S2 Maths | Pythagoras Theorem Lesson Prep Guide for Tutors | Six Triples, Three Question Types, 15-Question Diagnostic

2026-11-08

Tutor lesson-prep guides · Secondary maths

Students can recite the formula but freeze when asked which side is the hypotenuse. The whole lesson turns on one action: circle the side opposite the right angle first.

Objectives — Students can: (1) circle the hypotenuse in any orientation and justify it (opposite the right angle, longest side); (2) apply one formula to three question types: find the hypotenuse, find a leg, and test whether a triangle is right-angled; (3) recall six triples and score at least 11 of 15 in the diagnostic.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Squares and roots	What is 7 squared? What is the square root of 169?	49; 13
Angle sum of a triangle	What do the three angles add up to?	180 degrees
Right-angle mark	What does the small square in the diagram mean?	A 90 degree right angle

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-6 min	Predict three cases	Show 3-4-5, 5-12-13 and 6-8-10 and ask which form right triangles	Vote and write a reason
6-16 min	Circle the hypotenuse	Model three triangles in different orientations: the side opposite the right angle is c	Circle the hypotenuse in red
16-30 min	Three uses	Write a squared plus b squared equals c squared and model each type	Copy one worked example per type
30-42 min	Six triples	Board 3-4-5 up to 9-40-41 and show the same-family scaling to 6-8-10	Write the six and square them
42-56 min	15-question diagnostic	Five questions per type, marked immediately: 11 right and 4 wrong	Work independently, closed book
56-60 min	Exit check	Three oral questions; note which act to reteach	Answer orally

03 Teaching script (read it out)

Scene 1: The hypotenuse sits opposite the right angle

Start with the eyes, not the numbers: the side facing the small square is the hypotenuse, and it is also the longest side. Rotate the same triangle twice and ask again, so students see that the hypotenuse has nothing to do with being horizontal. Most wrong answers are not arithmetic errors, they are mislabelled sides. Only after circling the side do you write the formula.

Student task: circle the hypotenuse on three differently oriented triangles and say the reason out loud.

Scene 2: One formula, three uses

To find the hypotenuse, add the squares of the two legs and take the root: 81 plus 144 is 225, and the root of 225 is 15. To find a leg, subtract the square of the known leg from the square of the hypotenuse: 42.25 minus 6.25 is 36, and the root is 6. To test for a right angle, square all three sides and check whether the longest square equals the sum of the other two.

Student task: name which two sides are known, then state whether the unknown is the hypotenuse or a leg.

Scene 3: Take the root, then sanity-check

The most common lost mark is stopping at c squared: a student computes 225 and writes it as the answer. Second check: the hypotenuse must be longer than either leg. Third check: families. 3-4-5 doubled is 6-8-10 and still works, while 4-5-6 does not, because 16 plus 25 is 41 and not 36.

Student task: compute both square sums for 4-5-6 and 6-8-10 and say which one is right-angled.

04 Question types and worked steps

Type	Example	Steps	Answer
Find the hypotenuse	Legs 9 and 12	Add squares then root: 81 plus 144 is 225	15
Find a leg	Hypotenuse 6.5, leg 2.5	Subtract then root: 42.25 minus 6.25 is 36	6
Test for a right angle	Sides 6, 8, 10	Compare 100 with 64 plus 36	Right-angled
Real context	Ladder base 2.5 m, wall 6 m	Sketch, label the hypotenuse, then add and root	Ladder 6.5 m

Sanity check (teach this): Sanity checks: the hypotenuse is always longest; a squared answer is not the answer; scaling a triple keeps it valid; 4-5-6 is not right-angled because 41 is not 36; 11 of 15 is 73 percent, not 11 percent.

05 Independent practice (3 questions)

Question	Answer	What to watch
Three hypotenuses	9 and 12; 5 and 12; 7 and 24	Circles the hypotenuse first
Three legs	Hyp 13 leg 5; hyp 25 leg 7; hyp 41 leg 9	Uses subtraction
Two tests	6-8-10 and 4-5-6	Compares the longest square

06 Common mistakes and what to say

Mistake	Student writes	Say this
Wrong hypotenuse	Picks the longest side that looks horizontal	The hypotenuse depends on the right angle, not the orientation.
Stops at the square	Writes 225 as the answer	225 is c squared, so take the root: 15.
Adds when finding a leg	Uses 169 plus 25 to get 194	Adding is for the hypotenuse; here subtract to get 144, then 12.
Smallest side as c	Compares 36 plus 100 with 64	Always use the longest side as c: 100 equals 36 plus 64.

07 Homework (paper, section and question numbers)

Secondary maths chapter index — All chapters
Master prep index — Compare related chapters
Redo this chapter — Redo diagnostic questions 1 to 5, circling the hypotenuse first

08 Exit check (last 5 minutes)

Question	Answer	Decision
Where is the hypotenuse?	Opposite the right angle, longest side	Wrong answer: reteach act one
Hypotenuse 13, leg 5, other leg	12	Answer 18: reteach act two
Is 4-5-6 right-angled?	No, 41 is not 36	Answer yes: reteach act three

09 Tutor FAQ

Must students memorise six triples?

Memorise 3-4-5 first; the rest come from scaling and checking by squaring, so the reasoning is taught before the list.

Should answers keep a root sign?

When the root is not a whole number, keep the surd or round to two decimals, and stay consistent across the page.

How do students know when to subtract?

Look at what is given: two legs means add, while a hypotenuse plus one leg means subtract.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.