2. Modelling in Mathematics What do we mean by ‘Modelling’? Using mathematics to represent something in the real world to make something simpler to work with, or so that we understand it better, draw some kind of conclusion or make a prediction. Modelling can be quite simple or very complex. Often, we have to make assumptions about something, or make an educated guess about missing information.

3. Modelling in Mathematics Example: simple When we think about the Earth in Mathematics and Science, we model it as a sphere. It’s not really a sphere, it’s slightly flattened at the top and bottom and there are lots of lumps and bumps on its surface – but for most purposes, a sphere is close enough.

4. Modelling in Mathematics Example: complex Weather forecasters use complex models of weather systems to predict what is likely to happen in the next few days.

5. Challenge 1 By the age of 15, what percentage of their life has a person spent at school?

6. Challenge 1: assumptions What assumptions did you make? You will probably have had to make assumptions about some or all of the following: Regular attendance Length of school day Age to start school Number of weeks of the year spent at school Number of days in a school week

7. Challenge 2 Four minute mile In the late 19 th and early 20 th Centuries, people speculated whether or not it would be possible for a man to run a mile in under 4 minutes. If your school has a running track, that would be four times round the track in 4 minutes. Look at the data on the next slide and predict if and when it might have happened.

8. Challenge 2 Four minute mile data* (in seconds) YearTime1893257.8 18612861895255.6 18622731911255.4 1868268.8 1913 254.4 1873268.6 1915 252.6 1874266 1923 250.4 1875264.25 1931 249.2 1880263.2 1933 247.6 1882259.4 1934 246.8 1884258.4 1937 246.4 *IAAF data from 1913, amateur data pre-1913; only final record of any one year cited.

9. Challenge 2: Graph

10. Challenge 2 Four minute mile It actually happened in 1954. How close were you? Did you envisage a straight line or something else? Why? Bringing this up to date, on the next slide is a graph of the World Records since 1861. Do you think a man will ever run a mile in under 3 minutes?

11. Challenge 2: Graph 2

12. The Modelling Cycle This is a model of how we model in mathematics. There are many different versions in existence, the one on the next slide is a mixture of two classroom ones. It just captures what it is we’re doing when we’re modelling. You might find it helpful if you get stuck or are not quite sure what to do next.

13. (A version of) the Modelling Cycle

14. Choose a challenge On the next slide are some challenges. Remember, you may have to make a sensible estimate of something – don’t look it up! Information: There are approximately 65 000 000 people living in the UK, about 55 000 000 of whom live in England.

15. Choose one of the following problems to work on with a partner. Show your working to communicate your solution to others. How long is a line of a million dots? How heavy is the food that a person eats in a lifetime? How many pets are there in the UK? How many people are there on the Isle of Wight? (or your county). You may use a map of England for this one. Choose a challenge

16. You may have to think more carefully about the information you will need to have or to estimate for these problems. Some information is given which may be helpful, if you need it, but you might like to estimate it first. Longer challenges

17. Imagine working at a large theme park To help customers plan their time, information boards need to be placed to let them know how long they can expect to wait for a rollercoaster. Problem Where should you place signs to indicate a waiting time of 30 minutes ? Theme Park Queue

18. Millennium Force Ride at Cedar Point in Ohio Height 310ft Drop 300ft Length 6595ft Max Speed 93 mph Duration 2.00 minutes 3 trains with 9 cars (riders are arranged 2 across in 2 rows per car) Train leaves loading station every 1 minute 40 seconds Theme Park Queue: Facts and Figures

19. Charity marathons usually have a mass start. Several thousand runners assemble behind the start line. Problem How long would it take for all the competitors of a marathon to cross the start line? Starting a Marathon

20. The London Marathon Number of entrants: 40 000 Starting points:3 Width of start lines: 10 to 20 metres Wheelchair and paralympic athletes set off about an hour ahead of most of the rest of the field. Elite athletes lead the way at the main start time. Starting a Marathon: Facts and Figures

