3. Content and Skills Coverage and range: Level 1 Understand and use whole numbers and recognise negative numbers in practical contexts Add, subtract, multiply and divide using a range of mental methods Multiply and divide whole numbers by 10 and 100 using mental arithmetic Understand and use equivalences between common fractions, decimals and percentages Add and subtract decimal up to two decimal places Solve simple problems involving ratio, where one number is a multiple of the other Use simple formulae expressed in words for one- or two-step operations Solve problems requiring calculation with common measures including money, time, length, weight, capacity and temperature Convert units of measure in the same system Work out areas, perimeters and volumes in practical situations Construct models and draw shapes, measuring and drawing angles and identifying line symmetry Extract and interpret information from tables, diagrams, charts and graphs Collect and record discrete data and organise and represent information in different ways Find mean and range Use probability to show that some events are more likely to occur than others Understand outcomes, check calculations and explain results Understand and use positive and negative numbers of any size in practical contexts Carry out calculations with numbers of any size in practical contexts Understand, use and calculate ratio and proportion, including problems involving scale Understand and use equivalences between fractions, decimals and percentages Add and subtract fractions; add, subtract, multiply and divide decimals to a given number of decimal places Understand and use simple equations and simple formulae involving one- or two-step operations *Recognise and use 2D representations of 3D objects. Find area, perimeter and volume of common shapes Use, convert and calculate using metric and, where appropriate, imperial measures Collect and represent discrete and continuous data, using ICT where appropriate Use and interpret statistical measures, tables and diagrams, for discrete and continuous data using ICT where appropriate Use statistical methods to investigate situations Use a numerical scale from 0 to 1 to express and compare probabilities Title: Driving Lessons Content and skills covered Coverage and range: Level 2 At least 1 from each area
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5. Intro The legal age at which you can start driving in the UK is on your 17 th birthday and you must first have applied for and received a provisional driving licence ( cost £50 in 2008). On this date you may start your driving lessons. The DSA (Driver Standard Agency) say that “ There is no set number of driving lessons you must have before sitting your practical test. However, those who pass their driving test have had, on average, about 45 hours of professional training, combined with 22 hours of private practice. Candidates who combine professional instruction with private practice are also more successful on the test”. Apart from the practical test you must also sit a driving theory test. What maths might be involved in taking driving lessons?
6. Driving Schools Driving Lessons Peter is almost 17 years old and he has applied for his provisional driving licence. He is looking at taking driving lessons and has got some information on prices from three local driving schools. * Q1. How much would it cost him to take 10 lessons with FIRST CHOICE? £154 * Q2. How much would it cost him to take 10 lessons with “ACE”? £145 * Q3. How much cheaper is ACE than RAPID PASS for the first 10 lessons? £35 * Q4. What does a single 1 hour lesson cost with FIRST CHOICE when booked within the block booking special offer. £17.10 ACE DRIVING SCHOOL RAPID PASS DRIVING FIRST CHOICE MOTORING Single 1 Hour Lesson £20 First 5 lessons £9 each Double Lesson £40 10 hour block booking only £170!! Single 1 Hour Lesson £19 First 4 lessons £10 each Double Lesson £38 10 hour block booking 10% discount!! Single 1 Hour Lesson £20 First lesson Free Double £40 10 hour block booking £1 per hour discount!! *
7. Driving Lessons Sarah has already had some of the introductory lessons with FIRST CHOICE and she wants to book some more. * Q5. What will it cost her to book 4 double lessons with FIRST CHOICE? £152 * Q6. What will it cost her to book 2 double lessons and 5 single lessons with FIRST CHOICE? £171 * Q7. She notes that a FIRST CHOICE single lesson is 10% cheaper than RAPID PASS. True or false? False it is 5% cheaper * Q8. Who offers the best “block booking” value for money? ACE ACE DRIVING SCHOOL RAPID PASS DRIVING FIRST CHOICE MOTORING Single 1 Hour Lesson £20 First 5 lessons £9 each Double Lesson £40 10 hour block booking only £170!! Single 1 Hour Lesson £19 First 4 lessons £10 each Double Lesson £38 10 hour block booking 10% discount!! Single 1 Hour Lesson £20 First lesson Free Double £40 10 hour block booking £1 per hour discount!! *
