The wee Maths Book. Growth. Grow your brain. Book 2. N4 Expressions & Formulae. of Big Brain

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1 Grow your brain N4 Expressions & Formulae Book 2 The wee Maths Book of Big Brain Growth Guaranteed to make your brain grow, just add some effort and hard work Don t be afraid if you don t know how to do it, yet! It s not how fast you finish, but that you finish. It s always better to try something than to try nothing. Don t be worried about getting it wrong, getting it wrong is just part of the process known better as learning. Pythagoras, Enlargement and Reduction, Angles and the Circle

2 D Length and Area D1 I can use formulae to calculate the area of a rectangle and a triangle. 1. Find the area of each shape using the appropriate formula and showing all of your working. Don t forget to include units in your answer. (a) 5 cm (b) 20 cm 12 cm 150 mm (c) (d) 2 m 2 m 2 m 200 mm 2 m 150 mm Page 2

3 (e) (f) 18 m 16 cm 30 m 15 cm (g) 8 m (h) 9 m 5 cm 4 cm 3 cm (i) (j) 12 cm 6 cm Page 3

4 2. The side view of a wooden door wedge shows the height is 4cm and the length is 16 5cm. 4cm 16 5cm Calculate the area of the shaded part of the wedge. 3. The white sail of the yacht Ocean Voyager is in the shape of a right angled triangle with dimensions shown. 5 5m 4 6m Calculate its area in m². Page 4

5 4. Alan gets a driveway company to give him a quote for the cost of mono-blocking his 4 5m by 6m rectangular driveway. The price quoted is 85 per square metre including all materials and labour. If Alan chooses this company, what would be the cost of monoblocking his driveway? 5. Susan wants to turf her rectangular back garden. She orders the turf from her local garden centre who charge 2 70 per square metre and a delivery charge of m 9m How much would it cost Susan for the turf (including delivery)? Page 5

6 D2 I can find the area of composite shapes made up from rectangles and triangles. 6. Using a ruler, make a neat sketch of the shape shown below. All internal angles are right angles. 14 cm 6 cm A x cm B 4 cm (a) What is the value of x? 20 cm (b) Calculate the areas of rectangles A and B, showing all working. (c) Calculate the total area for the composite shape, showing all working. 7. The composite shape shown is made up from two rectangles. 8 m 24 m 8 m 12 m Calculate its area, showing all working. Page 6

7 8. Calculate the area of these composite shapes. (a) (b) 27 cm 80 cm 13 cm 25 cm 14 cm 70 cm (c) 24 cm 24 cm 50 cm (d) 2 4 m 3 8 m 25 m Page 7

8 9. The end face of a house has to be roughcast. Mr Smith has been quoted 6.30 per square metre to complete the job. 3m He has a budget of 450 can he afford the work to be done? 7m Justify your answer. 8m 10. A factory punches out equilateral triangle shapes from sheet metal for use in the car industry. 30cm 24cm 26cm How much metal is left over at the end of the process? Page 8

9 11. A garden is shaped as shown with grass surrounding a rectangular flower bed. 16m 12m 10m 7m 12m What is the area of the grass? 12. A piece of art is made from a square of metal, with length 70cm, which has 12 identical triangular holes cut out. Each triangle has a base of 10cm and a height of 8cm. Calculate the area of metal in the finished design (the shaded area)? Page 9

10 13. Jane is an artist. She is designing a square window pane made from coloured and clear glass. The window pane is 60 cm wide. Each coloured rectangle is 20 cm long and 10 cm wide. The four corners are congruent isosceles triangles. Jane thinks that she will use 1400 cm 2 of clear glass in the design. Is Jane correct? You must justify your answer. 14. The front, back and 2 sides of this shed need to be painted. 1 tin of paint can treat 25 m 2 of wood. Is one tin enough to paint this shed? You must justify your answer. Page 10

11 15. A Jawa sandcrawler has sucked up R2D2. The Jawas constructed their sandcrawler using a combination of rectangular and triangular sheets of metal, as shown in the diagram below. 3m 22m 10m 4m 7m 13m Can you work out the area available to Luke Skywalker to fire his blaster at? (Ignore the tracks) Page 11

12 D3 I can use pi to calculate the circumference given the diameter and vice versa. 16. Calculate the circumference of these circles. Make sure you set down all the working. Remember to provide units in your answers (a) (b) (c) 6m 12mm 25cm (d) (e) (f) 4 2cm 7mm 1 5m 17. The London Eye, is a giant Ferris Wheel with a diameter of 120 metres. How far does a passenger on the London Eye travel in one rotation? Page 12

