There are plenty of good examples on how to link proportional reasoning to the real world such as how to figure out from a recipe for 12 cookies the amount of each ingredient required for, say, making 24 cookies – students seem to react well to this kind of real-world situations.

A similar real-world situation can be used for proportional reasoning involving volumes, addressing possible misconceptions that could result from the kind of proportional reasoning used for doubling the number of cookies. It so happens that I love cooking (and friends tell me that I am good at it), and I had to address myself the misconception!

For example, if a recipe for a pizza that serves two has a 10-inch diameter, what should be the diameter of a pizza that serves four? Or if you have a recipe for a cake that requires a 12-inch tin and you want to scale it down to the 6-inch tin you happen to have, what proportion of the ingredients in the recipe should you use? In both cases, what is crucial when scaling up or down is that the height (thickness) of the pizza and the height of the dough in the tin remain the same to ensure that they cook evenly and do not end up undercooked or overcooked – the problem does not arise for the cookies because we double their number, not their size.

To work this out we need to realise that the proportion needs to be calculated relative to volume, which in both cases are those of cylinders:

Because the heights need to be the same, the proportion of the volumes of two cylinders with radiuses and is therefore:

where and are the corresponding diameters

Hence, in the case of the pizza, because we want to double the ingredients to serve twice as many people, we have

, i.e.

That is, the diameter of the pizza should be 14 inches, not 20 inches even if we are using the double of the ingredients. In the case of the cake, we have

That is, we would need to use a quarter of the ingredients even if the diameter of the tin is only halved. What would be the proportion of ingredients if we were using an 8-inch or a 10-inch tin?

Good baking!