### Doplication

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

### Round and Round the Circle

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

### Making Cuboids

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

# Factor-multiple Chains

## Factor-multiple Chains

Here is an example of a factor-multiple chain of four numbers:

Can you see how it works? Perhaps you could make some statements about some of the numbers in the chain using the words "factor" and "multiple".

In these chains, each blue number can range from $2$ up to $100$ and must be a whole number.

You may like to experiment with this spreadsheet which allows you to enter numbers in each box. Perhaps you can make some more chains for yourself.

What are the smallest blue numbers that will make a complete chain?
What are the largest blue numbers that will make a complete chain?
What numbers cannot appear in any chain?
What is the biggest difference possible between two adjacent blue numbers?
What is the largest and the smallest possible range of a complete chain? (The range is the difference between the largest and smallest values.)

### Why do this problem?

This problem offers opportunities to reinforce pupils' understanding of factors and multiples. They will become confident in using this vocabulary. It also gives them the chance to justify their solutions and to be creative by making their own chains.

### Possible approach

You may wish to show this spreadsheet to the whole class to introduce the problem. It is an interactive chain, which allows you to enter numbers in each box, giving feedback as to when factors and multiples occur. As you make different chains, ask the children to explain what is happening so that everyone fully understands the environment.

Encourage pupils to work on the different questions, ideally in pairs so that they have somone with whom to discuss their ideas. They could work on paper or mini-whiteboards, and access to calculators might be useful. You might like the class to be at computers so they can manipulate the spreadsheet themselves and check their solutions. You may have to discuss what is meant by 'largest' and 'smallest', and come to an agreement. You could ask two questions, for example 'What are the largest blue numbers that will make a complete chain?' and 'What is the largest possible first number in a chain?'.

When sharing solutions, encourage learners to justify their answers - how do they know that their chain contains the smallest/largest numbers etc.? Some children will be using trial and improvement, some will have developed a system for trying numbers in turn, and others may have been able to combine these with their knowledge of number properties.

### Key questions

Can any number start the chain? Why/why not?
Can any number be at the end of a chain? Why/why not?
What can you say about the numbers in the second and third positions in a chain?
How will you keep track of what you have tried?

### Possible extension

Children could investigate what would happen if the chain was made up of more than four numbers, or made up from a different range of numbers.

### Possible support

Enabling learners to manipulate the spreadsheet for themselves will help them access this problem and you can focus on justifying the solutions rather than pupils worrying about obtaining them. There is still a great deal of mathematical thinking taking place as children interact with the spreadsheet.