13 18. An ant is walking around the rim of a circular pot. The pot has a radius of 6 centimetres. (a) How far will the ant walk in one lap of the pot? The ant manages one lap every twenty seconds. (b) How far will it walk in a minute? Ben notices that the ant is still walking around the pot after an hour. (c) Assuming the ant has kept to the same pace, how far would it walk in an hour? 19. A bike has a wheel with a diameter 65 centimetres. How far would this wheel go in 250 revolutions (turns)? Give your answer in metres. 65 cm Page 13

14 20. Eva has a trundle wheel with a radius of 18 centimetres. How many complete metres are measured by 15 rotations of the wheel? 21. A farmer found a crop circle in his field one day. He measured the circle and found it had a radius of 20 metres. Calculate the circumference of the circle. 22. The average radius of the iris of a human eye is 5 9 mm. What is the average circumference of the human eye? 23. On Pi Day, Eva baked a circular cake. She wanted to put a ribbon around the cake. The cake was 30 centimetres in diameter and Eva had 1 metre of ribbon. Is this enough to go around the cake? Justify your answer Page 14

15 24. The ancient people of Scotland built stone circles. The precise date and function of the circles can be unclear, but some people think they may have been used in religious rituals. Ben measures the inside diameter of one of these circles and found it to be 13 metres. Ben found the outside diameter to be 15 metres. 13m 15m Ben tells Cara that if you walk around the outside of the circle you ll walk over six metres further than if you walk around the inside. Is Ben Correct? Justify your answer Page 15

16 25. Calculate the diameter of circles with the following circumference. (a) 12 centimetres (b) 53 millimetres (c) 1 5 metres 26. A circle has a circumference of centimetres. Calculate the size of the radius of this circle. 27. A roulette wheel has an outside circumference of 2 5 metres. Find the length of the radius from the centre of the roulette wheel, to the outer edge. Give your answer in centimetres. 28. The circumference of the Earth is kilometres. Calculate the diameter of the Earth. Give your answer to the nearest 1000 kilometres. Page 16

17 29. Mills were once common in Scotland and most communities would have had a mill. Mills were used to ground wheat to make flour for bread. The wheat was ground by large rotating millstones. The millstone shown in the picture has a circumference of 1 2 metres. What would be the diameter of the millstone? 30. In Cara s drum kit the circumference of the snare drum is half the circumference of the base drum The circumference of the bass drum is 180 metres. Find the diameter of the snare drum. 31. Find the perimeter of the semi-circles shown below (a) (b) (c) 38 cm 5 m 14 mm Page 17

18 32. Find the perimeter of the quarter circles shown below (a) (b) (c) 2 4 cm 38 mm 5 1 m 33. The opening of the fireplace is shown in the diagram opposite. The fireplace has a shape which comprises of a rectangle and a semicircle. 40 cm 45 cm A metal strip is to be placed around the fireplace opening. Calculate the length of metal strip. Page 18

19 34. Cara s garden, shown in the sketch, has two flower beds in the shape of quarter circles and one in the shape of a semi-circle. Cara plans to plant seeds all along all of the edges of the flower beds A packet of seeds can sow a line 3 metres long. How many packets of seeds does Cara need? Show all working. 35. A joiner is making tables for a new coffee shop. The shape of the top of a table is a semi-circle as shown. 120 cm The top of the table is made of wood and a metal edge is to be fixed to its perimeter. (a) Calculate the total length of the metal edge. (b) The coffee shop needs 16 tables. The joiner has 50 metres of the metal edge in the workshop. Will this be enough for all sixteen tables? Give a reason for your answer. Page 19

20 D4 I can find the area of a circle. 36. Find the area of each circle below (a) (b) (c) 25cm 6m 7mm 37. The cross section of an orange is shown. The cross section is a circle with diameter 7 centimetres. Calculate the area of the cross section of the orange. Page 20

21 38. The average radius of the iris of a human eye is 5 9 mm. What is the average area of a human iris? 39. Cara makes pizzas in a restaurant. The pizzas have a diameter of 18 centimetres. (a) Calculate the area of the pizza. (b) Cara has estimated that she has enough topping to cover 5000 square centimetres of pizza. How many pizzas can this make? 40. A farmer found a crop circle in his field one day. He measured the circle and found it had a radius of 20 metres. The crop circle was reported in the local newspaper with the heading huge crop circle with an area of nearly 5000 square metres found in a farmer s field. Comment on the accuracy of the newspaper headline. Page 21

22 41. Eva love circles. Her garden design, shown below, is very much influenced by circles. 2 6m 3 4m Summer House Flower Bed Stepping Stones Pond Decking Bird Table 0 7m Flower Bed 2 8m 4 8m Page 22 (a) Each stepping stone has a diameter of 30 centimetres. Find the total area of the 7 stepping stones. (b) The pond has a diameter of 1 5 metres. The path around the pond is 25 centimetres wide. What is the area of the path around the pond? (c) The two flower beds are semi-circular with dimensions shown. Find the combined area of the two flower beds. (d) The summer house and the deck are in the shape of quarter circles with dimensions shown. Find the areas occupied by the summer house and the deck? (e) The bird table has a radius of 0 7 metres. Find the area of the bird table.

