1. Pythagoras – Finding C – Complete Lesson
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2. Starter
A task at the beginning of the lesson that reviews a skill required for the learning.
Knowledge Check
Questions to assess students’ current understanding and to consequently show progress.
Real-Life Example
A ‘hook’ to raise interest and provide a concrete example.
Demonstration
Slides for a teacher to lead students – didactically or via questioning – through a mathematical method.
AFL Questions
Assessment For Learning Questions, used to assess students’ competency for independent tasks/activities.
Plenary
An opportunity for students to prove/evaluate their learning.

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4. 15
Square
+10 ÷2
Square -6 x3 √ 9 ??
+55 √ ÷3

5. Answers
Use a calculator to find the answers.
Give your answers to 2 dp
42 =
2.852 = =
3.52 =
27 = =
47 = =
(42+52) =
50 =
39 =
(62−102) =
10 =

6. square root of a negative number!
Answers
Use a calculator to find the answers.
Give your answers to 2 dp
42 = 16
2.852 = 8.12
= 54.1
3.52 = 12.25
27 = ± 5.20
= 29
47 = ± 6.86
= 30.29
(42+52) = ± 6.40
50 = +7.07, − 7.07
39 = ± 6.24
(62−102) =
−64
But! You cannot have a
square root of a negative number!
10 = ± 3.16

7. 15 July 2019
Pythagoras’ Theorem

8. 𝑎 𝐶 𝐵 6 cm 4 cm 3 cm 4 cm 1) Find the length, 𝑎.
Give your answers to 2 dp.
KNOWLEDGE CHECK
1) Find the length, 𝑎. 𝐶 𝑎
6 cm
4 cm 𝐵
3 cm
2) Find the length, 𝐵𝐶.
4 cm
Previous knowledge check to see whether students can already complete the learning objective. If they can’t, this provides an excellent opportunity to show progress at the end of the lesson.
3) A football pitch is 97 m long and 42 m wide.
Bill walks from one corner to another around the outside.
Jane walks diagonally across the pitch.
How much further does Bill walk?

9. Give your answers to 2 dp.
KNOWLEDGE CHECK
1) Find the length, 𝑎. 𝐶
6.71 cm 𝑎
6 cm
4 cm 𝐵
3 cm
2) Find the length, 𝐵𝐶.
4 cm
14.42 cm
Previous knowledge check to see whether students can already complete the learning objective. If they can’t, this provides an excellent opportunity to show progress at the end of the lesson.
3) A football pitch is 97 m long and 42 m wide.
Bill walks from one corner to another around the outside.
Jane walks diagonally across the pitch.
How much further does Bill walk?
33.30 m

10. 4 m Jane’s cat is stuck on top of a wall!
How long does a ladder need to be
to reach the cat safely?
4 m

11. 4 m Jane’s cat is stuck on top of a wall!
How long does a ladder need to be
to reach the cat safely?
4 m
What shape has been made?
What angle is here?
Is the ladder longer than 4 m?

12. A Greek philosopher who taught students about religion and politics,
Pythagoras of Samos
c. 570 – c. 495 BC
A Greek philosopher who taught
students about religion and politics,
and made mathematical discoveries.

13. 5cm 3cm 4cm 3cm Area = 9 cm2 Area = 16 cm2 5cm Area = 25 cm2 4cm
Pythagoras’ Theorem
(only for right-angled triangles)
3cm
Area =
9 cm2
5cm
Area =
16 cm2
3cm
5cm
Area =
25 cm2
4cm
4cm

14. c is always the longest side.
Pythagoras’ Theorem
(only for right-angled triangles)
Area =
9 cm2 c a
5cm
Area =
16 cm2
3cm
Area =
25 cm2
4cm b a2
+ b2
= c2
a & b are the shorter sides.
c is always the longest side.
(the hypotenuse)

15. a2 + b2 = c2 Perigal’s Dissection (1891)
A proof of Pythagoras’ Theorem. a c b
a2 + b2 = c2 15

16. Pythagoras’ Theorem Here is Triangle 1.
It is a right-angled triangle with
sides: 3 cm, 4 cm and 5 cm.
(not to scale)
Pythagoras’ Theorem
We can draw a square on each
side of the triangle.
5 cm
3 cm C
4 cm A
What is the area of square A? ___________
What is the area of square B? ___________
What is the area of square C? ___________ B
Continue the investigation…
Triangle 2
Area (cm2)
Triangle
Square A
Square B
Square C 1 2 3 4 5
13 cm
5 cm
12 cm
8 cm
Triangle 3
15 cm
17 cm
Triangle 5: sides of 12 cm, 35 cm and 37 cm.
Conclusion:
What is the relationship
between the length of the sides
of a right-angled triangle?
Triangle 4
25 cm
7 cm
24 cm

