Students open and close the angle between two sides to see that c² = a² + b² only at 90°, then use the comparison to classify triangles from their side lengths as right-angled, acute or obtuse, and hunt for Pythagorean triples (AC9M8M06, node 3).
Load these into Waypoints Prepare: choose Load URL and paste the copied address, or download the file and import it. In Prepare, the worksheet's Teacher edition shows the answers; the student copy hides them.
Predict and Run 8 min · Investigate 12 min · Modify 10 min · Practise and Make 10 min
Do the Run board yourself first. Students need Pythagoras (previous two lessons) and acute, right and obtuse triangles (Classifying triangles).
The one idea: compare a² + b² with the longest side squared; equal means right-angled, and the bigger angle opposite c pushes c² up. The common mistake is testing with a shorter side as c (Ava's mistake).
The test is the converse of Pythagoras' theorem: if a² + b² = c², the angle opposite c is a right angle. It is a separate fact from the theorem (which starts from a right angle); the board's angle slider shows why it holds.
The board's angle slider keeps a and b fixed and works out c, so students see c² change continuously.
Remembering which way the inequality goes. Ask students to open the angle on the board and say what happened to c² before they use the rule, instead of giving them the table.
| What you will see | A question to ask | The fix |
|---|---|---|
| A student uses a shorter side as c (Ava's mistake). | Which side is opposite the biggest angle? | c is always the longest side. |
| A student mixes up acute and obtuse. | If the angle opens wider, does the opposite side get longer or shorter? | Wider angle, longer c, bigger c²: obtuse. |
| A student compares the sides, not the squares. | What does the theorem say: sides or squares? | Square all three first. |
One possible solution for each task. Other correct programs are fine.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Stretch: Euclid's recipe: for whole numbers m > n, the numbers m² − n², 2mn and m² + n² make a triple. Test it for m = 2, n = 1 and m = 3, n = 2.
Support: Use the decision table and a list of squares to 20²; check with the board.
For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.
You are a patient maths coach for a Year 8 student in Australia. Help me understand ONE idea only, from the lesson "Right, acute or obtuse?". The idea: Classifying triangles using side lengths: I can Students use the relationship between a² + b² and c² to classify triangles as right-angled, acute or obtuse. What understanding looks like: - Correctly classifies a triangle given three side lengths. - States the rule for each classification (right, acute, obtuse). - Identifies the longest side as c before applying the test. Mistakes students often make here: - A student uses a shorter side as c (Ava's mistake). - A student mixes up acute and obtuse. - A student compares the sides, not the squares. How to coach me (rules from the Waypoints Kit coach): - Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question. - Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit). - If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go. - If I seem to be missing the idea that comes before this one (Finding unknown side lengths using Pythagoras), check it quickly, then come back. - Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back. - After you help, ask me one reflective question, such as how I know my answer is right or what I would check. - Praise my effort and method, not my ability. - You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so. - Use simple words. - Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Right, acute or obtuse?".
Classify triangles with sides (a) 8, 15, 17 and (b) 4, 6, 8.
(a) 64 + 225 = 289 = 17²: right-angled. (b) 16 + 36 = 52 < 64: obtuse.
This completes AC9M8M06.
Where it leads: the converse of Pythagoras in proofs, the cosine rule (Year 11), and distances on the Cartesian plane.