Maths olympiad lesson: Pick's theorem, the area of a shape from counting dots

Pick's theorem: practice

Every polygon here has its corners on grid points and no holes. Show your working on paper. Questions 6 and 9 need a written explanation.

Diagnostic question: choose and explain

1. The triangle with corners (0, 0), (4, 0) and (0, 4). How many grid points are on its edges, corners included?
   1. 12
   2. 15
   3. 3
   4. 9

Practice questions

1. A polygon has 14 grid points on its edge and 2 inside. What is its area?
2. A polygon has 6 grid points inside and 10 on its edge. What is its area?
   1. 10
   2. 11
   3. 12
   4. 16
3. How many grid points lie on the line segment from (0, 0) to (8, 12), ends included?
4. The triangle with corners (0, 0), (5, 0) and (2, 3). How many grid points are strictly inside it?
5. Can a polygon with all its corners on grid points have an area of 2¼?
   1. No: I + B/2 − 1 is always a whole number or a half
   2. Yes, if it is long and thin
   3. Yes, if it has enough sides
6. Prove that Pick's theorem gives the right area for every a by b rectangle whose sides lie along grid lines.
7. The square with corners (0, 0), (4, 1), (3, 5) and (−1, 4). How many grid points are strictly inside it?
8. How many grid points (x, y) have |x| + |y| ≤ 10?
9. Stretch: prove that no equilateral triangle has all three corners on grid points. (You may use that √3 is not a fraction.)

Support: For question 1: put the numbers into A = I + B/2 − 1. For counting B: count the steps along each edge (the HCF of how far across and how far up), and add the steps: every dot on the edge, corners included, is counted once that way. For the tilted squares, check your area with side².

Pick's theorem: exit

A number, then a sentence.

1. A polygon with its corners on grid points has area 9 and 5 grid points inside. How many grid points are on its edge? Then say what must be true of a shape for Pick's theorem to work.

Exit check

One backwards use of the theorem with a reason shows who can rearrange it and knows when it applies.

A polygon with its corners on grid points has area 9 and 5 dots inside. How many dots are on its edge?

Pick's theorem: count, use, prove

Every polygon has its corners on grid points. Use squared paper. Find each area two ways where you can. Question 6 is a written proof: write it as you would in an olympiad.