Problem Teachers' Notes Hint Solution

This problem requires children to apply their knowledge of factors and multiples, and is a good way of making the link between sharing, division and multiples/factors. It may also be used to introduce learners to the fact that a problem can have more than one solution and that the solutions can be generalised. It can be approached in many different ways so can be a useful context in which to talk about different ways of recording and different methods of solving problems.

It might be useful to break this problem down into two stages and you could decide whether to introduce it as two parts or whether to give children the whole problem, then break it down. Either way, you could start by having two children at the front of the class and asking how a bag of lollies could be shared equally between them. (You could even have lollies to try this out!) Children might suggest counting the lollies first or giving each one in turn. What would happen if there was one left over? Ask the class how many lollies could have been in the bag if there was one left over. Encourage them to work with a partner, using paper or small whiteboards, and recording whatever they find helpful.

Bring the group together after a short time to discuss what they think. Many might have specific examples of numbers that would work so you could ask questions to help them generalise, like "Could there have been $24$ lollies? Why?" or "Could there have been $99$ lollies? Why?" Invite the two children back to the front again and this time model what would happen if three friends came along as well. After talking this through, set the group off again on finding possible solutions in this new context.

As you walk around the room listening to the children talking about their work, look for examples of different ways of recording and different ways of approaching the problem to share with everyone in a plenary. Some children might have tried numbers randomly then noticed a pattern, others might have had a system so found out how many lollies there would have been if each child had one each, then two each etc.

(You might find it helpful to adapt this Smart Notebook file for use on the interactive whiteboard. Thank you to Gemma for giving us permission to include it here.)

How many lollies would have been in the bag if each child had just one? What about if they had two lollies each? Three lollies ... etc.?

Can you see a pattern in your answers? Can you explain where the pattern comes from?

At the highest level, encourage learners to articulate a full generalisation about what kind of numbers of lollies there could be in the bag. This might be worded in terms of multiples, or in terms of the units digit of possible solutions. Growing Garlic is a similar problem which pupils could have a go at as a follow-up to this one.

Learners might benefit from having a hundred square available, or some counters/cubes to represent the lollies.