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Unit 7 – Area and Volume

Mathematics (9-1) - iGCSE 2018-20 Year 09.

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Unit 7 – Area and Volume

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  1. Mathematics (9-1) - iGCSE 2018-20 Year 09 Goods are packaged in shapes that stack easily on shelves, have space to show the name of the product and attractive pictures, and are not easily knocked over. Packaging also needs to be cheap to produce, and should not leave too much empty space around the item inside. A manufacturer can choose either a cylindrical box or a cuboid to package 4 tennis balls. Each tennis ball has a diameter of 6.5 cm. Work out the possible dimensions for each box. Unit 7 – Area and Volume

  2. Contents Page iv 7 Area And Volume 203 Prior knowledge check 203 7.1 Perimeter and area 204 7.2 Units and accuracy 207 7.3 Prisms 210 7.4 Circles 213 7.5 Sectors of circles 216 7.6 Cylinders and spheres 220 7.7 Pyramids and cones 222 Problem-solving 225 Check up 227 Strengthen 229 Extend 232 Knowledge check 235 Unit test 237

  3. Contents Page v At the end of the Master lessons, take a check-up test to help you to decide whether to strengthen or Extend our learning Extend helps you to apply the maths you know to some different situations Unit Openers put the maths you are about to learn into a real-life context. Have a go at the question - it uses maths you have already learnt so you should be able to answer it at the start of the unit. When you have finished the whole unit, a Unit test helps you see how much progress you are making. Choose only the topics in strengthenthay you need a bit more practice with. You’ll find more hints here to lead you through specific questions. Then move on to Extend Use the Prior knowledge check to make sure you are ready to start the main lessons in the unit. It checks your knowledge from Key Stage 3 and from earlier in the GCSE course. Your teacher has access to worksheets if you need to recap anything.

  4. Contents Page vi

  5. Contents Page vi

  6. 7 – Prior Knowledge Check Page 203 • Numerical fluency • Calculate an estimate for 5.2 x 4.4 x 18.9 • Round to 1 decimal place (1 d.p.). • 3.57 b.2.06 • 4.99 • Round to 2 d.p. • 9.402 b. 13.9834

  7. 7 – Prior Knowledge Check Page 203 • Fluency with Measures • Match each object to the amount of liquid it can hold. • Copy and complete. • 5.2m = cm b. 24cm =mm • 1m = mm d.3.41km =m • 0.327 litres =  ml • 2400ml = litres 330ml 5ml 5 litres 1l

  8. 7 – Prior Knowledge Check Page 203 • Algebraic Fluency • When a = 3, b = 2 and c = 4, work out • 2a2b.4ac + 2b • ab2cd.ab • Make x the subject of • y = cx b.a = bxz • m=xy

  9. 7 – Prior Knowledge Check Page 203 • Geometrical Fluency • Sketch a circle. Draw and label its centre, radius and diameter. • a.Sketch the net of this triangular prism.Label the lengths on your net. • Work out the surface area of the prism.

  10. 7 – Prior Knowledge Check Page 204 • Geometrical Fluency • a.Sketch the solid formed by this net. • Draw sketches to show its planes of symmetry. • Calculate the volume of the solid.

  11. 7 – Prior Knowledge Check Page 204 Geometrical Fluency Work out the area of this parallelogram. Work out the volume of this 3D solid. Write your answer to 2 significant figures (2 s.f.).

  12. 7 – Prior Knowledge Check Page 204 * Challenge These chocolate bars are packed into cuboid-shaped boxes for transport.Sketch how you could pack 6 of these bars to take up as little room as possible.What size box do you need for 48 chocolate bars?What other shapes of box could you use to transport them? Which shape box would be most practical?

  13. 7.1 – Perimeter and Area Page 160 ActiveLearn- Homework, practice and support: Higher 7.1

  14. 7.1 – Perimeter and Area Page 204 • Work out the area and perimeter of each shape. • Solve: • 3x – 12 b. ½x -6 • 4(x + 2) = 28 Warm Up

  15. 7.1 – Perimeter and Area Page 205 a. Work out the area and perimeter. • Work out the perimeter • Work out the shaded area. Give your answers to the nearest square mm. • DiscussionHow did you work out the area in part b? How else could you have done it? Which way is most efficient?