23 42. The diagram below shows a birthday card. The card consists of a rectangle and a semi-circle. Find the area of the card. 43. In the diagram show, a square is inscribed by four quarter circles of radius 14 centimetres. 14 cm 14 cm 14 cm 14 cm Calculate the shaded area shown. Page 23

24 D5 I can identify quadrilaterals from their properties. Use the clues to identify each quadrilateral. Illustrate each quadrilateral with a sketch. 44. All of my sides are equal and all of my internal angles are right angles. 45. I have opposite sides which are equal and parallel and opposite angles which are equal. However, none of my angles are right angles. 46. All of my sides are equal and I have opposite angles which are equal. However, none of my internal angles are right angles. 47. I have one pair of parallel lines however none of my sides are equal and none of my angles are equal. 48. I have two pairs of equals and opposite sides and all of my internal angles are right angles. Page 24

25 49. I have two pairs of equal sides and my diagonals meet at right angles. 50. Use the following scales to draw a Cartesian diagram Vertical (x-axis) Minimum -6 and maximum 6, grid step 1, number step 2. Horizontal (y-axis) Minimum -6 and maximum 6, grid step 1, number step 2. Now plot the sets of points below on your coordinate diagram and join the points in order and finally close each shape to produce a quadrilateral. (0, 1) (2, 5) (0, 4) (-2, 5) Write down the name of this type of quadrilateral. 51. Repeat Question 4 with the following coordinates. Use a separate coordinate diagram for each quadrilateral. (a) (2, 0) (3, 3) (4, 0) (3, -3) (b) (2, -2) (1, -4) (-4, -4) (-2, -2) (c) (-3, 3) (-1, 1) (-3, -3) (-5, 1) Page 25

26 D6 I can find the area of a kite, rhombus, parallelogram and trapezium using composite shapes. 52. Find the area of the kites below. (a) (b) 2 3cm 7 2m 3 4cm 9 8m 4 2cm 10 6m 53. Find the area of the rhombuses below. (a) (b) 120mm 2 3m 80mm 120cm Page 26

27 54. Find the area of this parallelogram 40m 30m 110m 55. Find the area of this trapezium 110cm 35cm 70cm 40cm Page 27

28 56. Below are four quadrilaterals. For each shape, write down which type of quadrilateral it is, then calculate its area. (a) 3cm (b) 10cm 2cm 5cm 8cm 6cm 7cm (c) 2m (d) 5m 10m 8m 6m Page 28

29 57. Below are four quadrilaterals. For each shape, write down which type of quadrilateral it is, then calculate its area. (a) 5m (b) 6m 2m 7m 4m 2m 8m (c) 3cm (d) 4cm 12cm 7cm 10cm Page 29

30 58. After experimenting with some quadrilaterals, Cara made the following conjecture the area of a kite, or rhombus, is exactly half the area of the surrounding rectangle. (a) For each of the diagrams below find the area of the kite (or rhombus) and the surrounding rectangle. (i) (ii) (iii) (iv) Page 30

31 (b) Does Cara s conjecture hold for all four diagrams? You must clearly justify your answer with reference to calculations. (c) One of Cara s classmates suggested the following the area of a parallelogram is exactly half the area of the surrounding rectangle. Use the diagram below to test this statement. You must clearly justify your answer with reference to calculations. (d) Draw a trapezium on square paper and use it to test the following statement the area of a trapezium is exactly half the area of the surrounding rectangle. You must clearly justify your answer with reference to calculations. Page 31

32 E Surface Area and Volume E1 Working with others I have accurately made the net of a cube, cuboid and triangular prism and I can calculate surface area of a cube, cuboid and triangular prism. 1. Pick the correct net for the shape 2. Pick the correct net for the shape 3. Pick the correct net for the shape Page 32

33 4. Pick the correct net for the shape 5. Pick the correct net for the shape 6. Pick the correct net for the shape 7. Pick the correct net for the shape Page 33

34 8. A cuboid and its net are shown below. 2 cm 6 cm 3 cm 3 cm 3 cm 2 cm Calculate the surface area of the cuboid. 9. A triangular prism and its net are shown below. Calculate the surface are of the triangular prism. 6 cm 10 cm 10 cm 8 cm 6 cm 5 cm Page 34 5 cm

