Record Breakers This investigation will be assessed against Criteria A, C and D On the attached data sheet are details of records achieved between 1942 and 1993 in the men s one mile race. In this investigation, you are going to use what you know about linear and other graphs to analyse this information and answer questions on your findings. 1. Select 12 record holders and clearly mark the ones you have chosen by creating a table. Explain your system for choosing those 12 and why you thought that system was best. Could you have chosen them differently? How? Why did you not use this other method? 2. Now, without using technology, you are going to plot your 12 chosen values on a graph, plotting the year against the record time. To make this easier, first create a table containing the points you are going to plot. When drawing the graph, you should let the x- axis represent the year since 1900 (you will need to think carefully about the best way to do this!) and let the y-axis represent the time in seconds. Show all the working you had to do before you were able to record your chosen data in the table ready for graphing. How accurate do you think the time should be? Why do you think this is a good choice? 3. Choosing a suitable scale, plot your values on the graph paper provided. Once you have done this, try to draw a line of best fit for your data. How good a fit do you think your line is? 4. Find the equation of your line of best fit. Show clearly how you found the gradient and the y-intercept, either directly from your graph or by calculation. Looking at your graph, does this equation make sense? Explain. How could you check that your equation is correct? Do you think the rest of the class will get exactly the same equation as you? Explain. 5. Now use your equation to predict : (a) The record time in 2010. (b) In what year the record time will be 3 minutes 30 seconds. To what degree of accuracy should you give each of these predictions? Explain. Do you think your predictions would make sense in the real world of athletics? Again, explain your thinking.
6. Would it be reasonable to use your model to estimate when someone is likely to run a mile in 3 minutes? Explain your thinking. 7. Now, using Geogebra, try to find a linear model for your data. *8. Now try finding some non-linear models for your data. Include any graphs you draw and make sure you show the equations for them clearly. What kind of model do you think reflects the data most accurately linear or non-linear? Why do you think this? Do you think you are more likely to get the same model as the other students using technology? Explain. 8. Compare your model with the equation you got from your hand-drawn graph. Are they very different? Why? Do you think you would get a more accurate model if you used all the data? Explain. Now, try entering all the data in Geogebra to check. 9. Read this excerpt from the Guinness Book of Records: The fastest men s mile record is held by Hicham El Guerrouj of Morocco, who set the world mark at that distance with 3 minutes 43.13 seconds in Rome, Italy on July 7 th, 1999. El Guerrouj is the lightest runner to hold the mile record. In the 2002 Olympic Games in Sydney, Australia he won silver in the men s 1500m. (a) Investigate whether El Guerrouj fits your model. (b) Would you expect someone to have beaten this record since 1999? Explain. Now do some research to see if your assumption is correct. Does what your research affect your thinking about whether he fits your model? Why? And finally - Can you think of some other real-life examples where you could use similar techniques to find models that describe real-life relationships between things? Give details of your ideas. - In general, are there any limitations to predicting results from the models in this task or from models in any of the other real-life situations you have just described? - Why might you want to use this technique to predict results in real life?
