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Growing Garlic


Ben is on the allotment with his Mum. They would like to grow some garlic and are deciding how to plant the garlic cloves.

Ben arranges the cloves into three rows and finds that he has one spare clove. How many cloves might he have had to start with?

Ben plants cloves of garlic in two rows and has one clove left over. So he tries again.
He plants cloves in three rows and has one left over. So he tries again.
He plants cloves in four rows and has one left over. So he tries again.
He plants cloves in five rows and has one left over. So he tries again.
He plants cloves in six rows and has one left over.
We know that he has fewer than $100$ garlic cloves. How many did he have?
You could think about how many cloves he might have had if there were more than $100$.

You might like to use this interactivity to investigate the problem. You can alter the number of cloves using the up and down arrows, and you can drag them into different arrangements using the little white square.
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Why do this problem?

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.

Possible approach

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 the interactivity, for example:
  • Ben has $15$ bulbs of garlic. (Scroll down to $15$)
  • Can he plant them in rows of two? Why? Check using the interactivity.
  • Can he plant them in rows of three? Why? Check using the interactivity.
  • Can he plant them in rows of four? Why? Check using the interactivity.
  • What else could you tell me?

Then, without using the interactivity, 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 or counters available for them to use. You might also want to have squared paper available. Take answers and list them, using the interactivity to check them. Put them in order of size to help pattern spotting. Encourage comments which indicate the children have spotted the pattern (it's one more than the four times table) rather than specific (it could be $41, 5, 25 \ldots$)

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 IWB 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. If you are working in a room where there are lots of computers the children can use the interactivity to check their solutions. Otherwise pegboards, cubes, or any other units such as bottle tops can be helpful.

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 $21$').

Key questions

What is the same about these answers?
How could we record our results to help us to spot a pattern?

Possible extension

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.

Possible support

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.