The inquiry starts with students trying to understand the statement and, particularly, the constraint placed on the data set. Questions and comments that have arisen in the classroom are:
- What do median, mode, mean, and range mean?
- What is a set of data?
- Why are there ‘greater than’ signs in the sentence?
- What does it all mean?
- How many numbers are there in a set of data?
- Do the median, mode, mean, and range have to be in that order?
- If the mean is higher than the mode, you have to start with the mean, then work out the mode.
- It must be possible.
- When we tried this, it worked for the median and mode, but went wrong on the mean.
- You can’t do it because the range is too small.
The prompt leads students into trying to construct sets of data that fulfil a certain condition. The teacher might orchestrate a discussion on how many numbers are appropriate in the data set and guide students towards using five or seven to simplify the start of the inquiry. If students are unfamiliar with the terms in the prompt, or need to revise them, they have in the past chosen a regulatory card that gives them time to practice finding the median and mode and calculating the mean and range before exploring the statement itself. However, it is possible to develop fluency in these procedures during the inquiry.
In the search for a set of data described in the prompt, students begin to create sets that satisfy different inequalities. So, for example, students found this set that has mode > mean > range > median:
Thus, a period of exploration can lead to a question about how many permutations of median, mode, mean, and range exist. In
one inquiry, a group of students found the 24 permutations and
explained the result by using four ‘blocks’ based on the one they demonstrated to the class, which started with the median (see box).
Another class working collectively found that, along with the constraint in the prompt, there are four other permutations for which it is impossible to create a data set:
This final result, if confirmed by other classes, leads onto other questions: Why are the five permutations impossible to achieve? Do they have something in common? Do they remain impossible to create if we use more numbers in the data set?
Promethean flipchart download
Smartboard notebook download