24112020 Pythagoras MATHSWATCH CLIP 150 217 GRADE 4

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 L. O. To be able to § Find the missing side of a right-angled triangle, given the other two sides § Solve problems using Pythagoras’ theorem, such as finding the diagonal of a rectangle § Solve problems using Pythagoras’ theorem in 3 dimensions Key Words hypotenuse, right angle, square root, theorem Starter

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 CONTENTS – Click to go to… INTRODUCTION FINDING THE HYPOTENUSE FINDING ANY SIDE PROBLEM SOLVING PYTHAGORAS IN 3 D REVISION

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 INTRODUCTION

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 SQUARES AND SQUARE ROOTS On your mini whiteboards… Quick Questions!

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 2 Evaluate : 7

24/11/2020 Pythagoras x Square Area = 49 m 2 MATHSWATCH CLIP 150, 217 GRADE 4, 8/9

24/11/2020 Pythagoras 5 m Square Area = ? MATHSWATCH CLIP 150, 217 GRADE 4, 8/9

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Evaluate : 81

24/11/2020 Pythagoras x Square Area = 144 m 2 MATHSWATCH CLIP 150, 217 GRADE 4, 8/9

24/11/2020 Pythagoras 10 m Square Area = ? MATHSWATCH CLIP 150, 217 GRADE 4, 8/9

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 2 Evaluate : 1

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Evaluate : 400

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 1/2 Evaluate : 25

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Evaluate : 1

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Identifying the hypotenuse § A right-angled triangle contains a right angle § The side opposite the right angle is always the LONGEST § It has a special name - HYPOTENUSE hypotenuse

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9

MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Pythagoras 24/11/2020 Which side is the hypotenuse? f e a b c d g h i

MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Pythagoras 24/11/2020 Discovery task! § Make accurate constructions of the three triangles below 3 cm 6 cm 5 cm 4 cm 12 cm 8 cm § MEASURE accurately the missing length on each one. Copy and fill in this table… 3 4 5 12 6 8 9 § Can you spot the pattern that links the last three columns?

24/11/2020 Pythagoras proof activity From your worksheet… § Colour in square 5 one colour, then cut it off § Colour in the square with pieces 1 -4 a different colour, hen cut it off, also cutting along the dotted lines to get four shapes from that square § Try to arrange the five pieces you have so that they fit exactly into the larger bold square. Stick them down. § Write a sentence that describes what you have discovered from this activity “I have shown that…___________________________” MATHSWATCH CLIP 150, 217 GRADE 4, 8/9

24/11/2020 MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Pythagoras’ Theorem 2 a h b 2 a + b = h 2

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9

c b a a 2 X X X X + c 2 X b 2 X = X c 2

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Pythagorean trees! – Do you think you could draw one? !

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 FINDING THE HYPOTENUSE

MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Pythagoras 11/24/2020 Pythagoras’ Theorem – ALGEBRAIC METHOD § Learn the formula! § Substitute the two values that you know into the formula § Solve the equation to find the unknown value a h b 2 2 a + b = h 2

MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Pythagoras 24/11/2020 Example – Algebraic method 1 x 5 cm 12 cm 2 4. 6 cm x 9. 8 cm

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Pythagoras – QUICK METHOD § Decide if the side you are LOOKING FOR is the HYPOTENUSE, or one of the two SHORTER SIDES § Use one of these quick and easy methods to work out what you need Looking for the LONGEST SIDE? Looking for a SHORTER SIDE? (the hypotenuse!) (not the hypotenuse!) § Square it § Add it § Square root it

Pythagoras 24/11/2020 MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Example – Quick method 1 3 cm x 4 cm 2 4. 6 cm x 9. 8 cm

Pythagoras 24/11/2020 Task – Find the missing side, to 2 d. p. 1) a 10 cm b 2) 4 cm 6 cm 5 cm c 3) 5. 3 cm 8. 1 cm 4) d d = 9. 68 cm c = 7. 21 cm 7 cm 10. 1 cm 12 cm 8. 5 cm e b = 15. 62 cm 12 cm a = 7. 21 cm 5) MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 e = 8. 60 cm 6) 9. 7 cm f f = 15. 43 cm

24/11/2020 MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Pythagoras Task – Find the missing side, to 2 d. p. 9. 85 m 6 m 4 m 5) m 6) x 8. 2 14 c x m 9. 76 m m 7. 81 m x x 14. 5 m 9 m 16. 12 cm 8 cm x 11 m 4) 3) m 7 m x 2) 5. 3 1) 13. 04 m 16. 46 mm

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 FINDING ANY SIDE

MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Pythagoras 11/24/2020 Pythagoras’ Theorem – ALGEBRAIC METHOD § Learn the formula! § Substitute the two values that you know into the formula § Solve the equation to find the unknown value a h b 2 2 a + b = h 2

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9

24/11/2020 MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Pythagoras Example – Algebraic method x m 25 m 7 m

