At a glance. One lesson of about 60 minutes. The converse, Pythagorean triples and their multiples, and acute and obtuse triangles by comparing a² + b² with c².
Calculator: scientific calculator. The test itself needs only squares and a comparison; the calculator is for the larger numbers.
What students will learn
Know
- If a² + b² = c² for the sides of a triangle then the angle opposite c is a right angle.
- A Pythagorean triple is a set of whole numbers satisfying a² + b² = c².
- More than c² gives an acute angle, less gives an obtuse one.
Understand
- The theorem and its converse are different statements.
- Equal means exactly equal; nearly equal does not give a right angle.
Do
- Decide whether a triangle is right-angled and justify with working.
- Find the third number in a triple.
- Scale a triple.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: applying the converse and the triple vocabulary (questions 2 to 6).
- Criterion B. Investigating patterns: students look for which numbers make right angles and notice that multiples of a triple also work.
- Criterion C. Communicating: the conclusion is stated in words, naming the angle and the longest side, and the converse argument (same three sides, so congruent) is explained.
- Criterion D. Applying mathematics in real-life contexts: the carpenter's check (question 6).
The one thing to do with the figure
Set 8, 15, 17 and ask for the verdict before reading it. Change the last side to 16, then 18, and ask what each move does to the corner.
Then ask for a triple using only numbers larger than 10. Let them try; 12, 16, 20 is a multiple of 3, 4, 5 and a good first discovery. Do not give the pattern yet.
Before the worked examples, ask students to say the converse argument back in their own words: same two short sides and a right angle give the same third side, so the triangle is the same.
A possible order
| 5 min | Warm-up, then answers. |
|---|---|
| 8 min | Hook: the rope; ask how to check with the theorem. |
| 12 min | The figure; right, acute and obtuse in words. |
| 12 min | Worked examples; 20, 21, 29 in pairs. |
| 5 min | Question 1, the check. |
| 13 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 10; the hypotenuse; 36 is bigger than 25. |
|---|---|
| Worked examples | 3, 4, 5: right angle. 7, 8, 11: not right-angled, the largest angle is obtuse. |
| Faded example | 841 = 841: right-angled. |
| 1. Applying the converse | B. A confuses “can make a triangle” with “right-angled”; C accepts nearly equal. |
| 2. Sum of squares | 1369 = 144 + 1225, and 37² = 1369. |
| 3. Choosing a right-angled set | B. 10, 24, 26 is 2 times the triple 5, 12, 13. |
| 4. Missing side of a triple | 41. |
| 5. Counting right-angled sets | 3: (9, 12, 15), (11, 60, 61), (16, 63, 65). |
| 6. Square corner | 150 cm. |
| 7. Scaling a triple | 6.5. |
| 8. Reasoning with a triple | 5: the sides are 5, 12, 13. |
| 9. Acute, right or obtuse | C. A is the nearly-equal slip; D reverses the triangle inequality. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A in question 1 | Treats the triangle inequality as the test. | Ask: could a long thin triangle with 6, 8, 11 have a 90 degree corner? |
| 47 or 420 in question 2 | Adds or multiplies the lengths. | Write each square on a separate line first. |
| 49 or 31 in question 4 | Adds or subtracts the lengths. | Square first, then ask which is the hypotenuse. |
| 4 or 5 in question 5 | Accepts a near miss. | Ask: is 130 equal to 144? |
| 12 in question 8 | Reports the wrong side. | Ask which side the question names x. |
Questions worth asking
- Why must we put the longest side on the c² side? Listen for “the right angle is opposite the longest side”.
- If 3, 4, 5 works, why does 6, 8, 10? Listen for “every side is doubled, so every square is four times bigger”.
- Does 3.1, 4.1, 5.1 make a right angle? No: adding the same amount changes the shape.
Differentiation
A way in for anyone stuck
- Provide a two-column table: shorter sides squared and added, and longest side squared.
- Give the first three triples 3, 4, 5; 5, 12, 13; 8, 15, 17 on a reference card.
- Let students lay out strings of 3, 4 and 5 units on a table and check with a set square.
Stretch
- Find a triple with 20 as one of the short sides (20, 21, 29 or 15, 20, 25).
- Why is 1, 1, 2 not a triangle? What does the figure say about 3, 4, 9?
Students learning in English as an additional language
Pair the words “acute” and “obtuse” with sketches of angles smaller and larger than a right angle.
| Word | Plain meaning | Example |
|---|---|---|
| converse | the statement turned around: if the squares match, then the angle is 90 degrees | the converse of Pythagoras' theorem |
| Pythagorean triple | three whole numbers a, b, c with a² + b² = c² | 5, 12, 13 |
| acute angle | an angle less than 90 degrees | the corner in a 6, 7, 9 triangle |
| obtuse angle | an angle between 90 and 180 degrees | the large corner in 7, 8, 11 |
| right angle | an angle of exactly 90 degrees | the corner of a door frame |
Sentence starters for explaining aloud or in writing:
- The longest side is … so c = …
- The two squares add to … and c² is …, so the angle is …
- It is not right-angled because … is not equal to …
Two things not to say
“It is nearly a right angle.” Use acute or obtuse and say which way the squares differ.
“The theorem works both ways, so use it without checking.” The converse is a separate statement and needs its own justification; the lesson gives one.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- TestIs a triangle with sides 9, 12 and 15 right-angled?
Yes: 81 + 144 = 225 = 15². - TestIs a triangle with sides 4, 5 and 6 right-angled?
No: 16 + 25 = 41, but 36 is less, so the largest angle is acute.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- GardenA gardener pegs a rectangle with sides 6 m and 8 m. What must each diagonal be to be sure the corners are square?
10 m. - TriplesWrite down a triple that is double 8, 15, 17.
16, 30, 34.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningFind a triple whose smallest number is 20 and largest is 29 (hint: Your go).
20, 21, 29. - From Lesson 2Hypotenuse 25 cm, one leg 7 cm. Find the other leg.
24 cm.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard and by touch; “Show me the steps” gives five cases in words.
On paper: a loop of string with 12 knots and a set square reproduces the hook.