  16. 7.1 – Perimeter and Area Page 205 Communication hint - Trapezia is the plural of trapezium. Key Point 1 This trapezium has parallel sides, aand b, and perpendicular height, h.Two trapezia put together make a parallelogram, with base (a + b) and perpendicular height, h.Area of 2 trapezia = base x perpendicularheight = (a + b)h. Area of a trapezium = ½(a + b)h

  17. 7.1 – Perimeter and Area Page 205 Calculate the areas of these trapezia.Round your answers to 1 decimal place (1 d.p.) where necessary. Q4 hint - Use the formula: area = ½(a + b)h

  18. 7.1 – Perimeter and Area Page 206 Calculate the area and perimeter of this isosceles trapezium. Q5 communication hint - An isoscelestrapezium has one line of symmetry. Its two sloping sides are equal. Reflect How can you remember how to find the area of a trapezium?

  19. 7.1 – Perimeter and Area Page 206 5 – Exam-Style Questions Here is a diagram of Jim’s garden. Jim wants to cover his garden with grass seed to make a lawn. Grass seed is sold in bags. There is enough grass seed in each bag to cover 20 m2 of garden. Each bag of grass seed costs £4.99. Work out the least cost of putting grass seed on Jim’s garden. (4 marks) Nov 2012, Q25, 1MA0/1F, Exam hint Show your working by writing all the calculations you do on your calculator.

  20. 7.1 – Perimeter and Area Page 206 • Problem-solving – Here is the plan of a play area. Work out its area. • The area of this trapezium is 96 cm2. • Substitute the values of a, b and A into the formula A= ½(a + b)h • Simplify to get an equation,  = h • Solve to find h.

  21. 7.1 – Perimeter and Area Page 206 Problem-solving – A trapezium has area 32cm2, and parallel sides 5.5 cm and 10.5 cm. Work out its height. Q9 strategy hint- Sketch and label the trapezium. Use the method from Q8. Example 1 This trapezium has area 70m2Find the length of the shorter parallel side.70 = ½(a + 12) x 7 = 10 = ½(a + 12) 2 x 10 = 20 = a + 12 a = 8 cm Substitute the values of h, b, and A into the formula A = ½(a + b)h Divide both sides by 7 Multiply both sides by 2.

  22. 7.1 – Perimeter and Area Page 207 ReasoningFind the missing lengths, Problem-solving One corner of a rectangular piece of paper is folded up to make this trapezium.Work out the area of the trapezium.

  23. 7.2 – Units and Accuracy Page 160 ActiveLearn- Homework, practice and support: Higher 7.2

  24. 7.2 – Units and Accuracy Page 207 • Round each number to the level of accuracy given. • 3.567 (1 d.p.) • 320.6 (2 s.f.) • 8.495 (2 d.p.) • 15.721 (3 s.f.) • Work out • i. 10% of 25 kg ii. 10% less than 25 kg iii. 10% more than 25 kg • i. 5% of 40 m ii. 5% less than 40 m iii. 5% more than 40 m Warm Up Q2 all hint- Subtract 10% of 25 kg from 25 kg.

  25. 7.2 – Units and Accuracy Page 207 • a. Explain why these two squares have the same area. • Work out the area of each square, • Copy and complete. 1cm2 = mm2 Key Point 2 To convert from cm2 to mm2, multiply by 100. To convert from mm2 to cm2, divide by 100.

  26. 7.2 – Units and Accuracy Page 208 • Reasoning • Sketch a square with side length 1m and a square with side length 100cm. • Copy and complete. 1m2= cm2 • How do you convert from cm2 to m2? • Convert • 250mm2 to cm2b.5.2 m2 to cm2 • 7000cm2 to m2d. 3.4 cm2 to mm2 • 8.85 m2 to cm2f.1246mm2 to cm2 • 0.37 m2 to mm2 • 2 800 000 mm2 to m2 Q5g hint - Convert m2 to cm2 then to mm2.

  27. 7.2 – Units and Accuracy Page 208 • Calculate these areas. • ReflectIn part c, which was the easiest way to find the area in mm2? • Find the area in cm2, then convert to mm2. • Convert the lengths to mm, then find the area. • ReflectWhich was the easiest way to find the area of part d in cm2?

  28. 7.2 – Units and Accuracy Page 208 Here is the plan of a playing field.Work out the area of the field in hectares. Key Point 3 1 hectare (ha) is the area of a square 100 m by 100 m. 1 ha = 100 m x 100 m = 10000 m2 Areas of land are measured in hectares.

