How tall is a tree, or how wide is a river, when you cannot reach across? This unit builds the tools that answer it, and starts by asking what it means to be sure. A pattern that works many times can still be false, so the first lesson separates evidence from proof. Then the right-angled triangle does most of the work: Pythagoras' theorem and its converse, the same idea on a coordinate grid, similar triangles with a scale factor, and finally the sine, cosine and tangent ratios that turn an angle into a length.
Lessons 2 to 4, 6 and 7 use a scientific calculator, with a note on the page about the keys.
The lessons, in order
- Lesson 1 · Conjecture and proof Conjecture, counter-example, axiom and theorem; testing cases versus proving for every case. Slide n along a row of tests and watch a formula that looks like a prime-maker. No calculator
- Lesson 2 · Pythagoras' theorem The theorem a squared plus b squared equals c squared; finding a hypotenuse and a shorter side. Change two legs and watch the three squares and the length of the hypotenuse. Scientific calculator
- Lesson 3 · Is it a right angle? The converse The converse of the theorem, Pythagorean triples, and acute and obtuse triangles. Set three side lengths and see whether the corner opposite the longest side is a right angle. Scientific calculator
- Lesson 4 · Distance and midpoint The distance between two points as Pythagoras on a grid; the midpoint as an average of coordinates. Move two points on a grid and see the hidden right-angled triangle, the distance and the midpoint. Scientific calculator
- Lesson 5 · Similar triangles Similar triangles, scale factor, the AA, SSS and SAS conditions, and finding a missing length. Enlarge a triangle by a scale factor, then stretch one side and watch the ratios and angles disagree. No calculator
- Lesson 6 · Sine, cosine and tangent Naming opposite, adjacent and hypotenuse; the ratios sin, cos and tan; finding a missing side. Change an angle and a hypotenuse and watch which ratios move. Scientific calculator
- Lesson 7 · Finding an angle Finding an angle with inverse sin, cos and tan; angles of elevation and depression; choosing the ratio. Set the opposite and adjacent sides and see sin, cos and tan each give the same angle. Scientific calculator
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.