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First, Ahmed used interlocking cubes to make a rod four cubes long:
How many cubes did he need to make a rod twice the length of that one?
How many cubes did he need to make one three times the length?
How many cubes did he need to make one four times the length?
How many cubes did he need to make a rod half the length of his first one?
How many cubes did he need to make a rod a quarter of the length of his first one?
These rods are the ones Ahmed made:
Which one is twice the length of Ahmed's first rod?
Which one is three times the length?
Which one is four times the length?
Which one is half the length of his first rod?
Which one is a quarter of the length of his first rod?
Which one is the same length as his first rod?
Making Longer, Making Shorter
Age 5 to 7
Challenge Level 
Making Longer, Making Shorter
First, Ahmed used interlocking cubes to make a rod four cubes long:
How many cubes did he need to make a rod twice the length of that one?
How many cubes did he need to make one three times the length?
How many cubes did he need to make one four times the length?
How many cubes did he need to make a rod half the length of his first one?
How many cubes did he need to make a rod a quarter of the length of his first one?
These rods are the ones Ahmed made:
Which one is twice the length of Ahmed's first rod?
Which one is three times the length?
Which one is four times the length?
Which one is half the length of his first rod?
Which one is a quarter of the length of his first rod?
Which one is the same length as his first rod?
Why do this problem?
This problem brings in doubling, halves and quarters in a very practical way using rods made from interlocking cubes. It gives children a practical context in which to explore simple multiplying and dividing, even if these particular terms are not used explicitly. It can provide a very useful context for introducing and using the vocabulary of halves
and quarters.
Having multilink cubes for each child is essential.
Possible approach
It might be best, if it is possible, to work on this problem with a small group of children so that you can listen carefully to their conversations.
You could start with the children on the carpet with free play and some may make rods of different lengths. You could look out for a four-cube rod or, alternatively, you could ask the children to each make a rod four cubes long. (Of course the task does not necessarily have to start from a four-cube rod - you may wish to use whatever length happens to be made by a child as your starting point.)
You could start with the children on the carpet with free play and some may make rods of different lengths. You could look out for a four-cube rod or, alternatively, you could ask the children to each make a rod four cubes long. (Of course the task does not necessarily have to start from a four-cube rod - you may wish to use whatever length happens to be made by a child as your starting point.)
Once you have some four-cube rods, you could use the prompts given in the problem, encouraging the children to make the rods as you go along:
Can you make a rod twice the length of that one?
Can you make a rod three times the length of the first rod?
Can you make a rod four times the length of the first rod?
Can you make a rod half the length of the first one?
Can you make a rod a quarter of the length of the first one?
You could then encourage some discussion which involves reflecting back on what they have made. For example, choose two rods and ask learners to tell you what they know about those two rods.
You may like to suggest that children record what they have done. Squared paper may be useful but don't insist that children use it - let them decide for themselves how they would like to record. Suggest that they work with each other when a difficulty arises rather than seeking your help immediately.
At the end they could show the rods and their illustrations as an opportunity for you to reinforce the vocabulary that you have been using. Some children may count cubes and rely on their knowledge of number bonds or multiplication facts, others may use the cubes to make different rods using a system of trial and improvement along with counting.
Key questions
If Ahmed's second rod is twice as long as the first, how many of his first rod did he need to make it?
If Ahmed's third rod is three times as long as the first, how many of his first rod did he need to make it?
If Ahmed's fourth rod is four times as long as the first, how many of his first rod did he need to make it?
If Ahmed's fifth rod is half as long as the first, how could he break his first rod to make it?
If Ahmed's sixth rod is a quarter of the length of the first one, how could he break his first rod to make it?
How do you know that rod is twice the length of the four-cube rod?
Possible extension
Learners could investigate halves and quarters of other length rods using multilink.Possible support
Having the cubes available to make the rods will help all children access this problem.You may also like
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Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?