  29. 7.2 – Units and Accuracy Page 208 STEM / Problem-solving A farmer counts 2 wild oat plants in a 50cm by 50cm square of a field. The whole field has area 20 ha.Estimate the number of wild oat plants in the field. Q8 strategy hint- Work out the area of the square in m2. How many of these would fit in the field? Key Point 4 A 10% error interval means that a measurement could be up to 10% larger or smaller than the one given.

  30. 7.2 – Units and Accuracy Page 208 • A factory makes bolts 30 mm long, with a 10% error interval. • Work out the largest and smallest possible lengths of the bolts, • Write the possible lengths as an inequality. mm ≤ length ≤ mm • Sweets are packed in 20g bags, with a 5% error interval. Work out the possible masses of the bags of sweets. Q10 hint - g ≤mass ≤ g ^

  31. 7.2 – Units and Accuracy Page 209 • Reasoning • Each measurement has been rounded to the nearest cm.Write its smallest possible value, • 36cm ii.112cm • Each measurement has been rounded to 1 d.p. Write its smallest possible value, • 2.5 cm ii.6.7 kg • Discussion What is the largest possible value that rounds down to 36 cm? Key Point 5 Measurements rounded to the nearest unit could be up to half a unit smaller or larger than the rounded value. The possible values of x that round to 3.4 to 1 d.p. are 3.35 ≤ x≤3.45

  32. 7.2 – Units and Accuracy Page 209 • Each measurement has been rounded to the accuracy given.Write an inequality to show its smallest and largest possible values. Use x for the measurement, • 18 m (to the nearest metre) • 24.5 kg (to 1 d.p.) • 1.4 m (to 1 d.p.) • 5.26 km (to 2 d.p.) 12.5 ≤ x ≤ 13.5 lower upper bound bound Key Point 5 The upper bound is half a unit greater than the rounded measurement. The lower bound is half a unit less than the rounded measurement.

  33. 7.2 – Units and Accuracy Page 209 • Write • the lower bound • the upper bound of each measurement, • 8 cm (to the nearest cm) • 5.3 kg (to the nearest tenth of a kg) • 11.4 m (to 1 d.p.) • 2.25 litres (to 2 d.p.) • 5000 m (to 1 s.f.) • 32 mm (to 2 s.f.) • 1.53 kg (to 3 s.f.)

  34. 7.2 – Units and Accuracy Page 209 Example 2 The length of the side of a square is 5.34 cm to 2 d.p. Work out the upper and lower bounds for the perimeter. Give the perimeter to a suitable degree of accuracy.21.34 = 21.3 to 1 decimal place 21.38 = 21.4 to 1 decimal place 21.34 = 21 (nearest cm) 21.38 = 21 (nearest cm) Perimeter = 21 cm Use the upper and lower bounds of the side Length to calculate the upper and lower bounds of the perimeter. Round the upper and lower bounds to 1 decimal place. Do they give the same value? Round to nearest cm. They both give the same value so we can be sure that to the nearest cm, the perimeter is 21 cm.

  35. 7.2 – Units and Accuracy Page 209 • ReasoningA rectangle measures 15cm by 28cm to the nearest cm. • Work out the upper and lower bounds for the length and width. • Calculate the upper and lower bounds for the perimeter of the rectangle. Give the perimeter to a suitable degree of accuracy. Q14 strategy hint- Sketch a diagram. Which two possible lengths will give the largest possible perimeter?

  36. 7.2 – Units and Accuracy Page 210 • ReasoningA parallelogram has base length 9.4 m and height 8.5 m. Both measurements are given to 1 d.p. Work out the upper and lower bounds for its area. • Reasoning A parallelogram has area 24cm2 to the nearest whole number.Its height is 6.2 cm (to 1 d.p.). • Write the upper and lower bounds for the area and the height, • Work out • upper bound for area upper bound for height • lower bound for arealower bound for height • What is the upper bound for the base of the parallelogram? Q3 hint - Which calculation in part b gives the higher value?

  37. 7.3 – Prisms Page 210 ActiveLearn- Homework, practice and support: Higher 7.3

  38. 7.3 – Prisms Page 210 Warm Up • Work out the volume of this 3D solid made from two cuboids. • Solve • 9b = 72 b.h = 50 Key Point 1 The surface area of a 3D solid is the total area of all its faces.

