- start with good questions;
- use vertical non-permanent surfaces (VNPSs);
- and use visible random groups.

This week, I went into a grade 8 class who had been working on measurement. I began the lesson by showing them a game called 'Prism or No Prism!' which involves me holding up a shape and the class deciding if it is a prism or not. For the most part, they were correct but about half the class said that a cube was not a prism. When I asked why, they said because it is a cube! As they weren't too clear about what a prism is, I shared with them my 'loaf of bread' analogy:

If a shape can be sliced like a loaf of bread from front to back and give exactly the same size and shape slice, then it is a prism.

"So that would make a cube a square-based prism then!" said one student.

Next I wanted to ascertain that they knew how to get the volume of a prism. The 'loaf of bread' analogy works well here to as we can connect it to layers (or slices) that can be made thinner which leads us to develop the idea that the volume of a prism is the area of one 'slice' multiplied by the 'number of slices' into the more generalised formula, V=Axh

In all of these discussions, we did not look at cylinders.

I then showed them the opening act of Dan Meyer's Popcorn Picker:

Decide which way you want to make your cylinder. It will then be filled with popcorn!

I used playing cards to create visibly random groups of three students each and then gave each group one marker each and had them work at VNPSs.

The students got stuck into the task immediately, even though they have never been shown the formula for the volume of a cylinder. Whilst there was the occasional dead end (one group got stuck on using V=lxwxh before realising that this wouldn't work with a cylinder!), the students soon reasoned that since the cylinder is a prism, they could work out the area of the base circle and multiply this by the height for each cylinder. Getting the area of the circle requires knowing the radius and some did this by direct measurement whilst others measured (more easily I'd suggest) the circumference of the circle (that is, of course, one of the sides of the rectangle and then divided this by 2π to get the radius.

One of the great things about VNPSs is that as a teacher, it makes it easier for me to see what students are thinking and any errors that they might make.

When we were satisfied that the students had reached a conclusion, we noticed that seven groups opted for the shorter, wider cylinder and one group opted for the taller, narrower one. I filled one of each of these cylinders and, by then emptying the popcorn on the table, we could see that visually most groups had got it correct. It turned out that the group that didn't had the right idea but made a calculation error.

With the students merrily munching on popcorn, I was able to summarise the lesson by using their work on the VNPSs around the room and got them to tell me the formula of a cylinder:

The use of good questions, VNPSs, and visible random groups certainly proved effective in getting these grade 8s thinking. I wondered how it would be for younger students.