The vegetable garden
Problem
Ben is in the vegetable garden 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 try out your ideas:
Getting Started
Have you tried out your ideas using the interactivity? Or you could use some counters or cubes to stand for the garlic cloves and arrange them in rows.
If Ben had only planted one clove in each row, how many cloves would he have had?
What other numbers of cloves could he have planted in each row? So how many cloves would he have had altogether then?
Student Solutions
We had lots of correct solutions to this problem and many of you explained your thinking very clearly. Well done!
Musab from Orchard School answered the first part of the problem:
Ben might have started with a number 1 more than a multiple of 3 because it says when Ben put the garlic into groups of 3 there was 1 garlic plant left. For example Ben could have started with 7.You're right, Musab. What other possibilities could there have been then? Pupils at St Mary's in Tetbury approached the second part of the problem in a very logical (or systematic) way. They said:
We started with 6 rows with one left over, and wrote down all the multiples of 6, add 1, up to 100.
Then we looked at 5 rows with one left over, and wrote down all the multiples of 5, add 1, up to 100.
We rubbed out all the numbers that weren't in both.
Then we looked at 4 rows with one left over, and wrote down any multiples of 4, add 1, that appeared in our lists for 5 rows and 6 rows.
Then we looked at 3 rows with one left over, and there was only one number that worked for all of them. We think the solution is 61.
PJ, Josh and Michael tried a few things out and gradually got close to the answer. This can be a very useful way of solving a problem (we call it trial and improvement):
Qiuying of Wimbledon High School explained the solution in slightly higher level maths. (Don't worry if you haven't come across these ideas yet.)
Teachers' Resources
Why do this problem?
The idea of this problem is to encourage children to identify and describe a pattern, and to extend this into a general rule. Depending on their experience, they can also be encouraged to 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 also having in-built challenge for the most confident.
You may want to start by asking some simpler questions, for example:
Ben has 15 bulbs of garlic.
- Can he plant them in two equal rows? Why or why not?
- Can he plant them in three equal rows? Why or why not?
- Can he plant them in four equal rows? Why or why not?
- What else could you tell me?
In each case, encourage learners to use practical resources, to show their solution.
Then, ask the children to find how many bulbs Ben could have if he plants them in three equal rows but has one left over, as in the first part of the problem. Have cubes, counters, bottle tops, squared paper, hundred squares... available for everyone to use, should they wish. List children's suggestions and use the interactivity, or counters/magnets on the board, to check them. The interactivity offers a visual representation of factors, multiples and remainders by arranging numbered garlic cloves into a rectangular array, i.e. into rows of equal size). The 'left over' cloves are circled in red.
Rearrange the solutions in order of size on the board and ask learners what they notice. Encourage comments which indicate the children have identified the pattern ("It's one more than the three times table") rather than them offering specific examples that 'fit' (it could be 22, 4, 31...). Many will make comments about the "numbers going up in threes", which is absolutely correct and an excellent starting point. However, 'one more than the three times table', or 'one more than a multiple of three' allows us to identify instantly whether a particular number is in the sequence, as opposed to having to list all the numbers in order, going up in threes.
Pose the rest of the problem and leave the class to pursue their ideas, working in pairs. You may wish them to have access to the interactivity in pairs as well as access to a range of practical resources. Emphasise that you are interested in their answer or answers, but you are particularly interested in how they go about finding solutions. As they work, look out for pairs who have developed useful ways of recording their thinking. Some learners will be recording to help them find a solution (e.g. drawing blobs for the garlic bulbs), some will be recording to keep track of what they have tried - what has worked and what has not! Use a mini plenary to draw attention to some of the different ways of recording and encourage children to share their reasons for recording in this way.
Bring everyone together to share solutions and help 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 7"). It is interesting to note that there are far fewer multiples of 6 between 1 and 100 than multiples of 2, so starting with numbers that are one more than a multiple of 6 would give fewer possibilities compared with starting with odd numbers. Look out for learners that realise this, thus demonstrating their deep understanding of the number system.
Key questions
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 and encourages them to work systematically.
Possible extension
Children could create their own question with a unique answer for a partner to solve.