4. Draw a new set of similar triangles and check that the ratios of their sides are approximately the same.

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1 SMILE CENTRE MIDDLE ROW SCHOOL KENSAL ROAD LONDON W 5DB Tel You will need Worksheet 92a. Cut out all the triangles on worksheet 92a and sort them into 2 sets of similar triangles by comparing angles. 2. Measure the sides of each triangle and complete the tables on your worksheet. 3. You should have found that the ratio Short - Middle is approximately the same for all the triangles ^ in each set. Is this true for the other 2 ratios? 4. Draw a new set of similar triangles and check that the ratios of their sides are approximately the same.

2 Smile Worksheet 92a Length of sides Ratio of sides Setl Short Middle Long Sf-M S^L M-L Set 2 Short Middle Long S-J-M S-L M-L Your own set Short Middle Long S-nM S-L M-L RBKC SMILE 2

3 Find the operation Smile Worksheet 94 In this operation table * a * b means treble a then add b 2 so a * b = 3a + b 6 a Complete the table. 4 3 What number patterns do you notice? Find the operation in these tables. * 2 3 * 2 3 * b Id a* b = a * b = a * b = Use the sheet below to make a puzzle for a friend. Find the operation Designed by for Solution a *b = * RBKC SMILE 2

4 Smile Worksheet 97 Work with someone else. a. Discuss how long it takes to do each of these. Then time yourself. How close were you? You will need a clock or a watch. "I J Count up to 2 j Count down from, 2, 3, 4,...,99,98... S Actual time 3 ) Write your name 4 / Do 2 x 4 without a calculator ;J Take the register I Actual time Make one big cube using 27 smaller cubes

5 b. Choose 3 of these and then do some of your own. D^/ Boiling a kettle '\ 2 } Eating a packet of crisps 5 j Getting home from school RBKC SMILE 2

6 Smile Worksheet 9 Dissection Pairs Shape,.. I can be cut and rearranged to give... Shape. Find 5 other pairs of shapes. 2) 3) 4) 5) ' 6) 7) 8) ' ' ' ' 9) ) ' ' ' ' ) 2) ' Check your answers by comparing areas. RBKC SMILE 2

7 Smile 92 Painted Tyres While riding my bicycle along a path I went through a small patch of paint. A little later I looked back at the paint marks left by my tyres. What did I see? There are some hints on the back if you need them.

8 Hints It may help to make a scale drawing of any patterns formed showing the original patch of paint. * What happens if you change the distance between the wheels? * What happens if the diameter of the wheels is different? * What happens if you turn a corner? * Do both the wheels travel the same distance?

9 Smile bo This is part of an addition table in Bengali number script. Can you extend it?

10 Smile Worksheet 94 Adding Counters You will need counters.

11 Add one more counter to make 6 lines of 3 counters. Add two more counters to make 3 lines of 3 counters.

12 Add two more counters to make 5 lines of 4 counters. turn over

13 Add four more counters to make 8 lines of 3 counters. RBKC SMILE 2

14 Smile 96 Domino Trick Ask a friend to choose any domino. Let's try this one... Ask her to do the following calculations. # Multiply either of the numbers by 5 # Add 8 Multiply by 2 Add the other number from the domino Ask her for the answer. Subtract 6 from it. The two digits left are the numbers on the domino. Will it work with any domino? Can you prove it? Can you find another set of rules with the same effect?

15 i i i i i \ c Smile 97 i i i ^--! vv ^i j i ip \ -* \ t"j\ -/ w -/ \ ov> tfv ^ t i i i,\i >^\N & i i i i ~~r i^ ^' J5 \ 3 > it / ** i I i i - i i j i i - H V r ir x" i ^ f* \ \c s X h ^r *\ * \» j ^ f ^* w --* ^",<^ -r; 3^ i S* ' x,^ I ietf ^ I i i X* > i i i i i i i i % (*< v\o ^< G< ^ \c a /^\c x ^^ ^ X - ^ ^-» s -* / f *f p n \^\ -fjs* -4-! I -! iii i i^f^ ' i. i : - i i ; i ^i. * ii i i

16 Gradient measures how steep aslope is. Gradient = he'9ht base So for this triangle 8cm gradient = 2.7 -=- 8 =.34 Check that all of these have a gradient of approximately.58

17 . By drawing right angled triangles find the gradients for other angles. Copy and complete the table. Angle Gradient Draw a triangle to show a gradient of exactly. What angle has this gradient? Turn over

