# Teaching Notes. Contextualised task 35 The 100 Metre Race

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2 Task A: The Runner and the Cyclist Outline Phase 1 Students are presented with Usain Bolt s world record time for the 100m sprint, and the world record for the cyclist René Enders to cover 250m from a standing start. They are invited to consider how to use this information in order to compare if the runner or the cyclist would reach 100m first in a race. This part of the activity is suited for about 20 minutes as part of one lesson. The learners work can then be collected in, and assessed ready for the next lesson, where they should be given the opportunity to improve their work from the feedback given. Phase 2 Learners initial work is returned to them to improve. They have the opportunity to address the questions / guidance offered in the marking, and then to evaluate other work presented on the PowerPoint. Having discussed what further information they might need in Phase 1, some data are now provided at this point: split times for every 10m of Bolt s 100m record and the half-lap split times for a cyclist. This data should help learners refine their calculations and any graphs they draw. You will need For phase 1: Teachers script PowerPoint 1 slides 1-4 A selection of plain, lined, squared and graph paper Calculators / rulers / pencils Assessment criteria: Phases 1 and 2: Learner Outcomes and Assessment. Additionally for phase 2: PowerPoint 1 slides 5-9 Learner Resource Sheet 1: Split times

3 Task B: An old hatchback Outline This is only suitable for higher tier. A graph of the performance data for a hatchback car is introduced (using imperial rather than metric measures), and learners are asked to confirm whether or not the car would outperform the runner and cyclist over 100m. As this is a graph of changing speed over time, the problem can be solved using the area under the graph, and the trapezium rule. The teaching notes allow for this task to be presented either for learners to tackle themselves or as a worked example for the teacher to demonstrate, if the learners have a limited experience of this type of problem. You will need Teachers script PowerPoint 2: An Old Hatchback Learner Resource Sheet 1: Split times (if needed as a reminder) Learner Resource Sheet 2: Hatchback performance Learner Resource Sheet 3: Some hints A selection of plain, lined, squared and graph paper Calculators / rulers / pencils Mark scheme provided

4 Task A Phase 1: Teachers script for PowerPoint presentation The text in the right-hand boxes provides a possible script to be read to students. This is in plain text, with questions for the class shown in bullet points. Possible (correct) student responses are written in bold; instructions are written in italics. However, it is preferable to use your own words and elaboration, and to manage the resources as best suits your class. When questions are asked, time for discussion in pairs / groups should be provided. Ensure that students are given opportunity to explain their reasoning in response to these questions, to enable them to understand the concepts so they can progress. Slide 1 Keep this slide on the screen until you are ready to start the presentation The 100 metre race THE WORLD S GREATEST SPRINTER, USAIN BOLT, AGAINST THE WORLD S FASTEST SPRINT CYCLIST, RENÉ ENDERS. WHO WINS? Slide 2 Usain Bolt vs Rene Enders over 100 metres: who wins the race? THE RUNNER: USAIN BOLT Men s sprint 100m World Record is 9.58 seconds stm Alternative: LMM THE CYCLIST: RENÉ ENDERS The men s cycling world record for a 250m lap from a standing start is seconds, by René Enders of Germany. umwu (This video is of the German team, which includes Enders. A video of the world record ride is not available). You may wish to show the video clip of Usain Bolt s world record in Berlin in There are two options for this. A standard lap in a velodrome is 250 metres. The video clip for the cyclist is of Germany vs New Zealand in the World championships of The German team includes Rene Enders, and also shows clearly the riders striving to achieve 17 seconds for the first lap: 250 metres. We do not have the video for the world record attempt. If you can rewind and pause the video at the moment the riders have completed the first lap one or two questions will help students understand what is going on. The lap time for Germany is seconds. Which rider is given that time? (The third and final rider in the team) What is the time given at the bottom of the screen? (The extra time taken by the final NZ rider) What is seconds in hundredths or thousandths? 38 thousandths of a second, or 3 hundredths and 8 thousandths of a second.

