### Boxed In

A box has faces with areas 3, 12 and 25 square centimetres. What is the volume of the box?

### 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?

### The Genie in the Jar

This jar used to hold perfumed oil. It contained enough oil to fill granid silver bottles. Each bottle held enough to fill ozvik golden goblets and each goblet held enough to fill vaswik crystal spoons. Each day a spoonful was used to perfume the bath of a beautiful princess. For how many days did the whole jar last? The genie's master replied: Five hundred and ninety five days. What three numbers do the genie's words granid, ozvik and vaswik stand for?

# Maths Filler

### Why do this problem?

This problem requires careful thought about the way the water level in a vessel changes when water is added at a constant rate. Through analysing the key features of a graph, students can figure out the shape of the vessel it represents.

### Possible approach

The first part involves working out volumes. The key is to realise that the cross-sectional area is proportional to the volume and then to work out the areas. There are obvious 'easy' candidates for this and some harder letters. There are various ways in which the areas of the cross sections of the vessels can be 'rearranged' to form rectangles. Students could work on finding the areas in small groups and then feed back to the rest of the class, sharing their approaches for finding the trickier areas.

To work out which letter the graph corresponds to, ask for suggestions for a 'story' relating the height-chart diagram to a vessel filling up. For example, what happens to the water level at the horizontal parts of the graph? What could be happening to account for this?

Once the class have identified the correct vessel for the graph, they could work on producing graphs for the other letters. Students could check each other's work by seeing if they can match the graphs with the vessels.

Small groups of students could also design some other letters in the same way and draw the resulting graphs, perhaps producing a card-matching activity to challenge other groups. The results could contribute to a classroom display.

### Key questions

What could be happening at the horizontal parts of the height-time graph?
What can you work out from the steepness of the lines on the graph?

### Possible extension

Would the graphs change if the holes were moved, or if water was poured into both holes where available?

The final part of the M graph should be a curve rather than a straight line. Can students justify why the graphs for V, A, and S will also contain curves?

Can students work out the functions which describe any of these curved parts?

See also Maths Filler 2 for a suitable follow-up challenge.

### Possible support

How Far Does it Move? might provide a useful introduction to interpretation of graphs.

Start by working on the letters without diagonal lines and work out how quickly they will fill up.