21. Each year Earth day raises awareness of environmental issues. Some things to work out: How much water do you use each day? Putting a brick in a toilet’s cistern saves 1.5 litres per flush. How much water would your school save a year if there was a brick in each cistern? Earth Day: April 22nd

23. Teacher notes: Modelling in Mathematics This issue looks at modelling in mathematics. Using real contexts can often act as a motivator for young people and help them to understand how mathematics is useful in a range of situations. With modelling, the emphasis is on the processes, reasoning and justification students give rather than on the answer, however, some guides to answers have been given as knowing the right magnitude of an answer is often helpful in the classroom.

24. Teacher notes: Modelling in Mathematics »Students should have the opportunity to discuss this with a partner or in a small group »Students should sketch or calculate (as appropriate)

25. Challenge 1 What percentage of a person’s life is spent at school by age 15? Assumptions: Start school at exactly age 5 Attend every day 39 school weeks a year School day from 8:30 to 3:30 At exactly age 15. Time at school: 10 x 39 x 5 x 7 = 13 650 hours Hours alive: 15 x 365 x 24 = 131 400 hours Approximately 10% (10.38%)

26. Challenge 2 Four minute mile The data for this problem is subject to dispute as there are different records available. Additionally, timing is more accurate in recent years. However, the data do give a sense that time is decreasing. Question: how come people are running faster now? It could be improved technology of running shoes, people are taller, people train harder, have a better understanding of nutrition etc. Question: Should we use a straight line or something different? Can’t be a straight line to extrapolate, otherwise at some point in the future it will take zero time or even negative time to run a mile. This would suggest that a curve such as an exponential decay curve might be helpful.

27. Choose a challenge How long is a line of a million dots? It all depends on the size of the dots and the spacing. If the dots are close together and created with a sharp pencil then one dot per mm should be achievable. 1 000 000mm = 1000m How heavy is the food that a person eats in a lifetime? http://wiki.answers.com/Q/How_many_lbs_of_food_does_a_person_ea t_in_a_lifetime#page2 suggests that we eat 30 600 pounds/ 14 000 kg of food in a lifetime

28. How many pets are there in the UK? Approximately 67 million according to the Pet Food Manufacturers Association, including:Pet Food Manufacturers Association 8 million dogs 8 million cats 20-25 million indoor fish 20-25 million outdoor fish 1 million rabbits …and 100,000 pigeons! Choose a challenge

29. How many people live on the Isle of Wight (or in your county)? 2011 census: 133 713 Increasing at approximately 0.7% per year (UK population growth rate) 138 459 expected in 2016 Choose a challenge

30. Where to place a 30 minute sign In the example given, a train with a maximum of 36 people leaves every 100 seconds. If we assume that it’s not always full, there are perhaps 30 riders each time. 30 minutes is 1800 seconds, so 18 cars leave every 30 minutes. 18 x 30 = 540 riders. Students will need to decide how long a queue of 540 people is. This will depend on how wide the queue is and how close people stand. People tend to not like standing too close to the people in front of them, so a metre per row of people would seem a reasonable estimate. If they were in threes (on average) then it would be 180m. Theme Park Queue

31. How long to start a Marathon? Assumptions: The start line is 10 to 20 m wide and in the picture shown there are approximately 30 runners crossing the start line. Most runners are in the main body of competitors, perhaps 36 000 of them. It takes 2 seconds to cross the start line – so 30 rows a minute The runners cross in a steady flow. This would mean: 36 000÷ 30 = 1200 ‘rows’ of runners 1200 ÷ 30 = 40 minutes If runners only take 1 second to cross, then it will take 20 minutes. London Marathon

32. Acknowledgements https://en.wikipedia.org/wiki/Four-minute_mile http://www.flemings-mayfair.co.uk/blog/2015/04/17/spectators-guide-to- the-best-spots-at-london-marathon-2015/