Hunting for Averages
1 out of 3
Can you find the missing number(s)? The square is the average (mean) of the numbers surrounding it.
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?
The challenge, in the interactivity below, is to find the missing number(s). All numbers are whole numbers.
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 webinar recording offers guidance on how to introduce the concept of mean average, and then goes on to discuss this task and how it might be used in the classroom.
This problem offers opportunities for learners to explore what is meant by the mean of four numbers. As learners 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.
Key ideas
Learners might take an exploratory approach initially, possibly developing a systematic form of trial and improvement as they become more familiar with the problem.
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 1x2 grid. 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.
Possible extension
Learners could move on to creating a similar problem with whole number solutions.
Learners who become confident with both levels in the interactivity may like to have a go at Connecting Averages, which includes larger grids for them to think about. The second page of the printable sheet includes a taster of the challenges provided by the larger grids.