8. Driving Lessons James does not feel confident about driving after doing some with his dad so he intends to book the recommended minimum of 45 hours of lessons right from the start with one of the schools above. He intends to utilise the block booking offers to keep the cost down. * Q9. How much would this cost with RAPID PASS? £840 * Q10. Which school is the cheapest for James’ 45 hours of lessons. ACE ACE DRIVING SCHOOL RAPID PASS DRIVING FIRST CHOICE MOTORING Single 1 Hour Lesson £20 First 5 lessons £9 each Double Lesson £40 10 hour block booking only £170!! Single 1 Hour Lesson £19 First 4 lessons £10 each Double Lesson £38 10 hour block booking 10% discount!! Single 1 Hour Lesson £20 First lesson Free Double £40 10 hour block booking £1 per hour discount!! *
9. Test Centre Test Centre Pass Rates * Q11. What percentage of people taking the test passed at the test centre in town C in in 2006 – 2007? 49% * Q12. What percentage of people taking the test failed at the test centre in town B in in 2007 – 2008? 58% * Q13. What is the modal pass rate for all 3 towns in all years? 49% * Q14. What is the range of the pass rates for this data? 13% * Q15. Work out the mean pass rate for the test centre in Town C. 48% James lives in a village in the countryside and has a choice of taking driving lessons in one of three small towns nearby. He looked up some information on pass rates at each of the three test centres in these towns. * 2005 - 2006 YEAR 2006 - 2007 2007 - 2008 51% Town A 49% 53% 41% Town B 40% 42% 46% Town C 49% 49%
10. * Q16. Work out the mean pass rate for the three centres in 2005-2006 46% * Q17. Which centre on average has the highest pass rate. Town A centre (51%) * Q18. What was the probability (scale 0 - 1) of a candidate for the test passing in 2006 – 2007 at the centre in town C? 0.49 * Q19. What was the probability of a candidate for the test failing in 2007 – 2008 at the centre in town B? 0.58 * Q20. James decides to sit his driving lessons and test in town A in 2009. If the probability of passing the test remains the same as 2007 - 2008 what is the probability that he will fail? 0.47 Test Centre Pass Rates James lives in a village in the countryside and had a choice of taking driving lessons in one of three small towns nearby. He looked up some information on pass rates at each of the three test centres in these towns. * 2005 - 2006 YEAR 2006 - 2007 2007 - 2008 51% Town A 49% 53% 41% Town B 40% 42% 46% Town C 49% 49%
11. Scatter Graph The table shown gives information on 12 people of various ages and the number of driving lessons that they had before passing their test. * Q21. Plot the scatter diagram for this data on the supplied grid. * Q22. Describe the type of correlation given by your scatter diagram. Negative * Q23. Give an estimate for the number of lessons that a 33 year old may need. 38 +/- 5 Number of Lessons 20 30 40 50 Age (years) 10 20 30 40 50 60 70 80 Age 20 22 23 25 27 30 Lessons 62 41 68 56 46 40 Age 35 36 40 42 45 50 Lessons 26 46 32 24 19 26 *
12. Stopping Distances Thinking Distance + Braking Distance Sarah is preparing for her theory test and is learning about stopping distances. Help her answer some questions on the chart below. * Q24. Describe in words how the thinking distances increase. Go up in 3’s * Q25. What is the missing braking distance at 40 mph? * Q26. What should it say below 50 mph? 24 m 80 km/h Stopping Distance = 24 m (80 km/h) 3 Car lengths 6 Car lengths 9 Car lengths 13 Car lengths 18 Car lengths 24 Car lengths * 20 mph (32 km/h) 30 mph (48 km/h) 40 mph (64 km/h) 50 mph (? km/h) 60 mph (96 km/h) 70 mph (112 km/h) 6 m 6 m 9 m 14 m 12 m 24 m 15 m 38 m 18 m 55 m 21 m 75 m 12 metres (40 feet) 23 metres (75 feet) 36 metres (118 feet) 53 metres (175 feet) 74 metres (240 feet) 96 metres (315 feet) TYPICAL STOPPING DISTANCES
15. Road Signs Road Signs James was studying the road signs shown below when he started to think about some of their mathematical properties. * Q31. Which road sign is an octagonal shape? Stop Sign * Q32. Which road sign has 4 lines of symmetry? * Q33. Which road signs have rotational symmetry of order 3? *
