This unit asks how we know that something is true for every case, not just the ones we tried. It starts with conjectures: how to test them, how one counterexample ends a claim, and why a long run of successes still proves nothing. Then it shows how a letter such as n stands for every whole number, so that algebra can prove a pattern.
The middle of the unit turns geometry into numbers. The distance between two points comes from Pythagoras, the midpoint is an average, and gradient measures steepness, with parallel lines sharing a gradient and perpendicular gradients multiplying to −1. A straight line can then be written in three forms, and each is quickest for a different fact.
The last lesson puts these tools to work as proof: squared lengths, midpoints and gradients decide whether a triangle is isosceles or right-angled and whether a quadrilateral is a parallelogram.
A graphic display calculator (GDC) is optional all the way through. Lessons 1, 3 and 6 have a closed “With a GDC” section to open if you have one; everything else is done by hand.
The lessons, in order
- Lesson 1 · Conjectures and counterexamples Conjectures, testing cases, one counterexample disproves a general statement, evidence is not proof, and predicting a general rule from a pattern. Step a value of n along a strip of tested cases and see when an expression that gave primes stops. GDC optional
- Lesson 2 · Proving with algebra Writing even, odd and consecutive numbers with a letter, expanding and simplifying, and showing a factor to prove a statement about every whole number. Move three equally spaced numbers along a number line and see their total stay three times the middle one. No calculator
- Lesson 3 · Distance and midpoint The distance between two points from Pythagoras, the midpoint as the average of the coordinates, and finding a missing end point from a midpoint. Drag two points on a grid and see the right-angled triangle, the distance squared and the midpoint. GDC optional
- Lesson 4 · Gradient, parallel and perpendicular lines Gradient as rise over run (fraction, decimal and percentage), horizontal and vertical lines, parallel lines and perpendicular lines with m₁ × m₂ = −1. Join two points and read the gradient, then turn a gradient triangle through a right angle and read the gradient of the new line. No calculator
- Lesson 5 · Equations of straight lines Gradient-intercept, point-gradient and standard form of a straight line, finding an equation from two points, from a point and a gradient, and from a parallel or perpendicular line. Set a gradient and a y-intercept and see the same line written in three forms. No calculator
- Lesson 6 · Proving shape properties with coordinates Using squared distance to show equal sides and right angles, midpoints to show that diagonals bisect each other, gradients for parallel and perpendicular sides, and equidistant points. Move the apex of a triangle on a grid and test it with squared side lengths. GDC optional
Every lesson has a teacher page with the answers, the misconceptions each wrong answer reveals, discussion questions, differentiation and support for students learning in English, and extra questions in three tiers.