5 Slide 3 Slide 4 Explore some ways to solve the problem. How could you use this information Usain Bolt s 100 metre World to decide who will win in a 100 Record is 9.58 seconds metre race from a standing start? René Enders 250 metre (one lap) What calculations could you make? cycling World Record is seconds, from a standing start. What diagrams could you draw? Explore some ways to solve the problem. What assumptions are you making? Usain Bolt s 100 metre World Record is 9.58 seconds What are the limitations of the data we have? René Enders 250 metre (one lap) cycling World Record is What further information do you seconds, from a standing start. need? Why does a rider peel away each lap? It s the final rider who completes three laps, and they are being helped by being behind the others which reduces the wind resistance. It is also possible that each member of the team specialises in one aspect of the race: the front rider for the start and generating initial pace, the second rider for the best time over two laps, and the third has the greatest endurance for all three laps. So, if Usain Bolt and Rene Enders meet in a 100 metres showdown: who will win? How could we work it out? I will give you a few minutes (specify your own time suitable for the class anything up to 10 minutes) to see how you get started. Some clues are in the questions on the left of the slide. We will then share some ideas. As you circulate around the class, you will notice good examples of working which learners may wish to share with the class. You will also pick up others who may be stuck. Ask them to consider What calculations can they make from the information? What diagrams could they draw? What would the diagrams look like? You may wish to hold up some examples of different calculations or graphs from the class to discuss. You may wish to give the class a few more minutes to finish their work after this discussion if seeing some examples prompts them to improve it. Ask each question from the slide: What assumptions are we making? E.g. some ideas could be: a straight line graph or an average speed does not take into account acceleration, but assumes both go straight into a steady speed. No fluctuations in speed. No reaction time at the start. If a curve is drawn, there will be an assumed rate of acceleration, as we do not have the data. What are the limitations of the data we have?

7 Phases 1 and 2: Learner Outcomes and Assessment (To aid comment-only marking) Reasoning strand Learners are able to: Assessment Guidance: Can learners - Transfer mathematical skills across the curriculum in a variety of contexts and everyday situations Get started by choosing a calculation to do (e.g. average speed, or distance travelled after 100m for the cyclist)? Select, trial and evaluate a variety of possible approaches and break complex problems into a series of tasks Get started by drawing a suitable distance-time graph, showing the progress of each of the athletes over their races? Select appropriate mathematics and techniques to use Identify what further information might be required Identify how they may represent acceleration on their graphs, and what implications that has about the shape, gradient and direction of their graph? and select what information is most appropriate Suggest or create a different diagram to represent the Explain results and procedures precisely using situation? appropriate mathematical language Identify the assumptions made in the calculations / Select and construct appropriate charts, diagrams graphs / diagrams? and graphs with suitable scales Suggest improvements they could make in their Interpret graphs that describe real-life situations, calculations / graphs? including those used in the media, recognising that Interpret their graph, to identify the winner? some graphs may be misleading Identify the limitations of the data they have, and Evaluate different forms of recording and presenting suggest what further information would be useful to information, taking account of the context and make their work more accurate? audience Evaluate the class s presentations of the data, suggesting Select and apply appropriate checking strategies improvements and better solutions? Interpret mathematical information; draw Use reliable methods to check calculations, and the inferences from graphs, diagrams and data, accuracy of their graphs or diagrams? including discussion on limitations of data

10 Solutions and Mark scheme: Phases 1 and 2 The suggestions below are only that. Different possibilities will arise from your group, giving the opportunity to evaluate their methods. Phase 1: A good solution is not going to be found with the initial data, so it is important that we focus our attention on deciding what information may be needed. Some learners may offer the kinds of calculations or graphs that are given in phase 2, both of which suggest that Enders would win the race fairly easily. However, this would assume that there is no acceleration that both athletes reach top speed immediately. Other learners may develop more complex responses e.g. they make assumptions themselves about the acceleration, and how it can be presented. They may therefore attempt a graphical solution but be uncertain about the direction of the curve, if the speed is changing. When assessing learners work, this could be a point to pick up. So we are looking for learners to identify some more information about the progress of the athletes races e.g. split times, data about how the athletes accelerate, when the athletes reach top speed, what their top speed may be. Phase 2: Using better data It is during phase 2, when there are more detailed data to work with that we would expect to see reasonable calculations or graphs that would lead to a solution, e.g. A cyclist reaches top speed at 90 metres, and combined with the information that they are expected to cover 125 metres in 6.75 seconds, we can suggest their estimated split times from 90m to 250m. (See spreadsheet extract below, and associated graph). Assuming that the final 6.75 seconds are all at top speed, this would be m/s. If the cyclist is riding at top speed from 90 metres, then we can subtract the time taken for the final 150 meters of his 250 lap, to give us a time for 100m. o 6.75 seconds x 150 / 125 = 8.10 seconds. This is the time for the final 150m seconds = seconds at 100m. (A longer spreadsheet version of this is included below, with a graphical representation of the data).