  39. 7.3 – Prisms Page 210 • Reasoning • Sketch a cuboid 4 cm by 5 cm by 7 cm. • Work out the area of the top of your cuboid.Which other face of the cuboid is identical to this one? • Work out the area of the front and side of your cuboid. Which other faces of the cuboid are identical to them? • Work out the total surface area of your cuboid. • DiscussionHow can you calculate the surface area of a cuboid without drawing its net? • Calculate the surface area of a cuboid 3 cm x 2 cm x 6cm. Q3 hint– 2 x  + 2 x + 2 x = cm2

  40. 7.3 – Prisms Page 211 • Communication • Is the 3D solid in Q1 a prism? Explain your answer • Work out the area of the cross-section of the solid in Q1. • Multiply the area of the cross-section by the Length of the solid. What do you notice? • DiscussionFor a cuboid, why is multiplying area of cross-section by the length the same as multiplying length x width x height? Key Point 8 A prism is a 3D solid that has the same cross-section all through its length.

  41. 7.3 – Prisms Page 211 • Work out the volume of each prism. • ReasoningThis triangular prism has volume 48cm3. • Work out its height. • Sketch its net and work out its surface area. Q7 hint - Write and solve an equation: volume 48 = x h x  Key Point 9 Volume of prism = area of cross-section x length

  42. 7.3 – Prisms Page 211 8 – Exam-Style Questions Jane has a carton of orange juice. The carton is in the shape of a cuboid. The depth of the orange juice in the carton is 8 cm. Jane closes the carton. Then she turns the carton over so that it stands on the shaded face. Work out the depth, in cm, of the orange juice now. (3 marks) June 2012, Q12, 1MA0/1H Exam hint Add the level of the juice to the diagram. Sketch the carton when it stands on the shaded face.

  43. 7.3 – Prisms Page 212 • Reasoning • Sketch a cube with side length 1 cm and a cube with side length 10 mm. • Copy and complete. 1cm3= mm3 • How do you convert from mm3 to cm3? • Reasoning • Work out the volumes of a cube with side length 1 m and a cube with side length 100 cm. • How do you convert from m3 to cm3? Key Point 9 Volume is measured in mm3, cm3 or m3.

  44. 7.3 – Prisms Page 212 • Convert • 4.5 m3 into cm3 • 52 cm3into mm3 • 9500000 cm3into m3 • 3421 mm3 into cm3 • 5200 cm3 into litres • 0.7 litres into cm3 • 175 ml into cm3 • 3 m3into litres. Q11h hint- Convert 3m3to cm3 first. Key Point 9 Capacity is measured in ml and Litres. 1cm3= 1ml 1000 cm3 = 1 litre

  45. 7.3 – Prisms Page 212 • Problem-solvingA water tank is a cuboid 140 cm tall, 80cm wide and 2 m long. Kate paints all the faces except the base, • Work out the total area she paints, in square metres, • 1 tin of paint covers 4 m2. How many tins of paint does Kate need? • Problem-solving / Reasoning - A cube has surface area 507.8 cm2 to 1 d.p.What is the length of 1 side of the cube? out the area of 1 face.Give your answer to 1 d.p. Q12a hint– Sketch the task Q13 hint– First work out the area of 1 face.

  46. 7.3 – Prisms Page 212 • STEM / Modelling A scientist collects a sample of leaf mould 20 cm deep from a 0.25 m2 area in a wood. • By modelling the sample as a prism, calculate the volume of leaf mould she collects, • In the leaf mould sample she counts 12 worms. Estimate the number of worms in the top 20cm of leaf mould in 2 hectares of the wood. • Reasoning The dimensions of a cuboid are 5 cm by 3 cm by 8cm, measured to the nearest centimetre.Calculate the upper and lower bounds for the volume of the cuboid. Q16 strategy hint- First calculate the upper and lower bound of each measurement.

  47. 7.3 – Prisms Page 212 • CommunicationShow that the volume of this prism is 20x3. Q15 hint - Workout the volume using the measurements in x. Simplify your answer.

  48. 7.4 – Circles Page 213 ActiveLearn- Homework, practice and support: Higher 7.4

  49. 7.4 – Circles Page 213 Warm Up • Solve • 35 = 7r • 75 = 3r2 • Make xthe subject of • y= mx • t2= x2 • p= x2 Key Point 13 The circumference of a circle is its perimeter.

  50. 7.4 – Circles Page 213 • ReasoningThe table gives the diameter and circumference of some circles. • Work out the ratio circumference for each one. What do you notice? diameter • The ratio of a circle is represented by the Greek letter π(pi).Find the πkey on your calculator.Write the value of πto 8 d.p. • Discussion How can you workout the circumference of a circle if you know its diameter? What if you know its radius? Key Point 13 For any circle circumference = πx diameter C= πdor C = 2πr

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