Pythagoras' theorem

Classroom resources

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.

• Slides Download Copy address for Prepare Present from the browser, or export to PowerPoint or PDF.
• Worksheet Download Copy address for Prepare Print the student copy; the Teacher edition has the answers.

Students will be able to

• I can find the hypotenuse of a right-angled triangle.
• I can explain Pythagoras' theorem with squares on the sides.
• I can check that an answer is sensible (longer than each short side, shorter than both together).

Timing

Predict and Run 8 min · Investigate 12 min · Modify 10 min · Practise and Make 10 min

Before class

Do the Run board yourself first. Students need squares and square roots (Number Foundations) and right-angled triangles (Classifying triangles).

The one idea: the areas of the squares on the two shorter sides add to the area of the square on the hypotenuse. The common mistake is adding the sides (Leah's mistake). The sense check (between the longer short side and the sum) catches most slips.

This is the curriculum's Year 8 content; it is placed here, after circles, because it needs Year 7 squares and roots and triangle names.

Running the lesson

1. Predict: take a vote; students who chose 7 added; 25 forgot the root.
2. Run: count the small squares in the 9 and 16 and see the 25 together.
3. Investigate: write 'square, add, root' on the board.
4. Modify: for √2, ask between which whole numbers it lies before computing.
5. Make: ask whether 6-8-10 is new or a scaled 3-4-5.

The hard part (on purpose)

Keeping the order square, add, root. Ask students to say the three steps before they touch a calculator, and to check the answer against the two shorter sides.

When students get stuck

Answers to the checks

1. sides 3 and 4: A right-angled triangle has sides 3 cm and 4 cm at the right angle. How long is the hypotenuse?
   5 cm. 3² + 4² = 9 + 16 = 25 = 5². Adding the sides (7) is too long: the straight route across is shorter than going round the corner.
2. squares on the sides: Make a = 3 and b = 4. Do the two smaller squares add up to the biggest? Was your prediction right?
   
   9 + 16 = 25: the biggest square, so c = 5 cm.
3. Square, add, root: A right-angled triangle has sides 12 cm and 16 cm at the right angle. How long is the hypotenuse, in cm?
   20
   12² + 16² = 144 + 256 = 400, so c = √400 = 20 cm.
4. Find the mistake: Leah finds the hypotenuse. Click the line where her method goes wrong.
   line 2
   Line 2: it is the squares that add: 36 + 64 = 100, so c = 10 cm. 14 cm would be longer than going round the corner.
5. The hypotenuse: Which side of a right-angled triangle is the hypotenuse?
   The side opposite the right angle, the longest
   The hypotenuse faces the right angle and is always the longest side.
6. 5 and 12: Make a right-angled triangle with sides 5 and 12 at the right angle. How long is the hypotenuse?
   
   25 + 144 = 169 = 13²: c = 13.
7. 9 and 12: The sides at the right angle are 9 m and 12 m. How long is the hypotenuse, in m?
   15
   81 + 144 = 225, √225 = 15 m.
8. Not a whole number: The sides at the right angle are both 1 m. How long is the hypotenuse, in m, to two decimal places?
   1.41
   1 + 1 = 2, so c = √2 ≈ 1.41 m: between 1 and 2, as it must be.
9. Always the longest: Edge case: why is the hypotenuse always the longest side?
   c² = a² + b² is bigger than a² and bigger than b²
   c² is a² plus something, so c² > a², and c > a; the same for b.
10. 8 and 15: The sides at the right angle are 8 cm and 15 cm. Hypotenuse, in cm?
    17
    64 + 225 = 289, √289 = 17 cm.
11. A decimal: The sides at the right angle are 2 cm and 3 cm. Hypotenuse, in cm, to two decimal places?
    3.61
    4 + 9 = 13, √13 ≈ 3.61 cm.
12. A shortcut across: A park is a rectangle 30 m by 40 m. How long is the path straight across, corner to corner, in m?
    50
    The diagonal is a hypotenuse: 900 + 1600 = 2500, √2500 = 50 m.
13. Review: roots: Review: work out $\sqrt{144}$.
    12
    12 × 12 = 144.
14. a hypotenuse of 10: Make a right-angled triangle with whole-number sides whose hypotenuse is exactly 10.
    
    6 and 8: 36 + 64 = 100.

Solutions

One possible solution for each task. Other correct programs are fine.

one change at a time › 5 and 12

Open task: any program that does what the step asks.

one change at a time › 9 and 12

Open task: any program that does what the step asks.

one change at a time › Not a whole number

Open task: any program that does what the step asks.

one change at a time › Always the longest

Open task: any program that does what the step asks.

a hypotenuse of 10

Open task: any program that does what the step asks.

Stretch and support

Stretch: Find three more whole-number triangles (Pythagorean triples) that are not multiples of 3-4-5.

Support: Use squared paper for the squares; keep a list of squares to 20² visible.

Coach prompt (optional)

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 "Pythagoras' theorem".

The idea: Understanding and stating Pythagoras' theorem: I can Students state Pythagoras' theorem (a² + b² = c²) and understand it as a relationship between areas of squares on the sides of a right-angled triangle.

What understanding looks like:
- States Pythagoras' theorem correctly and identifies the hypotenuse.
- Verifies the theorem for at least one specific case.
- States the converse: if a² + b² = c² then the triangle is right-angled.

Mistakes students often make here:
- A student adds the sides (Leah's mistake).
- A student stops at c² (25 for 3 and 4).
- A student cannot tell which side is the hypotenuse.

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.
- 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 "Pythagoras' theorem".

Exit ticket

A right-angled triangle has sides 10 cm and 24 cm at the right angle. Find the hypotenuse.

100 + 576 = 676, √676 = 26 cm.

What each step is for

• sides 3 and 4: Committing to a hypotenuse for legs 3 and 4 shows whether they add the sides (7), add the squares without rooting (25), or expect 5.
• squares on the sides: Building a 3-4-5 triangle with squares on its sides shows the two small squares' areas add exactly to the big one.
• square, add, root: Square both shorter sides, add, then root; Leah adds the sides, which is always too long. The hypotenuse is opposite the right angle.
• one change at a time: One change at a time: another whole-number triangle on the board, a new pair of sides, a hypotenuse that is not a whole number, then why the hypotenuse is always the longest.
• the hypotenuse: Whole-number and decimal hypotenuses and a real context, mixed; adding the sides is the planted slip; the review keeps square roots fresh.
• a hypotenuse of 10: Working backwards from c = 10 means finding two squares that add to 100: a search, not a formula.

Where it leads

Next: Finding a side (node 2), including a shorter side.

Where it leads: distance between points on a graph, trigonometry (Year 10), and 3D diagonals.