I love Christmas pudding, and boiled up a nice little 100g number for dessert tonight. The instructions contain cooking times for all three sizes available. The sizes are 100g, 400g and 900g. The cooking times for oven are 20, 35 and 45 minutes respectively. The cooking times for boiling are 25, 45 and 80 minutes respectively. This is clearly not a case of linear variation, as 100:900 does not equal 20:45. Kids are likely to be at least a little surprised by that. I can imagine all sorts of activities springing from this data, depending on the year group and purported topic.
Year 7 number plane. Plot the data on a number plane (determine the scale yourself!) and join the points with a curve. Should the curve go through the origin? Predict the required cooking time for other weights.
Year 10 ratio of volumes. This is the most obvious application. Given that 100g must be boiled for 25 minutes (and given none of the other data), predict the required boiling time for the other weights and see if it matches the instructions. Required insight/knowledge: volume is proportional to mass; the different puddings are very likely to be similarly shaped; cooking time is likely proportional to the width; the width varies with the cube root of the volume (and therefore the mass). The specific language in that sentence is advances for Year 10, but the idea is simple enough.
Year 11 functions. As per Year 10 above, but plot the points, join it up and determine the equation of the curve (it “should” be , for some value of , at least to my mind). Or work out what the equation “should” be with nought but algebra, then plot it on a computer and see if it goes through all the points.
Year 8 ratios. Give the 900g cooking time and ask for a prediction for 100g. When it’s (very) wrong, give the correct answer and talk about why it might be like that. Then solicit a cooking time for 400g. Observe that cooking time is not proportional to mass, but tell them it is proportional to something to do with mass, and see if they can guess what. (Hint: it’s something on your calculator…) If possible, give an introduction to ratios of volumes of similar solids using simple diagrams, algebra and equations.
The Year 8 idea is the most handwavy, but there’s nothing wrong with that: it’s stretching the Year 8 syllabus beyond breaking point, so it depends entirely on the ability and interest of the class.
I look forward to using this example in class this year.