11 Accuracy: Most world records timings are presented to the nearest 100 th of a second, but Enders world record is given in thousandths, and as the figures are presented to the learners both to 2d.p. and to 3d.p. accept either in their presentation of the work. However, there must be some conversation about the degree of accuracy we can use, when there is incompleteness about the data we are using for the cyclist. It is possible that learners will argue that Enders should be a better rider than the qualification standard for the Australian team (as he is the world record holder), and that he would ride faster than them over the last 150m. He may therefore have a 100m time that is less than seconds, but not by the full seconds that his world record time betters the Australian 17.4 seconds standard for a full lap. Learners may subtract a reasonable proportion of the seconds, and as soon as they do, then they are working with less accuracy. So there is good reason to accept a figure such as 8.7 seconds, if it is supported by an argument about the quality of data they are working with. Mark scheme Full Credit States that Enders will win the race, and gives an accurate time for him to complete the 100m. Offers an answer to a suitable degree of accuracy (i.e. 2 or 3 decimal places, as presented in the original world record data), but also states that the cycling data is not sufficiently accurate to give a definitive solution to this degree of accuracy. Gives the winning time as seconds, or less, but no lower than about 8.65 seconds. This will enable them to present an argument that Enders world record time is just over 0.4 seconds inside the Australian standards, and so considers subtracting a proportion of this from the time above. Accept a reasonable argument and a reasonable time, such as one between 8.65 to 8.88 seconds. May present their working in any of a variety of ways, but must draw conclusions and state their answers clearly. Partial Credit States that Enders will win the race, but rounds to the nearest second or does not form an argument about the quality of the data. OR Working is appropriate and accurate, but solution is not clearly stated. No credit Any other response.

12 Spreadsheet and graph Estimated Split Times (seconds) Distance (m) Enders Bolt

13 Task B: An old hatchback Outline This is only suitable for higher tier. Firstly, a graph of the performance data for a hatchback car is introduced using an imperial measure (mph), and learners are invited to consider how the information should be adapted to help them compare the athletes performance. They are then offered data that has been converted to metres per second. Learners are asked to confirm whether or not the car would outperform the runner and cyclist over 100m. As this is a graph of changing speed over time, the problem can be solved by finding the area underneath the graph, possibly using the trapezium rule. The learning resources include a hints sheet to offer some scaffolding for learners. You will need Teachers script Learners previous work from phases 1 and 2. PowerPoint 2 slides: An Old Hatchback Learner Resource Sheet 1: Split times (if needed as a reminder) Learner Resource Sheet 2: Hatchback performance Learner Resource Sheet 3: Some hints A selection of plain, lined, squared and graph paper Calculators / rulers / pencils Mark scheme provided

14 Task B: Teachers script for PowerPoint presentation The text in the right-hand boxes provides a possible script to be read to students. This is in plain text, with questions for the class shown in bullet points. Possible (correct) student responses are written in bold; instructions are written in italics. Slide 1 Slide 2 The 100 metre race THE WORLD S GREATEST SPRINTER, USAIN BOLT, AND THE WORLD S FASTEST SPRINT CYCLIST, RENÉ ENDERS... against this old hatchback over 100 metres. Will the hatchback win? Keep this slide on the screen until you are ready to start the presentation. Ensure that learners have their previous work on finding who wins between Bolt and Enders. You may wish to organise your class into pairs / groups for this activity, and ensure that learners have the equipment they need: calculators, paper / graph paper, pencils rulers. We are going to see if both Usain Bolt and René Enders will beat Bring in the next slide This old car! Will Enders beat this car? Will Bolt beat this car? Slide 3 Usain Bolt and René Enders over 100 metres: their world records. THE RUNNER: USAIN BOLT Men s sprint 100m World Record is 9.58 seconds. THE CYCLIST: RENÉ ENDERS The men s cycling world record for a 250m lap from a standing start is seconds, by René Enders of Germany. This suggests that he will cover the first 100m in 8.88 seconds. This is just to remind you of the work we have done so far. A reasonable time for René Enders over 100m is about 8.88 seconds. You may wish to hand out Learner Resource sheet 1: Split times to remind learners of the data.

17 Speed (m/s) Speed (m/s) 0 30 Hatchback Acceleration in m/s Time elapsed (seconds) 30 Hatchback Acceleration in m/s Time elapsed (seconds) Solutions and possible working These are only suggestions: learners may offer a variety of strategies which give a reasonable estimate of the distance covered by the car after certain times in its journey. Learners may begin by considering an estimate of the distance covered after a time that they know will beat Enders time of 8.88 seconds. The graph shows that after about 8.7 seconds the car will be travelling at approximately 20m/s. The area of the triangle from start to this point would be ½ x 8.7 x 20 = 10 x 8.7 = 87 metres. There remains an area under the graph not covered by the triangle, so this may be a good starting point. Learners may then wish to take each part of the graph at a time. This will be more accurate. After 2 seconds, the car has travelled 7 metres. The following parts use the formula for the area of a trapezium. After 4 seconds, the car has travelled 19 more metres. (½ (7 + 12)) x 2). Running total = 26 metres. After 6 seconds, the car has travelled (½ ( )) x 2) = 28 more metres. Running total = 54 metres. After 8 seconds, the car has travelled (½ ( )) x 2) = 35 more metres. Running total = 89 metres. To ensure that the car beats Enders, we need to look at times less than 8.88 seconds. After 8.5 seconds the car has reached about 19.7 m/s. After 8.5 seconds the car has travelled a further (½ ( )) x 0.5) = 38.7 / 4 = 9.7 metres, a running total of 98.7 metres.

