3 Doing and undoing
A powerful idea in teaching mathematics is ‘doing and undoing’. This idea says that learners can often be asked to work backwards as well as forwards with mathematical ideas. The act of ‘undoing’ can often lead to new and interesting questions, and test learners’ understanding of ideas (Mason, Graham and Johnston-Wilder, 2005, p.66).
The final section this week explores how ‘doing and undoing’ might contribute to the teaching of the mean, the median, the mode and the range.
The next activity asks you to work backwards through the definitions of the range and the median. As you tackle the four questions, try to notice what you are finding easy, what you are finding challenging, and how these questions are testing your understanding in a different way.
Activity 9 Data sets
What would a data set with a range of 0 look like? Think of at least two examples.
What might a data set with a median of 0 look like? Again, think of at least two examples.
What might a data set with a range of 0 and a median of 0 look like?
What would a data set with a range of -4 look like?
Discussion
A data set with a range of 0 would have to be made from a single number, such as 7, 7, 7, 7 or 12, 12, 12.
If there were an odd number of data points, a data set with a median of 0 would have 0 as the middle number, for example -4, 0, 1. If there were an even number of data points, it would have either 0 on both sides of the middle, or positive and negative versions of the same number on either side of the middle, for example -8, -3, 3, 11.
A data set with a range and median of 0 would have to be made entirely of zeroes. Can you reason why?
It is impossible to have a data set with a range of -4. The smallest possible range is 0.
The next activity brings together all four main summary statistics we have discussed and asks you to work backwards and forwards. These types of question can be extended in lots of different ways. You can include different types of number, involve different summary statistics, or change what is being asked.
Activity 10
These five boxes all contain integers (whole numbers which could be positive, negative or zero). The numbers have been written in order from left to right.
Question 1
The mean is 2, the mode is 9, the range is 16 and the median is 0. What are my numbers?
Question 2
The mode is 1, the median is 3 and the range is 6. How many different answers could there be?
Answer
Question 1
Question 2
The summary statistics allow us to fill in four of the boxes:
We cannot repeat the 3 or the 7, so the empty box could contain a 4, 5 or 6 and there are three possible answers to this question.