Pythagoras' Theorem

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• Math
• Secondary Education

Pythagoras' Theorem

Text

Use the opening image as the attention grabber instead of a call-and-response routine. Begin with a silent prompt such as: 'What do you notice about this triangle?' This supports a calmer start and reduces teacher talk.

Learning objective

I can apply Pythagoras' theorem to calculate a missing side in a right-angled triangle and justify my method.

This keeps the objective as a single student-friendly statement while using higher-order Bloom's verbs: apply, calculate, justify.

What words come to mind when you hear Pythagoras' theorem?

Use this straight after the learning objective to surface prior knowledge. Expect terms such as right angle, triangle, hypotenuse, square, theorem, formula, missing side. Briefly pick out correct mathematical language without turning it into a long explanation.

Success criteria

• Identify the hypotenuse in a right-angled triangle.

• Select the correct formula: $a^2 + b^2 = c^2$.

• Substitute values accurately into the formula.

• Evaluate the result to find the missing side.

• Justify each step using correct mathematical language.

Although the default structure asks for one learning-goal slide, the user explicitly requested success criteria, so include them. Keep this slide brief and visible during the lesson.

The key idea

In a right-angled triangle:

• the longest side is the hypotenuse

• the hypotenuse is always opposite the right angle

• Pythagoras' theorem links the three side lengths

$a^2 + b^2 = c^2$

Starter, around 3 minutes. Keep teacher input minimal. Ask pupils to spot the right angle and identify the longest side with a partner before taking brief answers. This increases student talk from the start.

What do the letters mean?

• a and b are the shorter sides

• c is the hypotenuse

• the formula only works for right-angled triangles

Address a common misconception: students may think c can be any side. Stress that c must be the hypotenuse. This is still part of the starter/build-up before modelling.

I do: one modelled example

If the shorter sides are 3 cm and 4 cm:

$3^2 + 4^2 = c^2$

$9 + 16 = c^2$

$25 = c^2$

$c = 5$ cm

Use this as the 5-minute teacher model. Keep explanation tight: name the hypotenuse, substitute, square, add, square root. Narrate briefly and avoid over-explaining. This is the main direct instruction segment.

Why the final step matters

The equation gives $c^2$, not $c$.

So after finding the total, take the square root to get the side length.

$25 = c^2$ means $c = 5$, not 25.

Flag the common error of stopping at c². This slide supports accuracy before pupils begin guided practice.

Which side must be labelled c?

A

the shortest side

B

the sloping side

C

the hypotenuse

D

either short side

Checks the key misconception from the starter and modelled example: c is always the hypotenuse, not just any side.

Complete the method

In a right-angled triangle, the longest side is the . Pythagoras' theorem is + = c². After adding the squares, take the to find the side length.

In a right-angled triangle, the longest side is the

. Pythagoras' theorem is

+

= c². After adding the squares, take the

perimeter

multiply

diameter

cube root

Correct answers in order: hypotenuse, a², b², square root. This reinforces vocabulary and the final-step error highlighted on the previous slides.

We do: guided example

Find the hypotenuse when the shorter sides are 6 cm and 8 cm.

$6^2 + 8^2 = c^2$

$36 + 64 = c^2$

$100 = c^2$

$c = 10$ cm

Use mini whiteboards or paired discussion for the 5-minute guided section. Ask pupils to complete each line before you reveal the next. This keeps student participation high.

Choosing the correct side

Before using the formula, always check:

• Where is the right angle?

• Which side is opposite it?

• Is that side labelled c?

If not, relabel first.

This slide prevents mislabelling errors during independent work. Encourage pupils to say the check aloud with a partner.

For sides 6 cm and 8 cm, what is c?

A

10 cm

B

14 cm

C

100 cm

D

2 cm

Tests whether pupils can follow the guided example and avoid confusing c with c² = 100.

How do you decide which side is c?

Expected points: identify the right angle first; the side opposite it is the hypotenuse; that side must be labelled c before substitution. Use answers to rehearse the checking routine aloud.

You do: independent practice

Use Pythagoras' theorem to find the missing side in right-angled triangles.

Remember:

• identify the hypotenuse first

• square the known sides

• add them

• square root at the end

This anchors the 15-minute independent phase. The worksheet or board questions can vary in difficulty, but the slide keeps the routine visible so pupils rely less on teacher support.

Check your working

A clear solution should show:

• the formula

• substitution of values

• the squared numbers

• the total

• the square root

• the final answer with units

Use this as a self-check scaffold during independent practice. It supports independence and reduces repeated teacher explanation.

Group task: error analysis

Worked answer A

$9^2 + 40^2 = c^2$

$81 + 1600 = c^2$

$1681 = c^2$

$c = 41$

Worked answer B

$5^2 + 12^2 = c^2$

$25 + 144 = 169$

$c = 169$

Use this for the group task. Pupils discuss which answer is correct and explain why. Answer A is incorrect because the square root step is missing. This structure promotes talk, reasoning and peer explanation.

Reasoning with Pythagoras

A correct method is more than an answer.

You should be able to explain why the side chosen is the hypotenuse and justify why a square root is needed at the end.

This slide raises the level of mathematical explanation and aligns with the requested higher-order language. Good point for circulating and listening rather than leading from the front.

Plenary: AI feedback check

Use an AI tool to compare your method with a model solution.

Check whether:

• your formula is correct

• your steps are complete

• your final answer is sensible

This satisfies the user's request for a plenary using AI. Keep it to about 5 minutes. If live AI is not available, simulate this with a prepared AI-style model response. Focus on pupils evaluating their own method rather than listening to another explanation.

Why is worked answer A incorrect?

A

Wrong formula

B

No units

C

Wrong hypotenuse

D

No square root

Anchored to the error-analysis group task. The error is stopping at 169 instead of finding c from c².

Complete the reasoning

Pythagoras' theorem can be used only in a triangle. First identify the , which is opposite the right angle. If the total is 169 = c², the missing side is because you must take the at the end.

Pythagoras' theorem can be used only in a

triangle. First identify the

, which is opposite the right angle. If the total is 169 = c², the missing side is

because you must take the

equilateral

shorter side

169

square

Correct answers in order: right-angled, hypotenuse, 13, square root. This brings together condition for use, side choice and the final operation.

Why can't we use this theorem for every triangle?

Look for reasoning such as: it only works in right-angled triangles; the theorem links the three sides in that special case; students should mention checking for the 90° angle before using the formula.

When can we use this theorem?

Pythagoras' theorem can be used only when:

• the triangle is right-angled

• two side lengths are known

• one side length is missing

It does not apply to every triangle.

Use this as the final mathematical takeaway before the glossary. It resolves the central idea by clarifying the condition for use.

What helped you understand this method best?

Metacognitive exit ticket. Invite short reflective answers about understanding, strategy or confidence.

Which step still feels hardest for you?

Metacognitive exit ticket. Useful for planning follow-up support on misconceptions such as choosing c or taking the square root.

Glossary

right-angled triangle - a triangle with one angle equal to 90°

hypotenuse - the longest side in a right-angled triangle, opposite the right angle

Pythagoras' theorem - the rule that states $a^2 + b^2 = c^2$ in a right-angled triangle

square number - a number multiplied by itself

square root - a value that gives a number when multiplied by itself

substitute - to replace letters with known numbers

justify - to explain clearly why a method or answer is correct