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Write your own maths test (Investigation)


It's hard to be a maths teacher. If you think doing a maths exam is difficult, imagine having to write one! When maths teachers write maths questions, they often start by working backwards, so that they know what the answer will be. In this exercise, we're going to have a go at making our own maths questions, based on the calculations of mean, median, mode and range. 



Let's say we are setting a question of calculating the mean of a group of five people's heights, and we want the answer to be exactly $150$150cm. How can we do this? Guess and check is not a good idea, maths teachers are much too busy for that!

If we look at the formula for calculating the mean, we can get some hints:

$Mean=\frac{\text{Sum of Heights}}{\text{Number of People}}$Mean=Sum of HeightsNumber of People

In this case, we know that the number of people is five. Therefore, the formula becomes:

$Mean=\frac{\text{Sum of Heights}}{5}$Mean=Sum of Heights5

So, we can see that any numbers which have a total of $750$750cm will give a mean of $150$150cm, since the total will be divided by $5$5

1. Write up a table of $5$5 people's heights that add up to $750$750cm, and check that the mean is $150$150cm. 

2. Make another table of $5$5 numbers with a mean of $150$150cm, with the different in students heights being as extreme as possible. How tall can the tallest person be, for the mean to still add up to $150$150cm?

3. Make another table of $5$5 numbers with a mean of $150$150cm, with all students being the same height. 

4. Based on this activity, do you think that being given the mean height alone of a class is enough for you to be able to imagine what the students look like? What other information would you need? 



Let's say that the next question you have to write for your students is a question on calculating the median. 

As you may already know, medians are calculated by taking the "middle number" from a group. As such, if we want to make a group with a particular median, we just have to make sure that number ends up in the middle.

1. Make a group of five students with a median of $150$150cm. 

2. If you tried to make the height difference as extreme as possible, how tall can the tallest student be in this example? How does this compare with the answer you got for the last section? 



The range is the difference between the lowest and highest entries. To make a particular range, just make sure that the difference between these two is equal to what you want it to be.

1. Make a group of five students with a range of $30$30cm.


Challenge: Mean, Median and Range

Let's say your trying to make a test which tests all these three things at once. 

1. Try to make a group of five students with a mean of $150$150cm, a median of $150$150cm and a range of $30$30cm. 

2. If you had to pick only two features (out of mean, median and range) to know about the class next door, to get a good idea of how tall people in that class are, which two would you pick and why? 

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