17. Answers 9 cm2 16 cm2 25 cm2 Pythagoras’ Theorem 9 16 25 144 169 64 225
Here is Triangle 1.
It is a right-angled triangle with
sides: 3 cm, 4 cm and 5 cm.
(not to scale)
Pythagoras’ Theorem
We can draw a square on each
side of the triangle.
5 cm
3 cm C
4 cm A
9 cm2
What is the area of square A? ___________
What is the area of square B? ___________
What is the area of square C? ___________
16 cm2
25 cm2 B
Continue the investigation…
Triangle 2
Area (cm2)
Triangle
Square A
Square B
Square C 1 9 16 25 2
144
169 3 64
225
289 4 49
576
625 5
1225
1369
13 cm
5 cm
12 cm
8 cm
Triangle 3
15 cm
17 cm
Triangle 5: sides of 12 cm, 35 cm and 37 cm.
Conclusion:
What is the relationship
between the length of the sides
of a right-angled triangle?
Triangle 4
Answers
25 cm
7 cm
24 cm

18. Pythagoras’ Theorem Pythagoras’ Theorem Here is Triangle 1.
It is a right-angled triangle with
sides: 3 cm, 4 cm and 5 cm.
(not to scale)
Pythagoras’ Theorem
Here is Triangle 1.
It is a right-angled triangle with
sides: 3 cm, 4 cm and 5 cm.
(not to scale)
Pythagoras’ Theorem
We can draw a square on each
side of the triangle.
We can draw a square on each
side of the triangle.
5 cm
5 cm
3 cm
3 cm C C
4 cm
4 cm A A
What is the area of square A? ___________
What is the area of square B? ___________
What is the area of square C? ___________
What is the area of square A? ___________
What is the area of square B? ___________
What is the area of square C? ___________ B B
Continue the investigation…
Continue the investigation…
Triangle 2
Triangle 2
Area (cm2)
Triangle
Square A
Square B
Square C 1 2 3 4 5
Area (cm2)
Triangle
Square A
Square B
Square C 1 2 3 4 5
13 cm
13 cm
5 cm
5 cm
12 cm
12 cm
8 cm
8 cm
Triangle 3
Triangle 3
15 cm
17 cm
15 cm
17 cm
Triangle 5: sides of 12 cm, 35 cm and 37 cm.
Triangle 5: sides of 12 cm, 35 cm and 37 cm.
Conclusion:
What is the relationship
between the length of the sides
of a right-angled triangle?
Conclusion:
What is the relationship
between the length of the sides
of a right-angled triangle?
Triangle 4
Triangle 4
25 cm
25 cm
7 cm
7 cm
24 cm
24 cm

20. R K T O H E P D C P N W A G A E Q D P S T A G E L S P C O B F N M
Write down the letter of each hypotenuse.
(They are only in right-angled triangles!)
Rearrange the letters to get an animal. K T O H E P D C P N W A G A E Q D P S T A G E L S P C O B F N M

21. R K T O H E P D C P N W A G A E Q D Elephant! P S T A G E L S P C O B
Write down the letter of each hypotenuse.
(They are only in right-angled triangles!)
Rearrange the letters to get an animal. K T O H E P D C P N W A G A E Q D
Elephant! P S T A G E L S P C O B F N M

22. a 𝑥 c 4 cm 5 cm b a2 + b2 = c2 42 + 52 = 41 c2 = 41 c = 41 = 6.40 cm
Example 1
1) Identify the hypotenuse
and label the sides.
What is the length of 𝑥?
Give your answer to 2 d.p.
2) Substitute the lengths
into the formula. a 𝑥 c
4 cm
3) Find c2
5 cm
4) Square root to find c.
Not to scale b
a2 + b2 = c2 42
+ 52
= 41
c2 = 41
We can leave the answer as
a surd: which is exact, or
round the number to 2 decimal places.
c = 41
= 6.40 cm
(2dp)

23. 𝑥 a c 4 cm 7 cm b a2 + b2 = c2 42 + 72 = 65 c2 = 65 c = 65 = 8.06 cm
Example 2
What is the length of 𝑥?
Give your answer to 2 d.p. 𝑥 a c
4 cm
7 cm
Not to scale b
a2 + b2 = c2 42
+ 72
= 65
c2 = 65
c = 65
= 8.06 cm
(2dp)