DATA SHEET: World Records for One Mile Race (Men) Year Date Place Athlete Nationality Time 1942 July 1 st Gothenburg Gunder Hagg Sweden 4m 6.2s 1942 September 4 th Stockholm Gunder Hagg Sweden 4m 4.6s 1943 July 1 st Gothenburg Arne Andersson Sweden 4m 2.6s 1944 July 18 th Malmo, Sweden Arne Andersson Sweden 4m 1.6s 1945 July 17 th Malmo, Sweden Gunder Hagg Sweden 4m 1.4s 1954 May 6 th Oxford Roger Bannister UK 3m 59.4s 1954 June 21 st Turku, Finland John Landy Australia 3m 58.0s 1957 July 19 th London Derek Ibbotson UK 3m 57.2s 1958 August 6 th Dublin Herb Elliott Australia 3m 54.5s 1962 January 27 th Wanganui, NZ Peter Snell New Zealand 3m 54.4s 1964 November 17 th Auckland Peter Snell New Zealand 3m 54.1s 1965 June 9 th Rennes Michel Jazy France 3m 53.6s 1966 July 17 th Berkeley, USA Jim Ryun USA 3m 51.3s 1967 June 23 rd Bakersfield, USA Jim Ryun USA 3m 51.1s 1975 May 17 th Kingston, Jamaica Filbert Bayi Tanzania 3m 51.0s 1975 August 12 th Gothenburg John Walker New Zealand 3m 49.4s 1979 July 17 th Oslo Sebastian Coe UK 3m 49.0s 1980 July 1 st Oslo Steve Ovett UK 3m 48.8s 1981 August 19 th Zurich Sebastian Coe UK 3m 48.53s 1981 August 26 th Koblenz Steve Ovett UK 3m 48.40s 1981 August 28 th Brussels Sebastian Coe UK 3m 47.33s 1985 July 27 th Oslo Steve Cram UK 3m 46.32s 1993 September 5 th Stuttgart Noureddine Morceli Algeria 3m 44.39s
Breaking Records Investigation Name Criterion A: Knowledge and Understanding Descriptor 1-2 The student attempts to make deductions when solving simple problems in familiar contexts. You had a valid system for choosing 12 points. You plotted your points on a graph. You drew a line of best fit and tried to find its equation. You tried to use Autograph or your GDC. 3-4 The student sometimes makes appropriate deductions when solving simple and more- complex problems in familiar contexts. 5-6 The student generally makes appropriate deductions when solving challenging problems in a variety of familiar contexts. 7 8 The student consistently makes appropriate deductions when solving challenging problems in a variety of contexts including unfamiliar situations. You were correctly able to find an equation of your line of best fit. You tried to use it to make predictions. You correctly used Autograph or your GDC to find a model. You tested your line of best fit. You made correct predictions and wrote down enough working to show this. You were correctly able to use either Autograph or your GDC to find more than one other model. You briefly discussed which model was best, but did not really back up your thinking with evidence. You stated whether or not El Guerrouj fitted your model. You showed complete confidence in modelling data using different methods and were able to come up with at least one correct, non- linear model for the data. You decided on the best possible model and discussed why it was the best. You provided detailed reasoning on whether or not El Guerrouj fitted the model, backing your thinking up with numerical evidence. Criterion C: Communication in Mathematics Descriptor 1-2 The student shows basic use of mathematical language and/or forms of You plotted some points on a graph and tried to draw a line of best fit. You tried to use basic mathematical representation. The lines of mathematical language to write models and reasoning are difficult to follow. explain some answers. You did not connect your graphs and answers very well and your thinking was difficult to follow. 3-4 The student shows sufficient use of mathematical language and forms of mathematical representation. The lines of reasoning are clear though not always logical or complete. The student moves between different forms of representation with some success. 5-6 The student shows good use of mathematical language and forms of mathematical representation. The lines of reasoning are concise, logical and complete. The student moves effectively between different forms of representation. You were able to make good use of mathematical language to write models and explain your answers and predictions. You chose suitable scales for your graphs and related your work to those graphs. You showed evidence of using technology. You presented your work clearly and logically, but you missed some steps. You were able to combine mathematical language, graphs and calculations easily to find models and make predictions and you explained everything clearly and in detail. Your work was easy to follow and you made excellent use of technology to help you.
Criterion D: Reflection in Mathematics Descriptor 1-2 The student attempts to explain whether his or her results make sense in the context of the You tried to discuss the choices you made and the methods you used. You tried to explain problem. The student attempts to describe whether your models and predictions made the importance of his or her findings in sense in real life. You tried to give other real- connection to real life. life examples of where this modelling technique could be used to predict outcomes. 3-4 The student correctly but briefly explains whether his or her results make sense in the context of the problem and describes the importance of his or her findings in connection to real life. The student attempts to justify the degree of accuracy of his or her results where appropriate. 5-6 The student critically explains whether his or her results make sense in the context of the problem and provides a detailed explanation of the importance of his or her findings in connection to real life. The student justifies the degree of accuracy of his or her results where appropriate. The student suggests improvements to the method when necessary. You correctly but briefly discussed your choices and methods. You discussed how happy you were with your models and predictions and how they might compare with your classmates results. You discussed the validity of using your equations to predict results. You tried to justify the accuracy to which you worked and that of your results and predictions. You gave a few valid examples of where this modelling technique could be used in real life. You explained in detail how valid your models were and how accurate any predictions made from them might be. You discussed what you might need to do to find more accurate models. You gave details of a number of situations where you could use such techniques in real life. You discussed in detail what the benefits, drawbacks and limitations were for your models.