Making Longer, Making Shorter
Ahmed is making rods using different numbers of cubes. Which rod is twice the length of his first rod?
Problem
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?
Getting Started
Try making Ahmed's rods out of cubes yourself. You could start by making his first rod out of four cubes.
If Ahmed's second rod is twice as long as the first, how many of his first rod did he need to make it? Try for yourself.
Student Solutions
Thank you to everyone who submitted solutions to this problem. Some, like the Primary Two Class at Glencairn Primary, used a practical approach; the children in this class used cubes to make the different rods. The children in Primary Three had been learning their times tables, and so they used these to help them solve the problem. For example, Ewan said:
I used my $4$ times table to find the answer when we were making longer rods. e.g. $4\times2 = 8$, $4\times3=12$ and $4\times4=16$.
Sophie said:
I used my knowledge of the $2$ and $4$ times tables to find the shorter rods. Half of $4$ is $2$ and a quarter of $4$ is $1$.
Ardonis, Chioma, Phoebe and Nala, from Holy Trinity C of E Primary School submitted a lovely and clear solution. They noticed that "twice the length" meant "multiply by two", and "half the length" meant "divide by two". Here are their answers:
Ahmed's rod is four cubes long
How many cubes did he need to make a rod twice the length of that one? $4\times2=8$
How many cubes did he need to make one three times the length? $4\times3=12$
How many cubes did he need to make one four times the length? $4\times4=16$
How many cubes did he need to make a rod half the length of his first one? $4\div2=2$
How many cubes did he need to make a rod a quarter of the length of his first one? $4\div4=1$
Elyse from Putney High also submitted a correct answer, as did Reema from the British School in Dubai. Reema looked at the rods that Ahmed made, and matched them up with the descriptions:
the first picture of blocks is equal to Ahmed's
the second is half of Ahmed's blocks
the third is a quarter
the fourth is twice Ahmed's blocks
the fifth is three times Ahmed's blocks
the last one is four times Ahmed's blocks
Well done also to Padraic and Aoibheann from Cloghans Hill NS in Ireland who sent in a correct solution.
Thank you very much to those who submitted solutions. Well done!
Teachers' Resources
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?
Possible approach
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.)
Key questions
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.