### Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

### At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

### Contact

A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

# Ratios and Dilutions

### Why do this problem?

This problem provides a context within which to explore fractions and proportionality. Seeking concentrations which can be made in different ways and justifying why some concentrations can't be made at all gives practice on working with equivalent fractions, ratios, factors and multiples.

### Possible approach

Begin by introducing the idea of a solution with strength 100000 cells/ml. Make sure everyone is clear about what this means.

Then ask the class to imagine mixing 100ml of this solution with 100ml of pure water. Students could record any working out and their answer on individual whiteboards. Now show students the interactivity, and explain how it can be used to perform a single dilution, then use it to check their answer. Take time to discuss how they got to the correct answer.

Demonstrate that the interactivity can measure multiples of 10ml of liquid, up to 100ml - the scientific context of this could be using a dropper that measures 10ml at a time.

Ask students to come up with questions they would like to explore using the interactivity - some suggested questions appear in the problem. Then allow them some time to investigate, using the interactivity to check the predictions that they make.

Once students are competent at working with solutions created using one dilution, use the interactivity to perform a series of two dilutions. Perhaps start by giving them a couple of concentrations to work out, using individual whiteboards as before, and using the interactivity to check. At the end of the problem there are some suggested concentrations they could be asked to make.

Pairs of students could take it in turns to create a concentration using two dilutions, and then challenge their partner to work out the dilutions they used.

Finally, the problem challenges students to investigate impossible dilutions.

### Key questions

If I combine $a$ ml of solution with $b$ ml of water, how does the concentration change?
What happens when several dilutions are performed one after another?
Does the order in which I do the dilutions matter?

### Possible extension

The problem Investigating the Dilution Series uses the same context but explores four steps of dilution instead of just one or two. Exact Dilutions extends these same ideas further.

### Possible support

Mixing Lemonade investigates the strengths of different solutions informally and may provide a useful starting point.