Challenge Level

The idea of this problem is to encourage children to spot and describe a pattern, and to extend this into a general rule. Depending on their experience, the can also use relevant vocabulary associated with factors and multiples.

You could use this problem with the whole class as it is an example of a low threshold high ceiling activity - accessible to all but challenging for the most confident.

You may want to start by asking some simpler questions and model them using counters, for example:

- Ben has 15 bulbs of garlic.
- Can he plant them in rows of two? Why? Check using counters.
- Can he plant them in rows of three? Why? Check using counters.
- Can he plant them in rows of four? Why? Check using counters.
- What else could you tell me?

Then, ask the children to find how many bulbs Ben could have if he plants them in rows of, say, four, but has one left over. Have cubes, counters, bottle tops... available for them to use. You might also want to have squared paper available. Take answers and list them, using counters on the interactive board, or magnets on a whiteboard, to check them. Put them in order of size to help pattern spotting. Encourage comments which indicate the children have identified the pattern (it's one more than the four times table) rather than specific (it could be 41, 5, 25...)

Pose the first part of the written problem and invite pupils to 'think, pair, share' - think on their own, then talk to a partner about what they think the solution might be, and/or how they might work it out. Take suggestions and if appropriate let one or two children come to the board to model their ideas.

Pose the rest of the problem and leave the children to pursue their ideas, working in pairs. Emphasise that you are interested in the answer, but also interested in the way that the children found it.

Bring the children together and share their findings, again encouraging working that is systematic and helping them to differentiate between specific answers and general ones ('if there is one left over with two rows it must be an odd number', rather than 'it must be 21 or 15 or 21').

What is the same about these answers?

How could we record our results to help us to spot a pattern?

Children who find this easy could make up their own questions for a partner. Devising a question which has a unique answer is quite difficult.

Physical apparatus helps to consolidate the idea of 'one more than a multiple'. Listing possible answers in order of size can also help the children to spot patterns - encourage them to work systematically.