18 The tangent button a scientific calculator will give you a gradient for any angle. 3. Use the tan button to check your results in the table. Explore the gradient for angles between 8 and 9-4. What happens to the gradient for angles very near 9? Hint: Look at angles between 89 and 9 Investigate the tangent of angles greater than 9

19 Smile 98 The Coin Problem Put down 5 coins all showing TAILS. Can you make all 5 coins show HEADS by turning them over two at a time? Try starting with 6 coins, 7 coins.,. Investigate turning over three at a time, four at a time

20 How many centimetre squares? Smile Worksheet 99 & RBKC SMILE 2

21 Trig Lines Smile 92 You will need graph paper and a scientific calculator >.4 *' opdasi side I _.i. ll I L ^35 ad acent side^ The red line (radius) is )lt rotates anti-clockwise. It has stopped at 35. unit long. rt w =a a. Look at the right-angled triangle formed by drawing a vertical line opposite to the 35 angle. The length of the side opposite to the 35 angle is.57. a. How long is the side adjacent to the 35 angle?

22 b. On your graph paper draw diagrams with the red line stopped at Measure the opposite and adjacent sides for each angle. opposite side c. How does the length of the opposite side change as the angle gets larger? How does the length of the adjacent side change as the angle gets larger?

23 When the radius is unit long the opposite and adjacent sides can be found using the [siit] and [cos] buttons on a scientific calculator. Press I 3 I s answer for a. cos) to check your Use the sine and cosine button to check that your answers to b. are approximately correct.

24 2. Find the length of the opposite and adjacent sides of these triangles. (They are not drawn to scale.) ^25 c. d..4 e. 5 C Turn over

25 The small triangle has been enlarged by Scale Factor What are the lengths of the sides on the larger triangle? a. 4. Find the lengths of the sides on these triangles. (They are not drawn to scale.) c. 35 l 63 2

26 Matrices and Area Smile _ (2, 3) (4) 3) ' (2,) iiiiiiiiiiiil iiiiiiiiiiiii (4,) i i L 23 i. Transform this square using the matrix 2 o o * 8 l 3 3 s What is the area of the original square? What is the area of the new shape? What is the ratio New Area: Original Area? Turn over

27 2. Transform the same square using the following matrices: ; i ) sisils^^ 99 RBKC SMILE

28 You will need a set of 2 wooden pentominoes. Smile 927 ( Pentomino Puzzles This 4x5 rectangle has been made using four of the pentominoes. Use 2cm squared paper to show how you solved the three puzzles inside.

29 Make a rectangle with these 4 pentominoes.

30 Make a rectangle with these 6 pentominoes. Turn

31 Make a rectangle using these 7 pentominoes. 99 RBKC SMILE

32 r~ You will need a set of 2 wooden pentominoes. Smile 928 four pentominoes

33 An enlarged copy of the n pentomino can be made using four other pentominoes.

34 . Make an enlarged copy of the pentomino using four other pentominoes.. Do the same for <v ^ ^ } and Draw your results on 2cm squared paper. 3. a) How do the sides of the enlargement compare with the sides of the single pentomino? b) How does the area of the enlargement compare with the area of the single pentomino? Turn over

35 One other pentomino can be enlarged in this way. Can you find it? 99 RBKC SMILE

36 Smile 929' You will need a set of 2 wooden pentominoes and 2cm squared paper. Select one pentomino. Use nine of the other pentominoes l^make an enlargement of it. Draw your result, Try some more. 99 RBKC SMILE

37 How many different numbers? How many different scripts? Which scripts? (.p O CO O H CO O) E O) If you want more information look at Worksheet 93 a

38 Smile Worksheet 93 a Which Scripts? There are 6 numbers written in 5 different scripts. Can you find them? Write 5 in each script. RBKC SMILE 2

39 Smile Worksheet 93 a Which Scripts? There are 6 numbers written in 5 different scripts. Can you find them? Write 5 in each script.

40 Smile 934 a m Wfou may need tracing paper. A translation moves every part of a shape the same direction the same distance This shape... is moved by this translation to this new position.