16. Road Signs * Q34. Which two circular road signs have two lines of symmetry? * Q35. Identify the triangular signs that have 1 line of symmetry. * Q36. Which circular signs have only 1 line of symmetry? James was studying the road signs shown below when he started to think about some of their mathematical properties. *
17. Road Signs * Q37. Express the proportion of road signs that are triangular as: (a) A fraction (b) A decimal (c) A percentage 9/20 0.45 45% * Q38. Express the proportion of road signs that are octagonal as: (a) A fraction (b) A decimal (c) A percentage 1/20 0.05 5% James was studying the road signs shown below when he started to think about some of their mathematical properties. *
18. Passing the Test 1 Number of attempts 2 3 4 5 12 Number of drivers 7 3 2 1 Total 25 Sarah managed to pass her test at the first attempt. She made a list of family and friends and recorded the number of attempts that each had before passing the practical test. * Q39. What percentage of the people passed at the first attempt? 48% * Q40. Determine the median number of attempts to pass. 2 Q41. Calculate the mean number of attempts to pass. 1.92 *
19. Teacher Q + A £154 £145 £35 £17.10 £152 £171 False it is 5% cheaper ACE £840 ACE 49% 58% 49% 13% 48% 46% Town A centre (51%) 0.49 0.58 0.47 *Q1. How much would it cost him to take 10 lessons with FIRST CHOICE? *Q2. How much would it cost him to take 10 lessons with “ACE”? *Q3. How much cheaper is ACE than RAPID PASS for the first 10 lessons? *Q4. What does a single 1 hour lesson cost with FIRST CHOICE when booked within the block booking special offer. *Q5. What will it cost her to book 4 double lessons with FIRST CHOICE? *Q6. What will it cost her to book 2 double lessons and 5 single lessons with FIRST CHOICE *Q7. She notes that a FIRST CHOICE single lesson is 10% cheaper than RAPID PASS. True or false? *Q8. Who offers the best “block booking” value for money? *Q9. How much would this cost with RAPID PASS? *Q10. Which school is the cheapest for James’ 45 hours of lessons. *Q11. What percentage of people taking the test passed at the test centre in town C in in 2006 – 2007? *Q12. What percentage of people taking the test failed at the test centre in town B in in 2007 – 2008? *Q13. What is the modal pass rate for all 3 towns in all years? *Q14. What is the range of the pass rates for this data? *Q15. Work out the mean pass rate for the test centre in Town C. *Q16. Work out the mean pass rate for the three centres in 2005-2006 *Q17. Which centre on average has the highest pass rate. *Q18. What was the probability (scale 0 - 1) of a candidate for the test passing in 2006 – 2007 at the centre in town C? *Q19. What was the probability of a candidate for the test failing in 2007 – 2008 at the centre in town B? *Q20. James decides to sit his driving lessons and test in town A in 2009. If the probability of passing the test remains the same as 2007 - 2008 what is the probability that he will fail? Teacher Q + A
20. *Q21. Plot the scatter diagram for this data on the supplied grid. (see worksheet) *Q22. Describe the type of correlation given by your scatter diagram. Negative *Q23. Give an estimate for the number of lessons that a 33 year old may need. 38 +/- 5 *Q24. Describe in words how the thinking distances increase. Go up in 3’s *Q25. What is the missing braking distance at 40 mph? *Q26. What should it say below 50 mph? 24 m 80 km/h *Q27. At 40 mph what is the ratio of thinking distance: stopping distance 1:2 *Q28. At 40 mph what fraction of the overall stopping distance is the braking distance. 2/3 *Q29. The stopping distance at 70 mph is 315 feet. How many yards is this? 105 *Q30. Pete told Sarah that the formula shown above will calculate stopping distances in feet (d). Check that this formula works for all speeds (v) in mph shown in the chart. (see Slide 14) *Q31. Which road sign is an octagonal shape? Stop Sign *Q32. Which road sign has 4 lines of symmetry? Slide 15 *Q33. Which road signs have rotational symmetry of order 3? Slide 15 *Q34. Which two circular road signs have two lines of symmetry? Slide 16 *Q35. Identify the triangular signs that have 1 line of symmetry. Slide 16 *Q36. Which circular signs have only 1 line of symmetry? Slide 16 *Q37. Express the proportion of road signs that are triangular as: (a) A fraction (b) A decimal (c) A percentage 9/20 0.45 45% *Q38. Express the proportion of road signs that are octagonal as: (a) A fraction (b) A decimal (c) A percentage 1/20 0.05 5% *Q39. What percentage of the people passed at the first attempt? 48% *Q40. Determine the median number of attempts to pass. 2 Q41. Calculate the mean number of attempts to pass. 1.92
