A: Here is a hat-making activity that is fun and will meet the national curriculum requirement to "know and use the formula for the circumference and area of a circle".
You can use it to make hats that are conical, cylindrical and conical frustum shapes. This could be an enterprise project, or pupils could make hats to sell for charity, or perhaps for local primary school pupils.
They will need materials from catalogues (a costing exercise), to look at percentage profit and loss, pricing, measurement and statistics. They will also need to measure people's heads, carry out surveys of colour preferences, how much people would be willing to pay, preferred styles, and gender and age differences. They will also need to find out how much such party hats currently cost.
In this case, we have distributions and the three statistical averages - mean, median and mode - being used for a live project. The data would be relevant to them and useful for learning about mathematical concepts. With guidance, they might decide on a realistic sum of money that they hope to raise, and then set up a "target thermometer" so that they can see their progress.
It would be useful to have one of each type of hat to begin with, perhaps bought or made by you in advance. Pupils will need to be reminded of the relationship between circumference and diameter. Use a circular waste paper bin and a piece of string that just fits around the top. Get pupils to identify the circumference, diameter and radius. Take the string and wrap it round the circumference, then demonstrate that when folded into about three sections it fits across the widest part (the diameter).
Remind them that the "about three" is in fact pi, which is on the calculator. You could ask someone to write the formula on the board, showing both C=9D and C9=D, and explaining that they will need this for the hats.
They might like to have the instructions for each hat design as a worksheet. They can decide on the height of the hat and any decorations, streamers, glitter and so on.
Above is an example with a 65cm head measurement (heads aren't round, but the card can be bent to fit them). They need to cut out a rectangle 65cm wide, and to the chosen height, then calculate the circumference of a circle to match the rectangle's longest side by dividing the head measurement (rectangle length) by 9.
When using a calculator, they may have to round to the nearest decimal place. Here, the calculation was D = 659 = 20.7cm. The radius is half of this: 10.35cm. They also need to allow for tabs, so the circle they cut out will have to be slightly bigger. Mistakes can be discussed and used to emphasise the importance of understanding the calculations.
Next, the conical hat. For this, they need to the head measurement, then add some to the final circumference to allow for cutting out a sector of the circle to make the hat stand up. The size of the diameter will determine the height of the finished hat.
They could have a hat with a flatter profile - like the hats worn in Vietnam and China, for example. Talk through an example with them. Suppose the head measurement is 65cm and a pupil wishes to cut out 15cm from the circumference. In this case, we are looking at a circle with an 80cm circumference. So they would need to calculate the diameter as D = 809 = 25.5cm. Half this value is about 12.7cm, so they will need to create a circle with a radius of 12.7cm.
Give pupils time to experiment with different heights of hat. The taller the hat, the less stable it is - a principle you could link with science and centre of gravity. A higher-ability group could investigate the best position for the centre of gravity to make the hat stay on the head, generalising their findings.
Calculations for the conical frustum are a combination of both the above approaches as the top of the hat will be smaller than the head measurement.
* Wendy Fortescue-Hubbard is a teacher and game inventor. She has been awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts (NESTA) to spread maths to the masses.
* Email your questions to Mathagony Aunt at firstname.lastname@example.org Or write to TES Teacher, Admiral House, 66-68 East Smithfield, London E1W 1BX