I have used Maths Venns before and for this particular topic. I tried to incorporate this particular venn into a lesson but my students at that time did not reach this venn.
This time I decided to use this venn diagram with a more able group. I was confident they would already know how to find square, cubes and primes but triangle numbers is something that I thought they may have forgotten. I used the definition of a sequence of the summation of natural numbers for triangular numbers and also defined natural numbers for them as well.
After a few examples of prime, cubes, squares and triangular numbers I set my class the task of the venn diagram below:
I allowed my students to work in pairs and asked them to write sentences explaining their reasons for if they believed any of the regions were impossible. The students certainly engaged well with this activity and lots of excellent discussions were taking place between the students. G was a very well placed option as it tackles the misconception of 1 being a prime number.
A lot of my students completed this venn and so I set them the following extension:
Using the knowledge from today’s lesson, can you create a three way venn diagram where:
a) All of the spaces can be filled
b) Some of the spaces are impossible
This is where some intriguing conversations took place. A lot of my students decided to use the headings of natural numbers, cube numbers and square numbers. At first some of the students thought that it was impossible to have a cube number that wasn’t natural but then remembered that it is possible to have a negative cube number. My students then decided that it was impossible to have a square number that was not natural. I am a big believer of trying to avoid teaching misconceptions to students based on the limits of their knowledge and so we had a discussion around imaginary numbers. I told the students that at GCSE level they are only expected to rely upon the knowledge of natural square numbers but it is possible to have square numbers that are not natural due to imaginary numbers. One student then asked why we had imaginary numbers. I was able to explain that imaginary numbers were needed to explain more complex situations in Maths and that engineers need to use imaginary numbers. I explained that Maths is a journey of discovery and that the Mathematical foundations that we use all needed to be discovered at some point and that there is still more to be discovered.
I certainly recommend using Maths Venns in lessons as a thinking task. The extension that I used was also simple to implement and yet stimulated powerful conversations. I will certainly be trying to implement more of these in my lessons.