Connecting Averages
Problem
For a simpler version of this problem, take a look at Hunting for Averages.
Hunting for Averages printable sheet
Connecting Averages printable sheet
In these challenges, the number in each square is the average (mean) of the four numbers surrounding it.
Can you explain why the numbers 5, 4 and 6 belong in the squares in these two examples?
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)?
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 students to use what they know about the mean as they explore approaches to the different challenges. As students move towards working systematically, they can extend or adapt their approaches to the more complex challenges with more unknown connected means.
Key ideas
There are multiple ways of approaching this problem. Learners might take a numerical approach, developing a systematic form of trial and improvement. Some might prefer to use algebra, writing letters to represent all or some of the unknowns 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 (e.g. discovering that a number must be one more than a multiple of 4, or reasoning that one of the unknowns must be bigger than the other).
Key questions
How could 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?
What is the smallest value this mean could be? (Remember that the means are whole numbers.)
For students who are using algebraic approaches:
If $A$ is the mean of the four numbers surrounding it, what is the total of those four numbers?
Possible support
Students who take an algebraic approach may find it difficult to set up equations involving the unknowns. They might need support with keeping track of what their unknowns are and the information they have about these unknown. Encourage students to write the letters they are using for unknowns in their grid and then work systematically to set up their equations. Some students may be concerned when both unknowns show up in their first equation if they are not expecting that. If so, encourage them to continue to set up the second equation before attempting to work towards solutions.
Possible extension
Students who are confident with using an algebraic approach might find it interesting to explore whether it is possible to use fewer unknowns, instead of putting a letter in each square of the grid.
An alternative way to extend the problem is to ask students to create a similar problem for someone else to solve (with whole number solutions).