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Plunge pool tiling

The Australian Plunge Pool Company offers a free tiled path around each plunge pool constructed. The standard width of each plunge pool is 2 metres but the length can vary (it must be a whole number of metres). The tiling is completed using 1 metre square tiles around all sides of the pool. This diagram shows the tiling for a pool of length 3 metres.

Present a report for the following tasks and ensure that you show clear mathematical workings and explanations where appropriate.

Routine problems

a Use the diagram above to count the number of tiles required for a 2 metre by 3 metre plunge pool.

b Draw an accurate diagram of a rectangle which is 2 metres by 6 metres. If possible, use grid paper.

c On your diagram, accurately draw 1-metre square tiles around the outside of the rectangle.

d Use your diagram to find how many tiles would be required for a 2 metre by 6 metre plunge pool.

Non-routine problems

a The problem is to find a rule relating the number of tiles required to the length of the pool. Write down all the relevant information that will help solve this problem, and use diagrams to illustrate the following four pools.

b Use a table like the one shown to summarise the number of tiles required for varying pool lengths.

c If the pool length increases by 1 metre, describe the effect on the number of tiles required.

d If a pool could have a length of only 1 metre, how many tiles would need to be used?

e Determine the rule linking the number of tiles and the length of the pool.

f Explain why your rule correctly calculates the number of tiles required. Give a geometric reason.

g Using your rule from part e above, find the number of tiles required for a plunge pool of length: i 8 metres ii 15 metres.

h Using your rule from part e above find the length of a plunge pool if the number of tiles used is: i 22 ii 44.

i Use your rule to explain why there will always be an even number of tiles required.

j Summarise your results and describe any key findings.

Extension problems

a If luxury plunge pools can have a width of 3 metres instead of 2 metres, determine the rule linking the number of tiles and the length of these pools. Compare it to the rule found when the width is 2 metres.

b The luxury plunge pools of width 3 metres now require the path to be two tiles wide. Investigate how this requirement affects the rule for the number of tiles required. Use diagrams to help communicate your solution.