A: I have assumed that your student is dealing with bricks and mortar and so suggest the following problem which I had to solve when creating my garden.
I bought a circular gazebo and needed to create a brick base for it. The gazebo has a diameter of 3.05m. The question is: how many bricks did I need to make the circular base?
Not as straightforward as it seems. First, a decision needs to be made on the position of the gazebo on the bricks. I chose the middle. So the circle we are considering needs to be one brick-length less in diameter than the gazebo for the inner part of the circle (half a brick-length less on either side). Your construction student will have learnt about bricks and might suggest a radial brick as opposed to a cuboid one. We chose the cuboid because we had some lying around the garden. A brick's dimensions are 102.5mm by 215mm by 65mm.
The bricks were to be laid as shown in the photograph, with the long edges next to each other and the largest face showing. So the diameter (D) of the inner edge is 3050mm - 215mm = 2835mm. To calculate how many bricks are required, the student needs to calculate the circumference (C) of the circle using C = piD, where pi = approximately 3.14. Thus C = 3.14 x 2835 = 8901.9mm. The width of the brick is 102.5mm, but when estimating the number of bricks the space taken up by the mortar between them also needs to be taken into consideration; this is usually 10mm. The circumference needs to be divided by 112.5 (102.5 + 10). The number of bricks is therefore 8901.9 112.5 = 79.128: roughly 80 bricks.
Testing the correctness of the calculations could be done by laying out the circle in the yard where your student is studying. Perhaps use differently sized circles. There will be discrepancies when the space between the bricks is too large.
This exercise can be replicated in the classroom. Your "bricks" could be multilink cubes, with the circles that you want to create being much smaller. For example, using two multilink cubes as a brick, the problem is: how many bricks are needed to make a circle of bricks, with the inner circle being 16cm diameter and 2mm between each brick for mortar?'
A multilink cube has sides of 2cm. The circumference of the inner circle would be C = 3.14 x 160 = 502.4mm. The width of brick that needs to be considered is 20 + 2 = 22mm. So the number of bricks required is 502.4 22 22.8 (about 23). I have tested this one to see if it worked as you can see from the photograph.
You might like to have some circles ready printed and laminated on sheets of A4 and A3 for students to test the accuracy of their calculations as an alternative to drawing them on paper. You could also use these as the question prompts.
This provides students with a practical context for circle calculations.
For students who think there is little maths in construction, a look at The Design of Curved Brickwork by MHammett J Morton offers an interesting perspective on building with brick.
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 email@example.com Or write to TES Teacher, Admiral House, 66-68 East Smithfield, London E1W 1BX