Test it: predictions about points and lines

Mathematicians rarely start with a rule. They start with a question, try some examples, notice a pattern, make a guess (a conjecture), and then test it hard, hunting for a case that breaks it (a counterexample).

In this layer you will do exactly that with points and lines. How many lines can four points make? How many segments hide on a line with ten points marked on it? Can three lines cross in exactly two places? Does measuring with a divider really beat measuring with a ruler? You will need a pencil, a ruler, some scrap paper and, for one experiment, a friend.

Now allow some points to line up. With four points, three different things can happen:

• All four collinear: every pair gives the same line, so there is 1 line.
• Exactly three collinear: those three give 1 line between them, and the fourth point joins each of the other three with its own line: 1 + 3 = 4 lines.
• No three collinear: all 6 pairs give different lines: 6 lines.

Here is the surprise: 2, 3 and 5 are impossible for four points. A careful computer search over every way of placing four points on a small grid finds only the answers 1, 4 and 6.

Mark some points on a straight line. Every pair of marked points is the two ends of a segment. How many segments are there?

Start small. With 2 points A and B there is 1 segment. Add a third point C: now there are AB, BC and AC, which is 3. Keep going and record the results.

Look at the last column: +2, +3, +4, +5… Each new point makes a new segment with every point already there. The 5th point joins 4 old points, so it adds 4 segments.

The totals 1, 3, 6, 10, 15, 21, 28 are the triangular numbers, the same numbers you get stacking bowling pins or oranges in a triangle. There is a shortcut: each of the n points is the end of (n − 1) segments, giving n × (n − 1) ends, and every segment has 2 ends, so

segments = n × (n − 1) ÷ 2.

What about rays? Count only rays that start at a marked point and pass through at least one other marked point. With points in a row, the two end points each start 1 such ray (pointing inward), and every point in the middle starts 2 (one each way). So with n points there are 1 + 1 + 2 × (n − 2) = 2 × (n − 1) rays. With 5 points: 2 × 4 = 8 rays.

Segments hide inside figures, and counting them carefully is a classic puzzle. The trick is to be systematic: count the segments on each straight line of the figure using the n × (n − 1) ÷ 2 rule, then add up the lines.

With four lines there are 6 pairs, so at most 6 crossing points. But not every number from 0 to 6 is possible. A computer check of many random arrangements finds 0, 1, 3, 4, 5 and 6 crossing points, and never exactly 2. Try to see why 2 is impossible: if two lines cross, a third line that crosses either of them must add a new point unless it goes through the first crossing, and then the fourth line faces the same problem.

Here is an experiment for two or more people. Draw one segment, about 7 to 8 cm long, on a sheet of paper. Without telling each other your answers, everyone measures it three ways:

1. With a ruler, looking at it from the side (sitting low, eye near the desk).
2. With a ruler, eye directly above each end.
3. With a divider, then reading the divider on the ruler from directly above.

Write each result to the nearest millimetre. Then compare: which method gives readings closest to each other?

Comparing segments by observation feels natural, but the brain uses shortcuts that can fool it. Test these on yourself before you measure.

A statement in geometry can be always true, sometimes true (true in some cases and false in others) or never true. To show something is not always true, one counterexample is enough. To show something is always true, you need a reason that covers every case.

A sheet of paper is a surprisingly powerful geometry tool. Every fold is a perfectly straight line, and folding a crease onto itself makes an exact right angle. Try these folds on scrap paper; each one is an investigation.

Draw 3 parallel lines across a page, then 4 parallel lines crossing them at right angles. You have made a small grid, like a window with glass panes or the lines of a badminton court. How many crossing points are there? How many rectangles can you find, counting big ones as well as small ones?

Now count rectangles. Every rectangle in the grid is fixed by choosing 2 of the horizontal lines (its top and bottom) and 2 of the vertical lines (its left and right sides). So:

rectangles = (pairs of horizontal lines) × (pairs of vertical lines).

With 3 horizontal and 4 vertical lines: 3 pairs × 6 pairs = 18 rectangles, although only 6 small panes are visible.

The two hands of a clock are like two rays from the same end point, the centre. As they turn, they are sometimes perpendicular (at 90° to each other), sometimes opposite (forming one straight line through the centre) and sometimes on top of each other. How often does each happen?

Take a notebook round your school or home and record every example you can find of each relationship. Then look at the evidence: which kind is most common? Why do builders and designers use so many right angles and parallel lines?

Real line markings make great counting problems. To keep the counting clean, we use simplified models: every lengthwise line runs the full length and every crosswise line runs the full width, so each lengthwise line crosses each crosswise line exactly once. Real courts are close to this but not exact, so treat the numbers as answers about the model.