Hunting for Averages
Problem
Hunting for Averages printable sheet
In these challenges, the number in each square is the average (mean) of the four numbers surrounding it.
What number should go in this square? Have a think, then take a look at the answer.
Answer
Can you see why the answer is 5?
Can you see why the numbers 4 and 6 belong in the two squares in the problem below?
Your challenge is to find the whole number that belongs in each square using the interactivity below. You can click on the purple cog to change the grid size.
Can you create a similar problem for someone else to solve (with whole number solutions)?
Connecting Averages offers further challenges.
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
This problem offers opportunities for learners to explore what is meant by the mean of four numbers. As students move from the challenge with one unknown to the more complex challenge with two unknowns, they will have the opportunity to develop and improve their strategies, working systematically and using known facts about the four times table to find the solutions.
Points to consider
There are multiple ways of approaching this problem. Learners might take a numerical approach, possibly developing a systematic form of trial and improvement. Some might prefer to use algebra, writing letters to represent each unknown and creating algebraic statements using the information they know. They might combine the two methods, using some algebraic ideas to refine their trial and improvement.
Key questions
How do you calculate the mean of four numbers?
If the mean of four numbers is a whole number, what can you say about the total of these numbers?
If I know three of the numbers, how can I use their total to work out possible values for the fourth number?
If a number is the mean of the four numbers surrounding it, what can you say about the total of those four numbers?
Possible support
Spending some time on the grid with only one unknown will help learners become more confident with the properties of the mean before they move onto the more complex grids. Once they've noticed that the numbers surrounding a square must sum to a multiple of four, you could encourage learners to write down the multiples of four.
Learners who want to take an algebraic approach may find it challenging to get started. They might need support with keeping track of what their unknowns are and the information they have about these unknowns.
Possible extension
For learners who become confident with both levels in the interactivity, the second page of the printable sheet provides some larger grids for them to think about. The problem Connecting Averages provides a further extension to this problem.
Learners could move on to creating a similar problem with whole number solutions.