A Rod and a Pole
A lady has a steel rod and a wooden pole and she knows the length
of each. How can she measure out an 8 unit piece of pole?
Age
7 to 11
Challenge level
Problem
A lady has a steel rod that she knows to be exactly $3$ units
long, and a wooden pole that she knows is exactly $13$ units long.
Her only tools are a pencil and a saw.
Image
She needs an $8$ unit long wooden pole. She finds that is not possible to mark the steel rod, but she can draw on the wooden pole.
How can she measure out the $8$ unit pole?
Getting Started
You could start by drawing a line $13$ cm long (to represent the pole) and cutting a strip of paper $3$ cm long (to represent the steel rod).
What total lengths can you produce by marking off lengths of the rod on the pole?
Teachers' Resources
Why do this problem?
This problem is one that combines logical thinking about lengths while using the operations of addition and subtraction in a practical way. It could be introduced during work on the measurement of length. It also provides an opportunity for learners to consider the effectiveness of alternative strategies.Possible approach
You could start with some simpler examples of this type of
problem with the whole group. Examples which could be used include
having a $3$ unit rod (which cannot be marked) and a $5$ unit pole
and cutting off $1$ unit, and (slightly more difficult) a $2$ unit
rod and a $5$ unit pole and finding out how to cut off firstly $3$
units, then $1$ unit and then $4$ units. Having some
sticks cut in the right proportions with which to demonstrate will
make the problem more accessible.
After this introduction the learners could work in pairs on
the actual problem from a printed sheet so that they are able to
talk through their ideas with a partner. Ideally, each pair, or
even each child, could have sticks to represent the rod and the
pole. It might be helpful to supply some squared paper
for them to work on.
At the end you could bring the whole group together to see how each pair solved the problem as there is more than one way to do it. They could consider whether some strategies were more effective than others. Explaining their thinking to each other can be a real learning situation. If they have had real sticks, you could even test out their methods to see whether they work.
Key questions
If you place the end of the rod next to the end of the pole,
what length is left?
What length is left when you have measured off the rod twice?