At a glance. One lesson of about 60 minutes. a² + b² = c² as areas; finding a hypotenuse and a shorter side; hidden right-angled triangles in rectangles and isosceles triangles.
Calculator: scientific calculator. Most answers are not whole numbers, and the order of the keys (square, add, then root) is part of the learning.
What students will learn
Know
- a² + b² = c² in a right-angled triangle.
- c is the hypotenuse, opposite the right angle and the longest side.
- A shorter side comes from c² − b².
Understand
- The theorem is a relationship between areas of squares, so it works with squares, not lengths.
- The theorem needs a right angle.
Do
- Find a hypotenuse or a shorter side.
- Find a hidden right-angled triangle in a rectangle, an isosceles triangle or a journey.
- Round to a stated accuracy.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: applying a² + b² = c² in routine problems (questions 2 to 5).
- Criterion B. Investigating patterns: with the figure, students find that doubling both legs doubles c, and that c is far below a + b.
- Criterion C. Communicating: the unknown is labelled, the working shows the squares, and the answer carries its unit. With the second figure, students explain in words why the two arrangements prove the theorem.
- Criterion D. Applying mathematics in real-life contexts: ladders, wires, screens and journeys (questions 5 to 7).
The one thing to do with the figure
Set a = 3 and b = 4, then ask for c before showing it. Many will say 7. Slide b to 8 and then back, and ask them to compare c with a + b every time.
Then set a = 1 and b = 8 and ask whether c can ever equal a + b. Do not move on until someone says that the shortest way between two points is a straight line, which is what the hypotenuse is.
Then use the second figure. Ask students to say, before pressing the second button, what must be true of the empty space and why. The key line is that the big square and the four triangles do not change.
A possible order
| 5 min | Warm-up, then answers. |
|---|---|
| 8 min | Hook: the 13 m ladder; take guesses and ask what they assumed. |
| 12 min | The figure with predictions; the rule in words. |
| 12 min | Worked examples; the tile 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 | 289; 12; the longest side, opposite the right angle. |
|---|---|
| Worked examples | Ladder: 12 m. Wire: 15 m. |
| Faded example | √193 = 13.9 cm. |
| 1. Setting up the equation | B. A is the c = a + b slip; C subtracts when the unknown is the hypotenuse. |
| 2. Finding a hypotenuse | 15 cm. |
| 3. Area of the square | 97 = 16 + 81. |
| 4. Finding a shorter side | 20 cm. |
| 5. Diagonal | 30 cm. |
| 6. Diagonal, 1 d.p. | 24.2 cm. |
| 7. Two legs from a journey | 39 m. |
| 8. Hidden triangle | 24 cm: 625 − 49 = 576. |
| 9. Choosing the operation | B. A is the add-always slip; C is the lengths-not-squares slip. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A in question 1; 21, 13, 42, 33 or 51 in questions 2, 3, 5, 6, 7 | Adds the lengths. | Ask: is the answer bigger than both legs and smaller than their sum? Use the figure. |
| 65 in question 3, 9 in question 6 or 21 in question 7 | Subtracts when the hypotenuse is the unknown. | Ask: is the unknown the longest side? Then the squares add. |
| 225, 900, 585 or 1521 in answers | Stops at c² and forgets the root. | Ask: what does 225 measure? Area. What do we want? Length. |
| 10 in question 4 | Subtracts the lengths 25 − 15. | Square first: 625 and 225. |
| 29.2 in question 4 | Adds when the hypotenuse is given. | Ask which side is the longest. |
| 11 or 26 in question 8 | Does not split the base, or adds. | Draw the height and mark the half-base 7. |
Questions worth asking
- Why can the hypotenuse never be longer than a + b? Listen for “the straight path is the shortest”.
- When do we subtract in Pythagoras' theorem? When the hypotenuse is known and a leg is missing.
- Does the theorem work if there is no right angle? No. Ask for a test with 3, 4 and 6.
Differentiation
A way in for anyone stuck
- Give a triangle template with c labelled and the right angle marked before the first question.
- Let students draw the squares on graph paper for 3, 4, 5 and count the unit squares.
- Give a sentence frame: “The unknown is the …, so I add / subtract.”
Stretch
- Find two different right-angled triangles with whole-number sides in which the hypotenuse is 1 more than a leg.
- A square has diagonal 10 cm. How long is its side? (7.07 cm.)
Students learning in English as an additional language
Draw a right-angled triangle on the board and label “hypotenuse”, “leg” and “right angle” before the lesson starts.
| Word | Plain meaning | Example |
|---|---|---|
| hypotenuse | the longest side of a right-angled triangle, opposite the right angle | 13 m in the ladder triangle |
| leg | one of the two shorter sides that meet at the right angle | 5 m along the ground |
| square (a number) | a number multiplied by itself | the square of 5 is 25 |
| square root | the number that squares to give a number | the square root of 144 is 12 |
| diagonal | a straight line joining opposite corners | the diagonal of a screen |
Sentence starters for explaining aloud or in writing:
- The hypotenuse is … because it is opposite …
- I add / subtract the squares because …
- My answer is sensible because it is longer than … and shorter than …
Two things not to say
“Root each side and add.” That gives a + b, and √(a² + b²) is not a + b: the squares are added first, then the root is taken.
“Plug in the numbers.” Students then add when they should subtract. Ask which side is the hypotenuse first.
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.
- HypotenuseLegs 6 cm and 8 cm. Find the hypotenuse.
10 cm. - LegHypotenuse 17 cm, one leg 8 cm. Find the other leg.
15 cm.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- RectangleA rectangular field is 40 m by 30 m. How long is the diagonal path?
50 m. - KiteA kite string is 26 m long and the kite is directly above a point 10 m from the holder. How high is the kite, ignoring the holder's height?
24 m.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningWithout calculating, which is longer, the hypotenuse of legs 7 and 11, or 7 + 11?
7 + 11 = 18 is longer: the hypotenuse is √170, about 13. - From Lesson 1Is “a² + b² = c² for every triangle” proved by testing 3, 4, 5? Why not?
No: testing cases only gives evidence, and the statement is only true for right-angled triangles.
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 four cases in words.
On paper: squared paper and coloured pencils reproduce the squares for a 3, 4, 5 triangle.