A question of averages
I teach a bottom set Year 9 group at a boys' school. About a third of pupils on the SEN register. I'm looking for an activity to explore averages, mode, median and mean that will enhance their understanding of different types of average.
One of my favourites is using heights. The pupils are measured and their heights in centimetres are written in large characters on a card. We discuss the different averages and how they're worked out, estimating the statistics for the class. Then we all go to the playing field (a digital camera is useful to record findings). First the median is found. Pupils line up in order of height, then sit down two at a time (one from each end) when the whistle is blown. This carries on until one (the median value) or two (the heights are added together and divided by two) remain.
Now get pupils of the same heights to stand in lines. The mode (the most frequent height) can easily be seen. The mean average is not so easy as all the heights have to be added together and then divided by the number of pupils. This does not really help them to understand what they are actually doing when they are finding the mean.
I used the following activity with a group who were 14 years and over. We used MultiLink cubes, but you could use anything that can be joined in a straight line. Before the lesson, rule out a set of axes for a bar chart on A3 paper. Label Monday to Friday twice with "week one" and "week two" on the vertical axis and "Frequency" along the horizontal axis. I used the cubes as a guideline for the correct scaling and made the maximum frequency 11.
Have a scenario ready, eg: "My daughter sells designer sunglasses and at the end of each day she records the number of pairs sold. The most she has ever sold in one day is 11. We will create data for sales that might take place over two weeks at Easter." (Try to have some sunglasses catalogues or information from the internet.) Ask pupils to imagine they are in the shop for a day and to decide how many glasses they have sold. Give them a day for their sales, eg Monday, week one. For the initial exercise one pupil may have to change their number to make sure that the total is a multiple of 10. Ask them to create this number using MultiLink cubes put together in a line, one cube representing one pair of sunglasses, and place it on the graph (above right), and to make an extra line for a later exercise.
Discuss the different averages (mode, median and mean) and ask them which would be the most interesting as far as the boss is concerned. It may be the mean average, as it can be used as a comparison from week to week. The mode would be useful, because it would show which day was the busiest extra staff may be needed.
Ask pupils how they would work out the mean (my group said they would add the numbers - then forgot what happened next). Tell them that when finding the mean we are really trying to make sure that each of the bars are the same frequency - we are levelling them out. They can move the blocks to do this. In the example, the mean average is four, as can be seen (above right) when the blocks are moved around.
Explain that this is what is meant by the mean and that, mathematically, for the mean the total number of sunglasses sold would have been divided by the number of days.
If you had to change one pupil's number for the initial exercise, refer back to this and the fact that you made the numbers divisible by 10. Ask how the answer would have been different if one of the blocks had not been divisible by 10 (they would have had to chop the block or blocks evenly over the days).
The mode can be seen as the biggest bar on the graph. To find the median, the duplicates they created are now put in order and the shortest bar discarded along with the longest bar until the middle is reached.
An interesting follow-up is to provide a worksheet of problems, eg the mean is 2, the mode is 6 and the median is 1, what might the sales have looked like over 10 days? The number of days could also be changed. Ask pupils to make up their own questions in pairs for the rest of the group to solve.
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.
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