Here is a $100$ grid with some numbers shaded:
What do all the numbers shaded blue have in common?
What do you notice about all the numbers shaded pink?
Can you work out why two of the numbers are shaded in a maroon colour?
Now, here is part of a $100$ square shaded in a different way:
Can you explain the shading this time?
Here are some more parts of the $100$ square, each one shaded according to different rules. Can you work out what the rules are for each?
Is there only one solution each time?
This problem is featured in Maths Trails - Excel in Problem Solving, one of the books in the Maths Trails series written by members of the NRICH Team and published by Cambridge University Press. For more details about the other books in the series, please see our publications page .
Why do this problem?
is an interesting way of reinforcing understanding of factors and multiples.
To start with, ask children to talk in pairs about why the numbers in the first 100 square are shaded blue, pink and maroon. Invite them to share their ideas and encourage correct use of vocabulary.
Learners could continue to work in pairs, perhaps using this sheet of the four parts of differently-shaded 100 squares. As they work on the problem, trying to find out which factors have been chosen in order to produce the shading, encourage them to justify their solutions to their partners, and perhaps then to the whole class. How
are they going about the task? It might be useful to discuss ways of working systematically so that no solutions are omitted.
This spreadsheet ,which shades the squares according to the chosen factors, can be used to check their hypotheses. In a plenary session, you could use the second sheet of the spreadsheet to pre-prepare some shaded sections of the 100 grid without numbers. If you tell them which multiples have been shaded, can the class work out where
the small part of the 100 grid is, i.e. which numbers it contains?
What do the numbers shaded blue have in common?
What do the pink numbers have in common?
Can you rule out some factors straight away? How?
How will you know you have found all the possible solutions?
Learners could explore the spreadsheet for themselves at a computer. Challenge them to make up their own questions to ask a friend.
A multiplication square may be useful for those children who find instant recall of multiplication facts difficult.