24. 𝑥 a c a 𝑥 c 4 cm 3 cm 7 cm 9 cm b b a2 + b2 = c2 a2 + b2 = c2 42 + 72
Example 2
Your Turn
What is the length of 𝑥?
Give your answer to 2 d.p.
What is the length of 𝑥?
Give your answer to 2 d.p. 𝑥 a c a 𝑥 c
4 cm
3 cm
7 cm
9 cm
Not to scale b
Not to scale b
a2 + b2 = c2
a2 + b2 = c2 42
+ 72
= 65 32
+ 92
= 90
c2 = 65
c2 = 90
c = 65
c = 90
= 8.06 cm
= 9.49 cm
(2dp)
(2dp)

25. 𝑥 a a c c 𝑥 4 cm 5 cm 7 cm 3 cm b b a2 + b2 = c2 a2 + b2 = c2 42 + 72
Example 2
Your Turn
What is the length of 𝑥?
Give your answer to 2 d.p.
What is the length of 𝑥?
Give your answer to 2 d.p. 𝑥 a a c c 𝑥
4 cm
5 cm
7 cm
3 cm
Not to scale b
Not to scale b
a2 + b2 = c2
a2 + b2 = c2 42
+ 72
= 65 52
+ 32
= 34
c2 = 65
c2 = 34
c = 65
c = 34
= 8.06 cm
= 5.83 cm
(2dp)
(2dp)

26. 𝑥 a a 5 cm c 8 cm 4 cm b 𝑥 7 cm c b a2 + b2 = c2 a2 + b2 = c2 42 + 72
Example 2
Your Turn
What is the length of 𝑥?
Give your answer to 2 d.p.
What is the length of 𝑥?
Give your answer to 2 d.p. 𝑥 a a
5 cm c
8 cm
4 cm b 𝑥
7 cm c
Not to scale b
Not to scale
a2 + b2 = c2
a2 + b2 = c2 42
+ 72
= 65 82
+ 52
= 89
c2 = 65
c2 = 89
c = 65
c = 89
= 8.06 cm
= 9.43 cm
(2dp)
(2dp)

27. Your Turn: Find the missing length (𝑥)
Example 2
Your Turn: Find the missing length (𝑥)
Give your answer to 2 dp
What is the length of 𝑥?
Give your answer to 2 d.p. A B 𝑥 𝑥 𝑥 a
5 cm
3 cm c
4 cm
7 cm
6 cm C
7 cm D
11.5 cm
Not to scale b 𝑥
4.5 cm
a2 + b2 = c2 𝑥
4 cm
3.5 cm 42
+ 72
= 65 𝑥 E
c2 = 65
6.4 m
c = 65
= 8.06 cm
6 m
(2dp)
7 m

28. Your Turn: Find the missing length (𝑥)
Example 2
Your Turn: Find the missing length (𝑥)
Give your answer to 2 dp
What is the length of 𝑥?
Give your answer to 2 d.p. A B
𝑥 = 7.81 cm 𝑥
𝑥 = 7.62 cm 𝑥 𝑥 a
5 cm
3 cm c
4 cm
7 cm
6 cm C
7 cm D
11.5 cm
Not to scale b
𝑥 = 5.70 cm 𝑥
4.5 cm
a2 + b2 = c2 𝑥
𝑥 = cm
4 cm
3.5 cm 42
+ 72
= 65 𝑥
𝑥 = cm E
c2 = 65
6.4 m
c = 65
= 8.06 cm
6 m
(2dp)
7 m

29. Using Pythagoras’ Theorem to Find the Hypotenuse
Remember!
a2 + b2 = c2
1) Find the length of 𝑥 for each triangle. Answer to 2dp. a) b) c)
10 cm 𝑥
6 cm
4 cm 𝑥 𝑥
6.5 cm
9 cm
8 cm
𝑥 = ______
𝑥 = ______
𝑥 = ______ d) e) f)
7.5 m 𝑥
8.3 cm
5 m
6.1 cm 𝑥
2.2 cm 𝑥
𝑥 = ______
𝑥 = ______
𝑥 = ______
2) Find the length of 𝑥 for each triangle but don’t use a calculator!
Keep your answer as a surd instead. a) b) c) 𝑥
5 cm
8 cm
4 cm 𝑥
7 cm
5 cm 𝑥
2 cm
𝑥 = ______
𝑥 = ______
𝑥 = ______
3) Sketch a diagram to help answer these questions. Answer to 2sf.
a) From his car, Mike walks 5 km north, and then 7 km east.
How far away is he from his car now?
________
b) A boat sails directly south for 20 km. Then the boat sails west for 35 km.
How far is the boat away from where it started?
________
c) A field is 100 m long and 75 m wide. Jack walks from one corner to
another around the outside. Jane walks directly across the field.
How much further does Jack walk?
________

