Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
At the beginning of the night three poker players; Alan, Bernie and Craig had money in the ratios 7 : 6 : 5. At the end of the night the ratio was 6 : 5 : 4. One of them won $1 200. What were the assets of the players at the beginning of the evening?
A garrison of 600 men has just enough bread ... but, with the news
that the enemy was planning an attack... How many ounces of bread a
day must each man in the garrison be allowed, to hold out 45 days
against the siege of the enemy?
This problem provides a real-life context for working with proportionality, speed, rates, and units of measurement. The final answer may be surprising and leads to interesting questions about the validity of the model suggested.
Split the class into groups of two or three, and give each group a copy of this worksheet.
"Your task is to work out which athlete burned the most calories in a triathlon. Before you do any calculations, discuss which athlete you think it might be, together with your reasons why, and make a note of it."
As they are working, circulate and listen to the conversations that students are having, to identify anyone with particular insights that it would be useful to share.
Here are some key questions that could be used to prompt groups who get stuck:
What do you need to know to work out each athlete's speed?
How can you work out the speed in kilometres per hour?
If his (swimming) speed is slower than (4.5)km/h, would you expect him to burn more or less than (600) kcal per hour?
Once you know the rate at which he burns calories, how will you work out how many calories he burns in (19 minutes and 12 seconds)?
Towards the end of the lesson, bring the class together and invite different groups to explain how they worked out the number of calories burned for each stage of the triathlon. Allow some time for discussion of the perhaps surprising answer to the question "Who burned the most calories?".
Ratios and Dilutions provides students with an opportunity to explore ratio and proportion in a different context.
Mixing Lemonade offers an introduction to ratio and proportion in a simple context. It might be useful preparation for working on this problem.