1 WelcomeWelcome The slides in this PowerPoint presentation are not in order of difficulty, and the show should be ordered to suit the needs of your target audience. Where appropriate, slides may be printed and used as paper based worksheets. Slides 15 & can be used as a workbook with this show. Some slides are intended to stand-alone as starters or plenary components to a lesson. They are not always intended to be sequential Show advances in response to mouse click.

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3 Pythagoras was born at Samos, in Greece, and lived from 580 to 500 B.C. How old did that make Pythagoras in 525 B.C. ? 580 – 525 = Pythagoras’s Theorem? Pythagoras would have been 55 years old.

4 The relationship we call Pythagoras Theorem was independently discovered in Egypt, China and India where similar pyramids had been built. Here similar may mean made from different materials, (the Chinese used wood and clay) or of a different design. In mathematics what do we mean by SIMILAR?

5 The relationship we call Pythagoras Theorem was independently discovered in Egypt, China and India where similar pyramids had been built. Similar: Shapes or objects where all corresponding angles are identical but where all lengths have been scaled up or down by the same amount. Here similar may mean made from different materials, (the Chinese used wood and clay) or of a different design.

6 By pressing 4 right- angled mosaic tiles into sand a square can be made a like this… The longest side of the triangle The HYPOTENUSE Is on the outside

7 The 4 right-angled tiles can now be re- arranged in this way….

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10 A large square whose edge is made from the second-longest edge of the tile…. If we cut this shape along the dotted line we are left with two squares… …and a small square whose edge is made from the smallest edge of the tile

11 These two composite shapes have the same area = HypotenuseShort side 1 Short side 2 What do we mean by COMPOSITE? Composite: A composite shape is one constructed or built using several smaller shapes.

is equal to the total area of the squares made by using the two smaller sides. 12 = Hypotenuse Short side 1 Short side 2 The area of the square made using the hypotenuse of the tile Rule:

13 The same tiles can be used in another tile pattern to show the same rule: Can you explain how?

14 The square frame of each illustration is the same size. Each frame contains 4 CONGRUENT right-angled triangles. The unused part in each frame, one large square or two smaller ones, must have the same area. What does the word CONGRUENT mean?

15 The square frame of each illustration is the same size. Each frame contains 4 CONGRUENT right-angled triangles. The unused part in each frame, one large square or two smaller ones, must have the same area. Congruent: Identical, exactly the same shape and size.

16 Chou Pei Here is a copy of an original print of Chou Pei’s work. Chou Pei lived in China around the same time as Pythagoras. He also noticed this relationship.

17 Arabic Mathematics Here is an explanation of the same relationship from an Arabic manuscript of 1258 A.D.

18 The Mayan civilisation in the jungles of South America were brilliant mathematicians who also built pyramids. Can you think why we might never find out? The right angles in their buildings are very accurate, but we don’t know yet if they had noticed this relationship.

19 The Mayan civilisation in the jungles of South America were brilliant mathematicians who also built pyramids. The right angles in their buildings are very accurate, but we don’t know yet if they had noticed this relationship. i.The Mayans used carvings rather than mosaics, so some geometrical patterns are less common. ii.In wet jungle environments documents written on wood or paper type fabrics rot and so don’t survive

20 A B C Looking at the squares around this right angled triangle we now know that area A + B = C

21 When we don’t know the area of a square we can calculate it from the length of it’s sides. Length  a Area a x a = a 2 If a = 3cm then area of square is 3cm x 3cm or 3 2 cm = 9cm 2

22 We can now use this relationship to calculate the length of any one side of a triangle if we know the other two. 3cm 4cm ?cm Area = 9cm 2 Area = 16cm 2 Area = 25 cm 2 To find the missing length we need to know what number when multiplied by itself equals x 5 = 25 So the missing side is 5 cm long.

23 Finding a number this way is called finding the SQUARE ROOT.  This is the symbol invented to mean find the square root.  36 = 6  25 =5  49 = 7  81 =9 Why is 3½ a good approximation for  12 ?

24 Finding a number this way is called finding the SQUARE ROOT.  This is the symbol invented to mean find the square root.  36 = 6  25 =5  49 = 7  81 =9 Since  9 = 3 and  16 =4  12 must be about half way between 3 & 4

25 Egg sample 1 5 cm 12 cm = = = 169  169 = 13 So the hypotenuse must be 13cm long Example 1. Calculate the length of the Hypotenuse

26 Egg sample 2 30 mm 50 mm 50 2 = = = 1600  1600= 40 The missing short side is 40mm long Example 2. Calculate the length of a short side

27 Exercises. On your worksheet answer the following timed questions GO

28 Calculate: 10 seconds each: 1) 4 x 4 = 2) 6 x 6 = 3) 7 x 7 = 4)13 x 13 = 5)15 x 15 = 6) 2·5 x 2·5 =      

29 10 seconds each: Calculate: 7)  64 = 8)  16 = 9)  196 = 10)  = 11)  6¼ = 12)  11 1/ 9 =

30 9 x 9 =12 x 12 = + =  = So hypotenuse iscm long 13) Calculate the length of the Hypotenuse: 30 Seconds 9cm 12cm ? 

31 13 x 13 = 5 x 5 = - =  = So missing side iscm long 14) Calculate the length of a short side: 60 Seconds 5 cm 13cm ? 

32 9 ·5 x 9 ·5 =11 x 11 = + =  = So hypotenuse iscm long 15) Calculate the length of the Hypotenuse: 60 Seconds 9·5cm 11cm ? Calculator needed 

33 2 · 3 x 2 · 3 = 4 ·5 x 4 ·5 = - =  = So missing side iscm long 16) Calculate the length of a short side: 60 Seconds 4 · 5cm 2 · 3cm ? Calculator needed 

34 3·6 x 3·6 = 2·5 x 2·5 = - =  = So shortest side iscm long 17) Calculate the length the shortest side: 60 Seconds 2 · 5cm 3 · 6cm ? Calculator needed 

35 Find the height of the following isosceles triangle B ? C 13 A 8

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37 A model power boat can travel at 0.75 m/s in still water. It is released from a point P on the bank of a river which flows at 0.4 m/s. The river is 15m wide. The boat is aimed continually in a direction perpendicular to the flow of the river, as shown in the diagram Find: i.The resultant speed of the boat. ii.The direction in which the boat actually travels across the river. iii.How far downstream from P does the boat actually land on the opposite bank? iv.How long does the boat take to cross the river? 15m 0.75 m/s 0.4 m/s NEAB 1995

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