After learning about indices recently, I asked my year 10s to respond to this question for homework: Among the natural numbers, are there more square numbers or cube numbers?
I want to teach them over the course of two years to express their mathematical thinking in words. I asked for a paragraph for homework. Some students wrote me a single sentence, others a few sentences. Some included examples.
The next lesson this led to a really interesting discussion about the size of an infinite set. How many items in the set of square numbers? Could you count all the cube numbers? If there is a cap on the size of the number, then there are certainly more squares than cubes. But when you let the numbers go to infinity, it’s a different story altogether.
I hope to do more activities like this with them this year. They should be able to express and justify their opinions.