Pythagoras’ Theorem

Teacher’s Guide: Delivering the Pythagoras’ Theorem Lesson

As a youngster aiming to pass an exam, I was expected to recall a² + b² = c² with c as the longest side.

When I teach Pythagoras’ Theorem now, I try to deepen understanding by showing where the formula comes from. We start with experimentation, then move to algebraic proof.

Differentiated Learning Objectives

• 🟡 All students should be able to identify the hypotenuse of a right-angled triangle as the longest side.
• 🟢 Most students should be able to calculate the hypotenuse length in a right-angled triangle.
• 🟣 Some students should be able to calculate the hypotenuse length in an isosceles triangle when given its base and perpendicular height.

Finding the connection between the sides

In the video, I start with a dynamic visual: squares drawn on each side of a right-angled triangle.

As the vertices move, the areas of the two smaller squares sum exactly to the area of the square on the longest side. Students see combinations such as 9 + 9 = 18, 9 + 25 = 34 and 16 + 16 = 32 before we write the formula.

In class, groups discuss where the connection might lie. One group will usually investigate squares on the sides.

Using Pythagorean triples first, they recognise that the sum of the squares on the shorter sides equals the square on the hypotenuse. We then test non-integer sides. Measurement can be slightly inaccurate, but the pattern still holds.

At this stage, ask students to write Pythagoras’ Theorem in their own words. Also explain the difference between a demonstration and a proof.

a^{2} + b^{2} = c^{2}

The hypotenuse (c) is always opposite the right angle. It is always the longest side.

Step-by-step method for Pythagoras’ Theorem

1. Sketch the right-angled triangle and mark the right angle.
2. Label the hypotenuse (opposite the right angle) as c, and the other sides as a and b.
3. Substitute into a² + b² = c². Rearrange if a shorter side is missing.
4. Calculate carefully on a calculator, then round to the required accuracy.

Quick reference: finding the hypotenuse vs a shorter side

c = \sqrt{a^{2} + b^{2}}

a = \sqrt{c^{2} - b^{2}}

⚠️ Common mistakes

• ❌ Calling the wrong side the hypotenuse. It must be opposite the right angle.
• ❌ Adding instead of subtracting when a shorter side is missing. Use c² − b².
• ❌ Forgetting to square root after finding a² + b².
• ❌ Rounding too early. Keep full calculator values until the final step.

Proving Pythagoras’ Theorem

There are many proofs of Pythagoras’ Theorem.

My favourite classroom approach is to cut out four congruent right-angled triangles of base b and height a. Arrange them to form a square whose side length is the hypotenuse c.

Students find the area of the large square in two ways: as c², and as the four triangles plus the inner square. Equating the expressions proves a² + b² = c².

The key teaching point is that the formula does not simply “exist”. It comes from ideas that GCSE grade 4 and 5 students can follow.

💡 Teacher tips for this lesson

• Starter: show a right-angled triangle and ask students to point to the hypotenuse before any calculation.
• Support: begin with integer Pythagorean triples so the square-root step lands cleanly.
• Challenge: set composite perimeter problems where Pythagoras unlocks one missing edge first.
• Common fix: if students use a² + b² = c² with the wrong side as c, ask them to mark the right angle and draw an arrow to the opposite side.