41 Show these translations on squared paper

42 Combined i ranslafions This shape is moved by these two translations... to this position, 5. Show the following translation on squared paper. Find a single translation to replace these two translations. Turn over

43 Find a single translation to replace those in each of 6 and RBKC SMILE

44 You will need filter papers. Smile 935 Angles in semi-circles Fold a filter paper in half. Draw different triangles using the fold as the base. Investigate the angles of your triangles, Does the size of the circle matter? Turn over

45 What happens to the angles if P is inside the circle?... outside the circle?

46 Smile 937 Panjabi Numbers tf = x tf + 8 = x tf 'J t Copy and complete this number pattern in Panjabi number script. 99 RBKC SMILE

47 Olympic Rlfedals Smile 938 o ~

48 f LEADING 5 MEDAL WINNERS IN THE LAST FIVE GAMES II MUNICH 972 i Soviet Union... M United States.,..,? Italy Finland......_ 3 Cuba. _... 3 S B Tot MONTREAL 976 Q Italy Franco..._.... s B Tot MOSCOW 98C Czechoslovakia... G fl F, fi ». 2...? } S B Tot LOS ANGELES G United States Italy Great Britain Australia Finland... " S B Tot SEOUL 988 G Soviet Union East Germany United States Sooth Korea West Germany Hungary Bulgaria Romania F'ance Italy... China... Great Britain... Kenya Japan Australia S B Tot _«_^ri ^^ ^M fc ^^^m^hib - Source: The Sunday Times, October 988

49 This graph shows the mecwlfe won by the Soviet Union. Key Bronze Silver Gold CO T3 ) D Z Years when Olympics were held.

50 . Which countries medals are shown by these graphs? TO 3 ) z Years when Olympics were held Years when Olympics were held. 2. Pick another country Ad draw your own graph. Ask someone to find out which country.

51 You will need a scientific calculator and graph paper. Smile 939 and fcoit graphs Draw a table showing the values of sine and cosine for angles between and 36. Turn over

52 Plot three graphs to show sine against angle cosine against angle sine against cosine. Describe your three graphs. What happens for angles greater than 36? What happens for negative angles?

53 Dividing Investigation Smile 94 f Which other numbers give answers ending in,25 when you divide them by 4? Try dividing by some other numbers: 5, or 8, or 3,,, Write about what you have found.

54 Smile 94 Differences The mapping n >3n 2 + n creates the sequence 4, 4, 3, 52, 8, 4... Taking the differences between the terms of the sequence gives and again 6666 wthis gives a constant difference of 6 after two rows. Investigate for other mappings. You may wish to use a spreadsheet. 99 RBKC SMILE

55 Smile Worksheet 942 Growing Patterns turn over

56 Now try with your own shapes. RBKC SMILE 2

57 Smile Worksheet 945 Square ^Diagonals This square has been filled in using the rule "Add " across and "Add " down d > 9

58 Complete this square in the same way. i ^ A- & r > 4 2 You should find numbers on thi diagonal. th ese w ;;«; ; gf W- ' What do you n otice about t hem? 4 What do you notice about the numbers on the other diagonal?

59 Complete this square using "Add 2" across and "Add 2" down What do you notice about the numbers on the diagonals? turn over

60 This sqgare has two different rules. "Add 2" across and "Add 3" down. k.. ^ T *- d r What do you notice about the numbers on the diagonals? If you enjoyed doing this you might like to investigate what happens for different rules and different squares. RBKC SMILE 2

61 Smile RBKC SMILE

62 A matrix can describe a framework by listing the connecting lines. Smile 947$

63 Which matrices opposite describe the frameworks below?. 4.

64 a. b. c. d. A F D E A B C D E e. A B C D E F G A B C D E F G A B C D le F A B C D E F f. A B C D E F A B C D E F A B C D E F A B C D E F g. A B C D E A B C D E A B C D E F G H A B c D E F G H Can you draw a network for the remaining matrix?

65 Smile 948 y = ax Draw graphs for different values of a. How does the value of a change the graph? What happens if a is negative? Try using a graphic calculator or a graph plotting program such as QUAD. 99 RBKC SMILE

66 Smile 949 Each player needs a counter.or centicube. You will need a dice marked N E W NE SW NW SE NW NE SW SE Take turns to throw the dice. Choose one of the directions shown. Move one square in that direction. The winner is the first person to reach a shaded square.

67 North (N) North ^-^^lifcs^ North West East (NE) West (W) East (E) South West (SW) South (S) South East (SE)

68 Start

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73 Smile 95 Sorting Triangles Sort the triangles in this pack into sets of similar triangles. What is special about similar triangles? RBKC SMILE 2