21. Student Questions *Q1. How much would it cost him to take 10 lessons with FIRST CHOICE? *Q2. How much would it cost him to take 10 lessons with “ACE”? *Q3. How much cheaper is ACE than RAPID PASS for the first 10 lessons? *Q4. What does a single 1 hour lesson cost with FIRST CHOICE when booked within the block booking special offer. *Q5. What will it cost her to book 4 double lessons with FIRST CHOICE? *Q6. What will it cost her to book 2 double lessons and 5 single lessons with FIRST CHOICE *Q7. She notes that a FIRST CHOICE single lesson is 10% cheaper than RAPID PASS. True or false? *Q8. Who offers the best “block booking” value for money? *Q9. How much would this cost with RAPID PASS? *Q10. Which school is the cheapest for James’ 45 hours of lessons. *Q11. What percentage of people taking the test passed at the test centre in town C in in 2006 – 2007? *Q12. What percentage of people taking the test failed at the test centre in town B in in 2007 – 2008? *Q13. What is the modal pass rate for all 3 towns in all years? *Q14. What is the range of the pass rates for this data? *Q15. Work out the mean pass rate for the test centre in Town C. *Q16. Work out the mean pass rate for the three centres in 2005-2006 *Q17. Which centre on average has the highest pass rate. *Q18. What was the probability (scale 0 - 1) of a candidate for the test passing in 2006 – 2007 at the centre in town C? *Q19. What was the probability of a candidate for the test failing in 2007 – 2008 at the centre in town B? *Q20. James decides to sit his driving lessons and test in town A in 2009. If the probability of passing the test remains the same as 2007 - 2008 what is the probability that he will fail? Student
22. *Q21. Plot the scatter diagram for this data on the supplied grid. (see worksheet) *Q22. Describe the type of correlation given by your scatter diagram. *Q23. Give an estimate for the number of lessons that a 33 year old may need. *Q24. Describe in words how the thinking distances increase. *Q25. What is the missing braking distance at 40 mph? *Q26. What should it say below 50 mph? *Q27. At 40 mph what is the ratio of thinking distance: stopping distance *Q28. At 40 mph what fraction of the overall stopping distance is the braking distance. *Q29. The stopping distance at 70 mph is 315 feet. How many yards is this? *Q30. Pete told Sarah that the formula shown above will calculate stopping distances in feet (d). Check that this formula works for all speeds (v) in mph shown in the chart. (see Slide 14) *Q31. Which road sign is an octagonal shape? *Q32. Which road sign has 4 lines of symmetry? Slide 15 *Q33. Which road signs have rotational symmetry of order 3? Slide 15 *Q34. Which two circular road signs have two lines of symmetry? Slide 16 *Q35. Identify the triangular signs that have 1 line of symmetry. Slide 16 *Q36. Which circular signs have only 1 line of symmetry? Slide 16 *Q37. Express the proportion of road signs that are triangular as: (a) A fraction (b) A decimal (c) A percentage *Q38. Express the proportion of road signs that are octagonal as: (a) A fraction (b) A decimal (c) A percentage *Q39. What percentage of the people passed at the first attempt? *Q40. Determine the median number of attempts to pass. Q41. Calculate the mean number of attempts to pass.
23. Worksheet 1 ACE DRIVING SCHOOL RAPID PASS DRIVING FIRST CHOICE MOTORING Single 1 Hour Lesson £20 First 5 lessons £9 each Double Lesson £40 10 hour block booking only £170!! Single 1 Hour Lesson £19 First 4 lessons £10 each Double Lesson £38 10 hour block booking 10% discount!! Single 1 Hour Lesson £20 First lesson Free Double £40 10 hour block booking £1 per hour discount!! Worksheet 1 2005 - 2006 YEAR 2006 - 2007 2007 - 2008 51% Town A 49% 53% 41% Town B 40% 42% 46% Town C 49% 49%
24. Worksheet 2 Worksheet 2 Number of Lessons 20 30 40 50 Age (years) 10 20 30 40 50 60 70 80 Age 20 22 23 25 27 30 Lessons 62 41 68 56 46 40 Age 35 36 40 42 45 50 Lessons 26 46 32 24 19 26
25. Worksheet 3 20 mph (32 km/h) 30 mph (48 km/h) 40 mph (64 km/h) 50 mph (? km/h) 60 mph (96 km/h) 70 mph (112 km/h) 6 m 6 m 9 m 14 m 12 m 24 m 15 m 38 m 18 m 55 m 21 m 75 m 12 metres (40 feet) 23 metres (75 feet) 36 metres (118 feet) 53 metres (175 feet) 74 metres (240 feet) 96 metres (315 feet) TYPICAL STOPPING DISTANCES Thinking Distance + Braking Distance Worksheet 3 3 Car lengths 6 Car lengths 9 Car lengths 13 Car lengths 18 Car lengths 24 Car lengths
26. Worksheet 4 Q 31 - 38 Worksheet 4 1 Number of attempts 2 3 4 5 12 Number of drivers 7 3 2 1 Total 25
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