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Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

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Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.

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Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

Pick's Theorem

Stage: 4 Challenge Level: Challenge Level:1

Why do this problem?

At its simplest level this problem allows students to consolidate their understanding of how to calculate the area of irregular shapes. The extra mathematical demand comes from requiring students to identify the relationship between three variables.

Possible approach

This printable worksheet may be useful: Pick's Theorem.

You may be interested in our collection Dotty Grids - an Opportunity for Exploration, which offers a variety of starting points that can lead to geometric insights.

Draw a polygon on a square dotty grid on the board. Clarify that we shall be interested in three variables: the number of dots on the perimeter, $(p)$, the number of dots in the inside, $(i)$, and the area $(A)$. Ask students to work out the $(p)$, $(i)$ and $(A)$ of the shape that you have drawn.

Display the five shapes from the problem. For each, ask students to work in pairs and agree on the values of $(p)$, $(i)$ and $(A)$.

Draw attention to the two shapes that have an area of 1. What do they notice about their $(p)$ and $(i)$? Is this true for all shapes that have an area of 1? Allow the students some time to draw and share results. Confirm that there are an infinite number of possibilities of shapes which satisfy these conditions.

"The size of the shape will determine the area, so $(p)$ and $(i)$ may well determine the area. Your challenge is to draw some more shapes and find out if there is a relationship between these three variables."

Identify a central place where students can post their conjectures or other observations and encourage students to check their validity.

At an appropriate time bring the students back together to discuss the relationships they have discovered. Use this also as an opportunity to discuss effective strategies for identifying relationships, eg keeping one variable fixed.

You may wish to use the interactive pinboard, found here , to support your presentation/discussion of the problem.
Here is an account of one teacher's approach to using this problem.

Key questions

If $(p)$ is fixed and $(i)$ increases by 1, what is the effect on the area?
If $(i)$ is fixed and $(p)$ increases by 1, what is the effect on the area?

Possible extension

Does the same relationship hold when shapes are drawn on isometric dotty paper?

Possible support

Suggest that students start with shapes with small areas. How many different shapes can they draw for an area of 2? What possible values of $(p)$ and $(i)$ can they find? What about an area of 3..4..5...?