Why do this problem?
This problem encourages students to see the maths underpinning a situation. In this case it is the importance of factors and multiples in what is at first glance a geometrical setting.
To avoid spoiling the surprise, it may be worth doing this activity at the start of work on the topic, without telling the students what the topic is...
You will find sheets of different dot-circles for printing out at the bottom of the problem page.
Ask students to draw a five pointed star starting and ending at one of the "points" or vertices of the star. They must do this without taking their pencil off the paper and without drawing over a line they have already drawn. Many learners will have met this before.
Ask the group to discuss in pairs a description of what they did that they can share with the rest of the group. "How would you explain to someone else, at the other end of a phone, how to draw the star?"
Look out for ideas such as step size and ways to describe positions.
When ready, demonstrate a five pointed star with the interactivity and discuss the notation that has been used (going anticlockwise, stepping by two leaves two gaps between the points on the circle). Alternatively, you might get a group of students to stand in a circle and make the stars with string (by passing a ball of string).
Discuss points of interest including:
- What happens if you move clockwise.
- What constitutes a star (in these notes a polygon created from a step size of 1 is not a star).
- There is only one star on a five-dot circle.
- Complementary step sizes produce the same star (step size two is equivalent to step size three in this five-dot context)
Ask students to make as many stars as possible on a seven-dot circle:
- How many stars can they make?
- How do they know they have them all?
Challenge them to conjecture how many stars will be possible on a nine-dot circle (without drawing them at this stage). Discuss in pairs before sharing what they can offer as a convincing argument.
Students can now focus on generalising their results for any dot-circle.
- What are the things that affect the number of stars you can draw?
- Can you find one rule to determine the number of stars or do you need different rules for different circumstances?
- Can you write a rule, or set of rules, that someone who had never seen the problem, could understand?
If working with a small group - ask each person to create a star based on a different step size and compare the group's results, encouraging the students to identify what is the same and what is different about their stars and putting them into an order they can justify.
Encourage individuals to draw a star and, without showing it to their partner, give instructions to draw the same star. Are the two stars the same?
The activity could be carried out using string on peg boards.
How many times would the string pass around the circle for different stars in different dot-circles?
Can you find the angles at the vertices of any star?