18 Calculating for 8.7 seconds and 20 m/s: (½ ( )) x 0.7) = more metres after 8 seconds: metres. A good estimate therefore, is about 8.6 seconds. (½ ( )) x 0.6) = more metres after 8 seconds, giving a total of metres after 8.6 seconds. Applying the trapezium rule (up to 8 seconds) where we have considered the width of each trapezium to be 2, and the side lengths of each trapezium are given by the speeds at each 2-second interval, the estimated area under the graph after 8 seconds would be: = ½ (2) (0 + 2( ) + 19) = 2(35) + 19 = 89. Learners could, and should argue that the data is limited, and what we have is probably not accurate enough to come to a final, definitive conclusion. They might suggest that the car just beats Enders, but it could be argued that we do not have his split times, and only the data from comparable (but slightly slower) cyclists from the Australian team qualification. Both car and Enders will beat Usain Bolt. Mark Scheme Full Credit Clear working to find the area under the graph or some other means to lead to a suggestion that the car will reach 100m after about 8.6 seconds, and therefore is likely to beat Rene Enders, but perhaps only just. Other means could also include: Making suitable calculations of distance covered from average speeds in each section of the graph. Adding this information to the graph of Bolt and Enders, to establish the progress of the car over time. States that the data is limited, and therefore accuracy to 2 or 3 decimal places is unnecessary. Offers a time of 8.6 seconds. Partial Credit Correctly calculates at least four sections under the graph. Offers a reasonable solution, but does not suggest that the data is limited. Correctly calculates all sections, but does not offer a clear final solution. No credit Any other response.

19 Progression in reasoning Identify processes and connections select, trial and evaluate a variety of possible approaches and break complex problems into a series of tasks Represent and communicate select and construct appropriate charts, diagrams and graphs with suitable scales Review select and apply appropriate checking strategies interpret mathematical information; draw inferences from graphs, diagrams and data, including discussion on limitations of data Start the problem by looking at some calculations, e.g. comparing mph with m/s. Perhaps makes some estimates of distance covered over given numbers of seconds. Choose suitable scales for any graphs, to aid clarity and ease of interpretation? Ensure the calculations they make are checked for accuracy. Identify what further working may be needed to solve the problem. Break the problem into parts e.g. identify that they will compare distances or speeds, and that they will need to convert one set of information to compare with the other either with speeds or distances. Identify further information that may be needed or useful, beyond that on the first slide i.e. a breakdown of Bolt s 100m at split times, and / or a breakdown of how the car accelerates. Organise work clearly. Draw suitable tables / charts / graphs that show comparison e.g. own conversion graph between mph and m/s? Interpret their charts appropriately. Check that they have answered the problem, and that their results are sensible. Identify a way of comparing the car speeds at various times, with the split times of the sprinter. This could be by creating a speed /time graph; a distance / time graph; a table of estimations; or other means where both sets of data are shown. Evaluate their methods of calculating and presenting their findings, with regard to clarity and efficiency. Discuss what limitations there may be in terms of accuracy from the data given.

20 GCSE Content GCSE Mathematics Numeracy and GCSE Mathematics Understanding number relationships and methods of calculation Using the facilities of a calculator, including the constant function, memory and brackets, to plan a calculation and evaluate expressions. Knowing how a calculator orders its operations. Solving numerical problems Giving solutions in the context of a problem, selecting an appropriate degree of accuracy, interpreting the display on a calculator, and recognising limitations on the accuracy of data and measurements. Rounding an answer to a reasonable degree of accuracy in the light of the context. Interpreting the display on a calculator. Knowing whether to round up or down as appropriate. Understanding and using functional relationships Construction and interpretation of conversion graphs. Construction and interpretation of travel graphs. Construction and interpretation of graphs that describe real-life situations. Constructing and using tangents to curves to estimate rates of change for nonlinear functions, and using appropriate compound measures to express results, including finding velocity in distance-time graphs and acceleration in velocity-time graphs. Interpreting the meaning of the area under a graph, including the area under velocity-time graphs and graphs in other practical and financial contexts. Using the trapezium rule to estimate the area under a curve. GCSE Mathematics only Key Foundation tier content is in standard text. Intermediate tier content which is in addition to foundation tier content is in underlined text. Higher tier content which is in addition to intermediate tier content is in bold text.

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