Patterns, Proof and Prediction

You need: a box of toothpicks or matchsticks · squared paper · table template

1. Build a row of 1, 2, 3 and 4 squares that share sides. Count the toothpicks each time.
2. Record step number (n) and toothpicks (t) in a table and find the difference between rows.
3. Write the rule in words, then with letters: t = 3n + 1.
4. Predict 100 squares, then solve 3n + 1 = 100 to find which row uses exactly 100 toothpicks.
5. Build a triangle pattern and a hexagon pattern and find their rules too.

How you'll know: Your rule gives the right count for a row you build but didn't use to make the rule, and you can point to the '3' and the '+ 1' in the picture.

Safety: Toothpicks are sharp: keep them on the table and away from faces.

Go further: A fence with posts and three rails between each pair of posts: find the rule for posts and for rails, and work out the materials for a 20-section garden fence.

You need: 10 identical plastic or paper cups · a ruler · table template

1. Measure the height of one cup, then stacks of 2, 3, 4, 5 and 6 cups, in millimetres.
2. Find how much each extra cup adds (the rim). Write a rule: height = first cup + rim × (number of cups − 1).
3. Predict the height of 50 cups.
4. Measure the height under a table and use your rule to predict how many cups would fit. Check with a real stack if you have enough cups.

How you'll know: Your prediction for a stack you haven't measured is within a few millimetres of the real height.

Go further: Use a different kind of cup and compare the two rules. Which number in the rule changes, and why? When will the two stacks be the same height?

You need: square tiles and multilink cubes · grid paper or Desmos

1. Build squares of side 1 to 5 and cubes of side 1 to 4. Record n, n² and n³ in a table.
2. Build staircases of 1, 2, 3, 4 and 5 steps (triangular numbers 1, 3, 6, 10, 15).
3. Put two copies of the same staircase together to make a rectangle. What are its sides? Use this to explain n(n + 1) ÷ 2.
4. Plot all three sets of numbers on one graph and describe the curves.

How you'll know: You can explain, using the rectangle, why the tenth triangular number is 10 × 11 ÷ 2 = 55 without building it.

Go further: Add two neighbouring triangular numbers (1 + 3, 3 + 6, 6 + 10…). What do you get every time? Show why with tiles.

You need: ruler · protractor · paper · scissors

1. Cut out a paper triangle, tear off its three corners and line them up along a straight edge. What do they make?
2. Draw a quadrilateral, pentagon and hexagon. Split each into triangles from one corner.
3. Count the triangles for each shape and multiply by 180°.
4. Find the rule for n sides and explain why it is n − 2 triangles.

How you'll know: Your rule works for a polygon you didn't draw, and you can explain the n − 2 using the diagram.

Safety: Use scissors carefully and pass them handles first.

Go further: Use the rule to find each angle of a regular 12-sided shape, then check that the exterior angles add up to 360°.

You need: compass or round objects to trace · ruler · coloured pencils

1. Draw five large circles. Put 2 points on the first, 3 on the second, 4 on the third and 5 on the fourth.
2. Join every pair of points with a straight line and count the regions inside the circle: 2, 4, 8, 16.
3. Predict how many regions 6 points will make.
4. Put 6 unevenly spaced points on the last circle, join every pair (make sure no three lines cross at one point) and count very carefully.

How you'll know: You get 31, not 32, and can explain in your own words why a pattern in the first few cases is evidence but not proof.

Go further: Count the regions for 7 points and research the real rule. Why does it look like doubling at first?

You need: groups of 2 to 6 people · table template · a real sports draw or fixture list if possible

1. Act it out: in groups of 2, 3, 4, 5 and 6, everyone shakes hands once with everyone else. Record the handshakes.
2. Find the rule. Hint: each of the n people shakes n − 1 hands, but each handshake gets counted twice.
3. Use it to predict the games in a round-robin league of 8 teams, and in one of 12 teams.
4. Check the 8-team prediction against a real fixture list or by drawing a table of who plays whom.

How you'll know: Your rule gives 28 games for 8 teams, matches the real fixture list, and you can explain the 'counted twice' part.

Go further: Explain why the handshake numbers are triangular numbers. If each game takes 40 minutes on one court, how long does a 12-team round robin take?

You need: calculator or spreadsheet · graph paper

1. Plan A: $5 a day. Plan B: 1 cent on day 1, 2 cents on day 2, 4 cents on day 3, doubling each day.
2. Make a table of the running total for each plan for 20 days.
3. Find the day when Plan B's total first beats Plan A's.
4. Graph both totals on the same axes. Describe the shape of each line.

How you'll know: You find that Plan A leads until day 12, but Plan B overtakes on day 13 and reaches over $10,000 by day 20, while Plan A has $100.

Go further: Write both rules with letters (5n dollars and 2ⁿ − 1 cents). Where does doubling happen in real life, and why can't it go on for ever?