### Do Unto Caesar

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? ### Plutarch's Boxes According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have their surface areas equal to their volumes? ### 3388 Using some or all of the operations of addition, subtraction, multiplication and division and using the digits 3, 3, 8 and 8 each once and only once make an expression equal to 24. # Mixing Lemonade ##### Age 11 to 14 Challenge Level: ### Why do this problem? This problem gives a clear context in which fractions, ratio and proportion can be investigated. When using the interactivity, students can develop strategies for comparing fractions or ratios while thinking about which strategies are most useful for different cases. ### Possible approach Start by showing the class the image of the two glasses of lemonade. Ask them to decide which would taste stronger, and then share different ways of working it out, collecting different representations on the board. Then use the interactivity to generate a couple more examples for the class to discuss in the same way. If students have access to computers, set them to work in pairs using the interactivity, and challenge them to get ten correct answers in a row. A possible approach, if computers are unavailable, is to generate some questions and write them up on the board for students to work on with their partner. Alternatively, you could use the examples in this worksheet. Once students have had a chance to develop clear strategies for working out which mixtures taste strongest, share with students the questions to consider from the problem and allow them some time to explore and discuss their answers. You can read about one teacher's experience of using this task in the classroom. ### Key questions Is there a strategy that always works? Is your strategy always the quickest way to work out which mixture tastes strongest? ### Possible support Spend plenty of time sharing alternative strategies for the first few simple examples on the worksheet, ensuring students understand why they work. Encourage students to work together using each other's methods to solve the harder examples. ### Possible extension The mediant of two fractions$\frac{a}{b}$and$\frac{c}{d}$is found by adding the numerators and the denominators:$\frac{a+c}{b+d}\$

How could the lemonade problem be used to convince someone that the mediant is always in between the two original fractions?

Ratios and Dilutions looks at solution strength at a more challenging level.