35 10. On squared paper draw three different nets of a cube. 11. Which of the following nets below represent a net of a cube (a) (b) (c) (d) (e) (f) (g) (h) (i) Page 35

36 E2 I can calculate the volume of a variety of 3D shapes including prisms by applying a formula 12. A fish tank is 55 cm long, 40 cm wide and 25 cm high, as shown. 25cm 55cm 40cm (a) What formula would you use to calculate the volume of the tank? (b) Calculate how much water you need to fill the tank to the top. (c) Another fish tank holds twice the amount of water. How much water do you need to fill it to the top? 13. A five litre jug of water is poured into this tank. 19cm Will the tank overflow? 30cm 9cm Page 36

37 14. Find the volume of the prisms shown. (a) (b) (c) (d) 15. The diagram shows a triangular prism. The dimensions are given on the diagram. (a) Calculate the area of the cross section of the prism. (b) Calculate the volume of the prism. Page 37

38 16. The maths department were having cheese. What is the volume of this wedge of Emmental (including the holes)? 60cm² 7 5cm 17. A rubbish skip is prism shaped as shown. The skip has a length of two metres and a cross sectional area of 3 6 m². Hops Skips for Hire 3 6m² 2m Find the volume of the skip in cubic metres. 18. A farmer turns a tap in order to fill this drinking trough, which is in the shape of a cuboid, with water. 75cm 120cm 30cm What volume of water will the trough hold? Page 38

39 19. This swimming pool is 25 metres long and the area of the cross section is 62 5 m². Calculate the volume of the pool in cubic metres when it is full. 10 m 62 5 m² 20. A tin of soup has a volume 1256 ml. Calculate the height of the tin, to the nearest millimetre, if the cross-sectional area is 150 cm 2. h cm Page 39

40 21. The front of a hut has an area of 4 6 m², it also has a volume of m³. Calculate the depth of the hut in millimetres. 4 6 m² Depth 22. A wedge of cheese has a volume of cm³. If it has a cross sectional area of 72 cm², then how thick is the cheese? Give your answer in millimetres. Thickness mm Page 40

41 23. A baker has been asked to make alphabet shaped cakes. The letter A cake mould has a uniform cross section of area 350 cm 2. The depth of the mould is 8.5 cm. The baker has 2.9 litres of cake mix. Is this enough to make the cake? You must justify your answer. 24. Physiotherapists often use exercise pools with movable bases to help rehabilitate injured patients. A physiotherapist wants to work with 4 patients simultaneously. Guidelines advise that for a group of 2 to 5 people the floor area of the pool should be between cm 2 and cm 2. The maximum depth of this pool is 2 metres. The floor is raised by 60 cm. If the volume of water in the pool is litres, will the floor area meet the guidelines? You must justify your answer. Page 41

42 25. An exotic fish is placed in an aquarium which is a rectangular prism with a base of area 1500cm cm 2 If the water level rises by 0.2cm, what is the volume of the fish? 26. A contestant on BBC MasterChef makes 500 ml of jelly mix in preparation to pour into a mould. The mould is a prism as shown below. 64 cm 2 7cm Page 42 Will she have enough mix to fill the entire mould?

43 F Rotational symmetry F1 I can describe the order, angle of rotation or fraction of a turn for a given shape with rotational symmetry. 1. Celtic Knots are perhaps the most notorious and recognisable artwork in Celtic history. Many Celtic Knots have rotational Symmetry. The two shapes below are modern interpretations of Celtic Knots. State the order of rotational symmetry. (a) (b) 2. Companies often use logos which have rotational symmetry. For each logo on the next page, state: (i) the order of rotational symmetry; (ii) the rotation symmetry as a fraction of a complete turn; (iii) the rotational symmetry as an angle of rotation about the centre of the shape. Page 43

44 A B C D E F G H I J K L M N O Page 44

45 3. Alloy wheels usually have spokes which are arranged so that the wheel has rotational symmetry. A very common arrangement for alloy wheels is rotational symmetry of order five as shown. For each of the alloy wheels below, state: (i) the order of rotational symmetry; (ii) the rotation symmetry as a fraction of a complete turn; (iii) the rotational symmetry as an angle of rotation about the centre of the shape. (a) (b) (c) (d) Page 45

46 F2 I can complete a shape to a given order of rotational symmetry, angle of rotation or fraction of a turn. 4. Copy each shape and complete so that the finished diagram has quarter Turn Symmetry about. (a) (b) (c) (d) Page 46

47 5. Complete the Rotational Symmetry worksheet so that each finished diagram has rotational symmetry of order three about. Page 47