P is a point on the circumference of a circle radius r which rolls,
without slipping, inside a circle of radius 2r. What is the locus
The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design. Coins inserted into the machine slide down a chute into the machine and a drink is duly released. How many more revolutions does the foreign coin make over the 50 pence piece going down the chute? N.B. A 50 pence piece is a 7 sided polygon ABCDEFG with rounded edges, obtained by
replacing AB with arc centred at E and radius EA; replacing BC with arc centred at F radius FB ...etc..
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
Many students are accustomed to using number patterns in order to generalise. This problem offers an alternative approach, challenging students to consider multiple ways of looking at the structure of the problem, through making sense of other people's approaches, an important part of working mathematically.
The powerful insights from these multiple approaches can help us to derive general formulae, and can lead to students' appreciation of the equivalence of different algebraic expressions.
This problem and Christmas Chocolates are essentially the same question but presented in a different way.
Start by showing this image:
"Cables can be made stronger by compacting them together in a hexagonal formation. Here is a 'size 5' cable. Can you work out, without counting every strand, how many strands it contains?"
Give students a short time to consider this and then discuss their ideas in pairs, before bringing the class back together to share their different methods.
Hand out this worksheet, together with these templates, and ask:
"How many strands are needed for a size 10 cable?"
"While you are working on this, keep in mind how your method could be adapted to work out the number of strands needed for any size of cable."
While students are working, circulate and observe any interesting methods that students are using. When the class is ready, bring them back together and invite those students with an interesting method to explain what they did to the rest of the group.
"Here is some work done by groups of 15 year old students who were asked to find a formula to work out the strands needed for a size n cable."
Arrange the class in groups of four, and hand out one of the four pictures from this worksheet to each student.
"I'd like each of you make sense of the picture drawn by one of the groups and work out how to solve the problem using their diagram. Then when you are ready, you need to explain the method to the rest of your group. Finally, the group needs to decide which of the four approaches makes the most sense to you, and be prepared to justify your choice to the rest of the class."
To finish off, each group can explain to the rest of the class which approach they chose, and why.
Here are some key questions that could be used to help groups who are struggling to make sense of the different methods:
"Is there a quick way of adding up all the numbers from 1 to n?"
"Look at the picture for a size 5 cable. What might the group have drawn for a size 6 cable?"
Challenge students to come up with alternative, elegant ways of computing the number of strands in a size n cable.
The problems Summing Squares and Picture Story lead to formulae for some intriguing sequences through analysis of structure.
Seven Squares gives lots of simple contexts where formulae emerge by looking at structure rather than number sequences.
Picturing Triangle Numbers, Mystic Rose, and Handshakes introduce students to summing consecutive numbers, which is a feature of several of the solutions used by the groups in the problem.
To read one teacher's account of using this problem, click here.