### Pebbles

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

### It Figures

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

### Bracelets

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

# Multiples Grid

## Multiples Grid

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?

This problem is an interesting way of reinforcing understanding of factors and multiples.

### Possible approach

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?

### Key questions

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?

### Possible extension

Learners could explore the spreadsheet for themselves at a computer. Challenge them to make up their own questions to ask a friend.

### Possible support

A multiplication square may be useful for those children who find instant recall of multiplication facts difficult.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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