11/24/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Pythagoras – QUICK METHOD § Decide if the side you are LOOKING FOR is the HYPOTENUSE, or one of the two SHORTER SIDES § Use one of these quick and easy methods to work out what you need Looking for the LONGEST SIDE? Looking for a SHORTER SIDE? (the hypotenuse!) (not the hypotenuse!) § Square it § Add it § Square root it

Pythagoras 24/11/2020 MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Example – Quick method 1 x 15 cm 9 cm 2 7 cm x 5 cm

Pythagoras 24/11/2020 Task – Find the missing side, to 2 d. p. 1) 10 cm i 5. 2 cm 14. 5 cm 4) k 9. 6 cm k = 9. 15 cm j 8. 4 cm j = 23. 86 cm 25. 3 cm 13. 8 cm 2. 9 cm h = 11. 26 cm h i = 4. 45 cm 5) 12. 4 cm 2) g 4 cm g = 9. 17 cm 3) MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 6) 23 cm l 30 cm l = 19. 26 cm

MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Pythagoras Task – Find the missing sides, to 2. d. p 1) 2) 12. 69 m 15 m m 2 1 8 m 3) x x 8 mm m 4 2 9. 75 m x 4) 18 m 5 m 17. 29 m 22. 63 mm 7 m 5) x 6) 14 m 20 m 14. 28 m m m 2 m 5 x x 28 m m 24/11/2020 43. 82 mm

MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Pythagoras 24/11/2020 Task – Find the missing sides, to 2. d. p 1) 34 x 13 20 5) x 1. 8 4. 48 4. 1 9 27. 50 14. 32 4) x x 23 10. 44 x 6) 14 18. 25 14 6 5. 3 3) 2) 10 9. 80 x

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9

24/11/2020 Pythagoras Exam question – Spot the errors! MATHSWATCH CLIP 150, 217 GRADE 4, 8/9

11/24/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 PYTHAGORAS CATCH PHRASE! On your mini whiteboards… Choose a question to answer. Get it right to reveal part of the picture and guess the catchphrase!

The diagram shows a trapezium ABCD. Calculate the length of AB. 6. 71 cm 8 cm A C 7 cm B 10 cm 151. 99 miles Hull 200 miles 130 miles 20 cm Pythagoras j 5 cm 13 cm D Cardiff is 130 miles west of London. Hull is 200 miles north east of Cardiff. How far is Hull from London? Cardiff This rectangle has a diagonal of 13 cm and a width of 12 cm. Calculate it’s height. 12 cm 8. 54 m 16. 52 cm 8 m 23 cm d London 16 cm 3 m f b k 12 cm 15 cm 12. 73 m 9 m 13. 23 cm 8. 94 cm a MATHSWATCH CLIP 150, 217 GRADE c 9 m 4, 8/9 8 cm 6. 93 m 4 m 8 m

24/11/2020 BAD HAIR DAY! Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 PROBLEM SOLVING

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Problem solving with Pythagoras § Draw yourself a diagram which includes a right-angled triangle § Identify which side is the hypotenuse § Use Pythagoras in the normal way to find the missing side

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9

Pythagoras 24/11/2020 MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Examples 1 Find the diagonal of the rectangle d 6 cm 2 9. 3 cm A rectangle has a width of 4. 3 cm and a diagonal of 7. 8 cm. Find its perimeter. 4. 3 cm 7. 8 cm x cm Perimeter = 2(6. 5+4. 3) = 21. 6 cm

Pythagoras 24/11/2020 MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Example A boat sails due East from a Harbour (H), to a marker buoy (B), 15 miles away. At B the boat turns due South and sails for 6. 4 miles to a Lighthouse (L). It then returns to harbour. Make a sketch of the journey. What is the total distance travelled by the boat? H 15 miles B 6. 4 miles Total distance travelled = 21. 4 + 16. 3 = 37. 7 miles L

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Example A 12 ft ladder rests against the side of a house. The top of the ladder is 9. 5 ft from the floor. How far is the base of the ladder from the house? 12 ft 9. 5 ft L

11/24/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 PROBLEM SOLVING WITH PYTHAGORAS On your mini whiteboards… Quick Questions!

11/24/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9

11/24/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9

11/24/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9

11/24/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9

11/24/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9

11/24/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9

24/11/2020 MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Pythagoras Finding the distance between two points The points A(– 4, 5), B(– 4, – 2), and C (6, – 2) form a right angled triangle. A(– 4, 5) 7 units What is length of the line AB? AB = 5 – – 2 BC = 6 – – 4 = 5 + 2 = 6 + 4 = 7 units = 10 units y 5 4 3 2 1 x – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6 – 1 – 2 10 units B(– 4, – 2) C (6, – 2) – 3 – 4 – 5