30. Answers 8.94 cm 10.82 cm 11.93 cm 9.01 m 6.48 cm 11.74 cm 20 cm 74 cm
Using Pythagoras’ Theorem to Find the Hypotenuse
Remember!
a2 + b2 = c2
1) Find the length of 𝑥 for each triangle. Answer to 2dp. a) b) c)
10 cm 𝑥
6 cm
4 cm 𝑥 𝑥
6.5 cm
9 cm
8 cm
10.82 cm
8.94 cm
11.93 cm
9.01 m
6.48 cm
11.74 cm
20 cm
74 cm
89 cm
8.6 km
40 km
50 m
𝑥 = ______
𝑥 = ______
𝑥 = ______ d) e) f)
7.5 m 𝑥
8.3 cm
5 m
6.1 cm 𝑥
2.2 cm 𝑥
𝑥 = ______
𝑥 = ______
𝑥 = ______
2) Find the length of 𝑥 for each triangle but don’t use a calculator!
Keep your answer as a surd instead. a) b) c) 𝑥
5 cm
8 cm
4 cm 𝑥
7 cm
5 cm 𝑥
2 cm
𝑥 = ______
𝑥 = ______
𝑥 = ______
3) Sketch a diagram to help answer these questions. Answer to 2sf.
a) From his car, Mike walks 5 km north, and then 7 km east.
How far away is he from his car now?
________
b) A boat sails directly south for 20 km. Then the boat sails west for 35 km.
How far is the boat away from where it started?
Answers
________
c) A field is 100 m long and 75 m wide. Jack walks from one corner to
another around the outside. Jane walks directly across the field.
How much further does Jack walk?
________

33. 𝑎 𝐶 𝐵 6 cm 4 cm 3 cm 4 cm 1) Find the length, 𝑎.
Give your answers to 2 dp.
KNOWLEDGE CHECK
1) Find the length, 𝑎. 𝐶 𝑎
6 cm
4 cm 𝐵
3 cm
2) Find the length, 𝐵𝐶.
4 cm
Previous knowledge check to see whether students can already complete the learning objective. If they can’t, this provides an excellent opportunity to show progress at the end of the lesson.
3) A football pitch is 97 m long and 42 m wide.
Bill walks from one corner to another around the outside.
Jane walks diagonally across the pitch.
How much further does Bill walk?

34. Give your answers to 2 dp.
KNOWLEDGE CHECK
1) Find the length, 𝑎. 𝐶
6.71 cm 𝑎
6 cm
4 cm 𝐵
3 cm
2) Find the length, 𝐵𝐶.
4 cm
14.42 cm
Previous knowledge check to see whether students can already complete the learning objective. If they can’t, this provides an excellent opportunity to show progress at the end of the lesson.
3) A football pitch is 97 m long and 42 m wide.
Bill walks from one corner to another around the outside.
Jane walks diagonally across the pitch.
How much further does Bill walk?
33.30 m

35. What is special about these triangles?
4 m
12 m
3 m
5 m

36. 4 m 12 m 3 m 5 m What is special about these triangles?
These are called Pythagorean Triples
because they are right-angled triangles with all integer lengths.
Here are just a few others…
(3,4,5)
(5,12,13)
(7,24,25)
(8,15,17)
(9,40,41)
(11,60,61)
(12,35,37)
(13,84,85)
(15,112,113)
(20,21,29)

37. Check your success!
I can calculate the length of a hypotenuse using Pythagoras’ Theorem.
I can use Pythagoras’ Theorem
without a calculator.
I can answer real-life questions using
Pythagoras’ Theorem.

38. Check your success!
I can calculate the length of a hypotenuse using Pythagoras’ Theorem.
I can use Pythagoras’ Theorem
without a calculator.
I can answer real-life questions using
Pythagoras’ Theorem.

39. Write a text message to a friend describing…
What Pythagoras’
Theorem is.

40. tom@goteachmaths.co.uk Questions? Comments? Suggestions?
…or have you found a mistake!?
Any feedback would be appreciated .
Please feel free to