem and the opportunity to solve it using a chart.
Some students may still need to draw some pictures to get started, but ultimately we want them to begin thinking about the number relationships. Once I know how many legs are on 5 ants, do I need to draw 10 ants or can I “double”. That’s multiplicative thinking!