Teacher Notes: Swimming Pool

Transcription

1 Teacher Notes: Swimming Pool Process skills FM Level Understand practical problems in familiar and unfamiliar contexts and situations, 1 some of which are non-routine Identify and obtain necessary information to tackle the problem 1 Select and apply mathematics in an organised way to find solutions to practical 1 problems for different purposes Use appropriate checking procedures at each stage 1 Interpret and communicate solutions to practical problems, drawing simple 1 conclusions and giving explanations Understand routine and non-routine problems in a wide range of familiar and 2 unfamiliar contexts and situations Identify the situation or problem and the mathematical methods needed to tackle 2 it Select and apply a range of mathematics to find solutions 2 Use appropriate checking procedures and evaluate their effectiveness at each 2 stage Interpret and communicate solutions to practical problems in familiar and 2 unfamiliar routine contexts and situations Draw conclusions and provide mathematical justifications 2 Coverage and range Understand and use whole numbers 1 Add, subtract, multiply and divide whole numbers using a range of mental 1 methods Multiply and divide whole numbers by 1 and 1 using mental arithmetic 1 Understand and use equivalences between decimals 1 Add and subtract decimals up to 2 d.p. 1 Solve problems requiring calculation, with common measures including money, 1 time and capacity Convert units of measure in the same system 1 Work out areas and volumes 1 Extract and interpret information from tables and graphs 1 Collect and record discrete data and organise and represent information in 1 different ways Find mean and range 1 Check calculations and explain results 1 Carry out calculations with numbers of any size in practical contexts 2 Add, subtract, multiply and divide decimals 2 Use, convert and calculate using metric and, where appropriate, imperial 2 measures Collect and represent discrete and continuous data, using ICT where 2 appropriate Use and interpret statistical measures, tables and diagrams, for discrete and 2 continuous data, using ICT where appropriate Use statistical methods to investigate situations 2

2 Cross-curricular links Citizenship Opportunity to discuss environmental issues (e.g. possibility of using solar heating to reduce carbon footprint, using a cover to reduce heating, the chemicals that may be needed for cleaning) Economic wellbeing and financial capability Opportunity to consider which product offers the best value and to realise that this may depend on the expected lifespan of the product Students may pick one or more pool sizes to work with, or may carry out the calculations for the full set of sizes. Estimate of time and cost to fill Time taken in minutes = volume in litres/8 Length Width Depth Volume (m 3 ) Volume (litres) Time (nearest min) Time (h and min) Cost (nearest p) h 36 min h 8 min h 25 min h 3 min h 28 min h 45 min h h h h 3 min h 15 min Students may give the results for the specific size(s) they are working with, or may carry out all the calculations and try to summarise. To summarise the full set of results, they could choose to use the median time/cost. Times in order: 2 h 36 min, 3 h 8 min, 1 h 25 min, 11 h 28 min, 12 h 3 min, 13 h 45 min, 15 h, 18 h, 21 h, 22 h 3 min, 26 h 15 min Median time = 13 hours 45 minutes Corresponding cost = If students decide on this approach in advance they may realise that there will be less calculations to make if they find the median volume then just work out the time/cost for that volume. Vol in order (m 3 ): 12.5, 15, 5, 55, 6, 66, 72, 86.4, 1.8, 18, 126 Median volume = 66 m 3 = 6 litres Time taken = 6/8 = 825 minutes = 13 hours 45 minutes

3 Corresponding cost = Or they could find the mean time/cost from the table. Sum of times = 153 hours 217 minutes = 9397 minutes Mean time = 9397 minutes/11 = 854 minutes (nearest minute) = 14 hours 14 minutes Sum of costs = Mean cost = 699.8/11 = (nearest penny) Again, they could start by finding the mean volume then just work out the associated time/cost. Sum of volumes = m 3 Mean volume = m 3 = litres (nearest litre) Time taken = 68336/8 = 854 minutes (nearest minute) = 14 hours 14 minutes Corresponding cost = x.93 = (nearest penny) Cost of heating The first graph gives a running cost of 19 per square metre of surface area for the gas boiler. The second graph gives a running cost of 8 per square metre of surface area for the electric heat pump. Student readings from the graphs may not be this accurate though. There are actually only 6 different values for the surface area; the depth is irrelevant. L W Surface area (m 2 ) Size Cost ( ) Running costs ( ) Size Cost ( ) Running costs ( ) Mini Mini Mini Mini Mini Midi Mini Midi Mini plus Maxi Midi Maxi plus It is clear from the table that a gas boiler has lower set-up costs but higher running costs than an electric heat pump. To analyse this further students could look at a particular pool size, finding out the total cost for 1, 2, 3 years etc, then using a graph to find when one option becomes cheaper than the other.

4 For a 5 m by 2.5 m pool: The gas boiler is significantly cheaper to buy so even taking into account the higher running costs, it is 19 years before the costs are about equal for the two types of boiler. It is only from 2 years onwards that the total cost becomes cheaper for electric. As it takes so long for the electric heat pump to become cheaper than the gas boiler you may need to take into account the expected life of each type of boiler and how long you are expecting to stay in the property

5 For a 5 m by 3 m pool: Once again, the gas boiler is significantly cheaper to buy and it is 16 years before the costs are about equal for the two types of boiler. From 17 years onwards the electric heat pump becomes progressively cheaper when compared to the gas boiler, as it has lower running costs

6 For a 1 m by 5 m pool: The gas boiler is cheaper for the first 5 years but becomes more expensive than the electric heat pump during the 6 th year. After that the electric heat pump quickly becomes much cheaper as the running costs are significantly lower each year

7 For an 11 m by 5 m pool: The gas boiler is cheaper at first but after 5 years the costs are about equal. From year 6 onwards the electric pump is the cheaper option and quickly generates big savings when compared to the gas boiler, since the running costs are significantly lower each year

8 For a 12 m by 6 m pool: The gas boiler is cheaper for the first 4 years but becomes more expensive than the electric heat pump during the 5 th year. After that the electric heat pump quickly becomes much cheaper as the running costs are significantly lower each year

9 For a 15 m by 6 m pool: The gas boiler is cheaper at first but after 4 years the costs are about equal. From year 5 onwards the electric pump is the cheaper option and quickly generates big savings when compared to the gas boiler, since the running costs are significantly lower each year Conclusion: Whatever the size of pool, it is cheaper to buy a gas boiler than an electric heat pump. Even adding in the running costs the total cost is cheaper for a number of years. At some point though the lower running costs of the electric heat pump make this the cheaper option overall, and from this point onwards the savings increase yearly. The bigger the pool the quicker the electric heat pump becomes the cheaper option.