24/11/2020 MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Pythagoras Finding the distance between two points The points A(– 4, 5), B(– 4, – 2), and C (6, – 2) form a right angled triangle. A(– 4, 5) 7 units What is length of the line AB? y 5 4 3 2 1 12. 21 units (to 2 d. p. ) x – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6 – 1 – 2 10 units B(– 4, – 2) C (6, – 2) – 3 – 4 – 5 Using Pythagoras, AC 2 = AB 2 + AC 2 = 72 + 102 = 49 + 100 = 149 AC = 149 = 12. 21 units (to 2 d. p. )

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Finding the distance between two points We can use Pythagoras’ Theorem to find the distance between any two points on a co-ordinate grid. Find the distance between the points A(7, 6) and B(3, – 2). We can add a third point at C to form a right-angled triangle. AB 2 = AC 2 + BC 2 = 82 + 42 = 64 + 16 = 80 AC = 80 = 8. 94 units (to 2 d. p. ) y 7 6 5 4 3 2 1 A(7, 6) – 2 – 1 0 1 2 3 4 5 6 7 x – 1 – 2 C (7, – 2) – 3 B(3, – 2)

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Task – Coordinate cards § Choose any TWO CARDS § Sketch them and join with a line. This is your hypotenuse. § Sketch in the horizontal and vertical sides to make a right angled triangle. § Use Pythagoras to find the length of the hypotenuse § Do as many as you can Difficulty Levels Blank Orange Green Red ( , ) coordinates in the first quadrant coordinates in the other three quadrants coordinates with decimal numbers make your own! POSITIVE ONLY POSITIVE AND NEGATIVE + AND - INCLUDING DECIMALS AS DIFFICULT AS YOU LIKE

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Finding the distance between two points - Algebraically What is the distance between two points with coordinates A(x. A, y. A) and B(x. B, y. B)? The horizontal distance between the points is The vertical distance between the points is x. B – x. A y. B – y. A Using Pythagoras’ Theorem the square of the distance between the points A(x. A, y. A) and B(x. B, y. B) is (x. B – x. A)2 + (y. B – y. A)2 The distance between the points A(x. A, y. A) and B(x. B, y. B) is (x. B – x. A)2 + (y. B – y. A)2

MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Pythagoras 24/11/2020 Challenge problem! Use Pythagoras’ theorem to find the length and width of the rectangle and hence find its perimeter and area in surd form. 1 2√ 10 3 Perimeter = √ 10 + 2√ 10 6 2 = 6√ 10 units √ 10 Area = √ 10 × 2√ 10 = 2 × 10 = 20 units 2

Pythagoras 24/11/2020 Find the diagonals of the kite 6 cm 5 cm x cm y cm 12 cm MATHSWATCH CLIP 150, 217 GRADE 4, 8/9

24/11/2020 MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Pythagoras Real life problem B An aircraft leaves RAF Waddington (W) and flies on a bearing of NW for 130 miles and lands at a another airfield (A). It then takes off and flies 170 miles on a bearing of NE to a Navigation Beacon (B). From (B) it returns directly to Waddington. Make a sketch of the flight. How far has the aircraft flown? 170 miles A 130 miles W

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 PYTHAGORAS IN 3 D

Pythagoras 24/11/2020 MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Example – Find the distance BG A B 3 cm C E D 13 cm G 5 cm H F 12 cm Find FG first FG 2 = 52 + 122 FG = (52 + 122) FG = 13 cm Use FG to find BG BG 2 = 32 + 132 FG = (32 + 132) FG = 13. 3 cm

Pythagoras 24/11/2020 MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Example – Find the distance BG A D E 10. 67 cm 5. 4 cm F B 3. 1 cm C 9. 2 cm Find EC first EC 2 = 5. 42 + 9. 22 EC = (5. 42 + 9. 22) EC = 10. 67 cm Use EC to find BG BE 2 = 3. 12 + 10. 672 BE = (3. 12 + 10. 672) BE = 11. 1 cm (1 dp)

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Problem solving My pencil case is in the shape of a cuboid. It measures 8 cm, by 7 cm by 5 cm. I want to put my new pencil into the pencil case. The pencil measures 12 cm. Does it fit?

24/11/2020 Pythagoras REVISION MATHSWATCH CLIP 150, 217 GRADE 4, 8/9

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 HYP Right Angled Triangle Topics (printable version) OTE NUS E! Area of the triangle • Questions involve a BASE and PERPENDICULAR HEIGHT • It will ask you for the area of the triangle • Use AREA = Pythagoras • Involves _____ sides, ____ given and ____ missing • If you want the LONGEST side (hypotenuse): square it, ADD it, then square-root it • If you want one of the SHORTER sides: square it, SUBTRACT it, then square-root it Trigonometry • Involves ____ sides and ____ angle • ____ of these things will be given and ____ will be missing O O A • Use SOH___TOA T × A S × H C × H

24/11/2020 Pythagoras MATHSWATCH CLIP 150, 217 GRADE 4, 8/9 Task – Complete the review worksheet of Pythagoras questions Click to reveal solutions… …and click again to reveal Qs 7 -10…