Modelling in GCSE Maths Paul Chillingworth FMSP

 Description of the modelling process- WHAT?  Requirements and benefits- WHY?  Opportunities- WHEN?  Examples- HOW? Outline of this session

A model is a representation of a real situation. A real situation will contain a rich variety of detail. A model of it will simplify reality by extracting those features which are considered to be most important. What is modelling? REAL WORLD MATHEMATICS MODELLING

Modelling Process Real-world problem Simplifying assumptions Mathematical model (equations etc) Analysis and solve Prediction Interpretation Validation Experiment

List of Key Steps in Modelling  Collecting and summarising data  Observing  Listing relevant factors  Deciding which are the most important factors and ignoring the others  Simplifying the situation  Making Assumptions  Assigning values  Sketching a diagram of the situation  Writing information as an equation or calculation that you can solve  Making sense of your solutions, interpreting and checking they are realistic  Changing some assumptions to get a more sophisticated or realistic solution

New Assessment Objectives AO2 make deductions, inferences and draw conclusions from mathematical information construct chains of reasoning to achieve a given result interpret and communicate information accurately present arguments and proofs assess the validity of an argument and critically evaluate a given way of presenting information Weighting 30% (Higher) 25% (Foundation) AO3 translate problems in mathematical or non-mathematical contexts into a process or a series of mathematical processes make and use connections between different parts of mathematics interpret results in the context of the given problem evaluate methods used and results obtained evaluate solutions to identify how they may have been affected by assumptions made. Weighting 30% (Higher) 25% (Foundation)

In practice? 18 (Foundation) or 6 (Higher) Modelling the planet Mercury as a sphere, it has a radius of 2440 km. Work out an estimate in square kilometres for the surface area of Mercury. 11 (Higher) The population of animals on an island increases exponentially from the start of the year 2010 at a rate of 20% per year. At the end of 5 years the size of the population was 2500 Work out the size of the population at the start of 2010.

New SAMs OCR Alexander, Reiner and Wim each watch a different film. Alexander’s film is thirty minutes longer than Wim’s film. Reiner’s film is twice as long as Wim’s film. Altogether the films last 390 minutes. How long is each of their films? 1 AO3.1d translate problems in mathematical or non- mathematical contexts into a process or a series of mathematical processes 1 AO3.3 interpret results in the context of the given problem

Better reasons  Showing –the usefulness (by solving real problems) –and the power ( to predict and generalise) of mathematics  Developing skills in a meaningful way  Promoting interest  Learning how mathematicians and scientists make sense of the complex world we live in  Introducing new concepts or applying those already learnt.

Opportunities  Short problems, such as Fermi estimations  Fitting functions to data  Interpreting formulae and functions  Real world problems and investigations

Modelling problems - short How many bananas do you need to make 100 banana sandwiches? By the age of 15 what fraction of your life have you spent at school?

Modelling problems - short How many bananas do you need to make 100 banana sandwiches? Bananas are about 20 cm long and 3cm in diameter Thickness of a slice = 0.5 cm giving 40 slices Bread dimensions 10cm by 12cm Each sandwich requires 3x4 = 12 slices 100 sandwiches = 1200 slices = 30 bananas 30 bananas = 4800 g = 4.8kg = 10 lbs

Modelling problems - short By the age of 15 what fraction of your life have you spent at school? School day = 8 hrs School year = 40 weeks School week = 5 days Age is exactly 15 years (10 years of schooling) Time spent at school = 8 x 5 x 40 x 10 = 16000hrs No. of hours in life so far = 24 x 365 x 15 = hrs Fraction = 12% = about an eighth

Fitting functions to data Data collected from real situations (experiments) provides opportunity for algebraic and graphical investigation m d Beam balance

Curve fitting and interpretation Beam balance

Fitting Functions: Tidal Flow Time (hrs) Height (m) The following table shows the recorded Times and Heights of the waters for a Harbour from 12 Midnight on April 20 th 2015

Tidal Flow  What questions might we ask?  How can we use this data in modelling?

Tidal Flow  Plot the data on Autograph/Geogebra  Describe the main features of the data  Data looks like it will fit a sine or cosine curve  What is the length of time between the high/low tide values?  What is the tide range of the data?

Tidal Flow  Use the technology to find a suitable curve of best fit.  Write down a model (mathematical equation) for Height ‘H’ in terms of Time ‘t’. Is it reasonable?  H = 0.25 sin(30t)  Use the model to predict the tidal height at a later time

Interpreting functions The following functions have been suggested as possible models for the height, h in feet of a person of age, x years. Arrange the formulae in order of merit and comment on the limitations and validity of each formula. Can you suggest a formula better than any of these?

Modelling height

Growth Chart: Girls

Growth Charts: Boys

Longer modelling problems  Queuing –Waiting times and length of queue (traffic jam, theme park, dinner time)  Evacuation and flow –Time to evacuate classroom, train etc. flow through road works, start of a marathon  Making a profit –Probability games, selling tea/coffee, business plans

Theme park queue Imagine working at a large theme park. To help customers plan their day you have been asked to place information signs to indicate how long the waiting times are from certain points in the line. Problem Where should you place signs to indicate a waiting time of 30 minutes? Wait time from here 30 minutes

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

Charity Cafe A group of students plan to sell coffee and tea at a fair to raise money for charity. They aim to make a 20% profit. Problem What should they charge for a cup of tea and a cup of coffee?

Useful sources  mathshell.org.uk  Nrich  101qs.com  National Stem Centre Archive – Spode Group work

The Millennium Force Ride Millennium Force at Cedar Point in Ohio  Height 310ft  Drop 300ft  Length 6595ft  Max Speed 93 mph  Duration 2.00 minutes  Capacity 1300 riders per hour  Max G Force 4.5  Trains 3 trains with 9 cars (riders are arranged 2 across in 2 rows per car)  Train leaves loading station every 1 minute 40 seconds

London Marathon  Number of competitors  Start time for non-elite runners10:00  Number of starting points3  Width of start line10 to 20 metres

£1.00 £0.79 £5.00 £6.75 Milk 4 Pints Sugar 1Kg Teabags 240 Coffee 500g

The Further Mathematics Support Programme Our aim is to increase the uptake of AS and A level Further Mathematics to ensure that more students reach their potential in mathematics. To find out more please visit The FMSP works closely with school/college maths departments to provide professional development opportunities for teachers and maths promotion events for students.