2.03 Lesson:

Modelling the problem

In Module 2.03, we will delve deeper into the art of problem-solving. You will be challenged to interpret and analyze complex problems, enhancing your ability to think critically and creatively.

This module will involve posing thought-provoking problems and engaging with an exam question designed to test your understanding and application of the concepts learned. By actively participating in these exercises, you will refine your problem-solving skills and prepare for real-world scenarios where critical thinking is essential.

Modelling the problem

Modelling means 'grabbing hold' of the problem and taming it by expressing it in a form that is manageable. Any prediction about the world requires a model and any problem that you have seen (or will see) in an exam paper has already been modelled to some extent - but almost any serious engagement with a problem like this involves further modelling.

Models can be physical, symbolic or computational.

For example, imagine you are playing in a snooker(pool/billiards) tournament this weekend. To help you win, you want to maximise your success at the opening shot (the break). You could choose to use a physical, symbolic or computational model to help prepare for this.

Physical

You make (or buy) a model of the balls and table on which you will be playing; this need not be exactly the same as the 'tournament equipment' but it needs to be similar in the relevant ways. In particular, the relative physical dimensions of the balls and table need to be correct. If the force with which you strike the balls is important, the weight also needs to be appropriate. This physical model will allow you to simplify the event (the break), perhaps beginning with a triangle of three balls, then considering a triangle of six, or to control certain aspects (you could strike the cue ball right in front of the triangle of balls, so that you have more control over the angle and force). You could use the model to train your fine motor skills to strike the cue ball with just the right force and angle to maximise the impact of that first shot.

Symbolic

You could study the logic behind that initial strike. In particular, the geometry that underlies the contact between the balls, and the laws of physics which apply in these cases. This kind of analysis would start with diagramming the relevant aspects of the initial collision of two balls and writing the relationship symbolically (or algebraically). Any basic relations can then be established and expanded to more complex cases.

Computational

A computer simulator could be used to investigate alternative options at the break. This might involve an interactive 2D picture of the table, which captured some of the aspects of the game: I may be able to control the angle and force of the cue ball, but may not be able to control the spin. The use of such a 'game' would depend on whether the model satisfies basic 'representational fidelity criteria' (requirements regarding how close key aspects of the simulation were to 'real life').

These three types of model reveal some of the choices that need to be made when simplifying a situation in order to experiment with it and achieve some goal. In this case the model may be deemed unnecessary - you could decide to just go out and play in the competition at the weekend, see how you do, learn from any mistakes, and play again next year. This is like treating the actual event as a model for future events.

Here is a list of some features of the situation which might be relevant when creating a symbolic model for an opening snooker break.

Whenever we make a model we choose to focus on the key variables: those that we think will allow us to determine the answer to our question, to a sufficient degree of accuracy. When we have chosen the wrong variables, or measured them with insufficient accuracy, we find the model is inadequate.

Discrete variables represent quantities that can be counted, rather than measured. Significantly, discrete variables can come only from a finite set of values. For example, you can have one child, or two or three or more children sitting on a sofa. You cannot have 1.5 children sitting on a sofa. Discrete variables avoid the gaps between the numbers that can lead to nonsensical answers.

The number of balls on a pool table (0, 1, 2, 3 ...) is a discrete variable, as is the century in which someone was born (nineteenth century, twentieth century, twenty-first century ...), or what shoe size you are (7, 7.5, 8, 8.5 ...).

We do not tend to define what a unit is for a discrete variable - although sometimes it is necessary to. For example, we may want to measure how big a city is in number of households, rather than number of individual people. Households and individuals are both plausible discrete units, and therefore we need to state which we are using. Normally, when dealing with discrete variables, the unit ('one thing') is the smallest object you can have one of.

Although discrete variables often feel nicer than continuous variables, because the numbers are easier, one must be more careful when dealing with them: if you are not careful you could accidentally 'fall into one of the gaps' when you give your answer, or in your working.

Diophantine problems - named after a mathematician from ancient Greece, called Diophantus - are puzzles which relate a number of discrete variables.

Discrete and Continuous Variables

Continuous variables represent quantities you can measure, rather than things you can count. These can take any value on a section of the number line, meaning there are no unfilled gaps between numbers. Examples include the distance between your finger and thumb (measured in millimetres), the weight of a feather (measured in grams), the length of time you have been alive (measured in seconds).

As you can see, unlike discrete variables, continuous variables are defined by their units (what counts as 'one'). Continuous variables may seem harder to deal with (since they are likely to produce fractions and decimals), but most of the mathematical tools you have learned are geared towards them - equation-solving techniques, as well as basic arithmetic operations, assume that any number on the number line is feasible.

Rounding

Rounding is the inevitable outcome of considering continuous measures. If you tried to give an absolutely precise answer in terms of continuous measures, you would be listing an infinite decimal: 'I am 1.824657918374928734982 7394 ... m tall.'

The relationship between rounding and measurement is peculiar - the diagram below shows a series of attempts to measure the length of the coastline of Great Britain, using different scales. As the unit of measurement becomes smaller (200 mile units, 100 mile units, 50 mile units ...), the perimeter increases: this is because, when you reduce the size of the unit, you always replace one straight line with some smaller zig-zagging sections of line, which will always increase the length. This paradox leads to the study of fractal geometry - which is not necessary for problem solving.

The paradoxical relationship between distance and rounding - the finer the detail, the longer the perimeter

The practical use of continuous variables does not produce such paradoxes and involves treating any measurement as an interval. For example, when I say I am 1.65 m tall, I mean that I am between 1.645 and 1.655 m tall (there is a reminder of how to find these limits and round to varying degrees of accuracy in Section 5.5).

It is important to remember, however, that rounding interferes with modelling at two different points in the process: when you start and when you finish.

Rounding is normally done 'to the nearest unit', but there are some cases where we naturally round up or round down. The most pervasive case is when we discuss someone's age. It is normal to say that someone is 17 years old until the day of their 18th birthday. Children and elderly people who claim they are 'nearly 9' or 'nearly 90' are quietly resisting this rounding mechanism.

With time comes change - and getting a grip on the rate of change (or the rate of rate of change) can be both critical and tricky. The problem solving tool that is best suited to managing and tracking changes in a problem over time is that of taking 'snapshots'. In this way you can control time, by pausing it.

The relationship between distance, speed and time is frequently presented as a triangle of variables, as shown in the diagram.

The triangle is not wrong: it certainly helps if you are trying to remember the formulae that link the three variables (you need to remember what order they are in, obviously!).

But the balanced symmetry of the triangle hides a deeper relationship:

Speed is a compound unit.
It measures the rate of distance covered over time.
It is proportional to the distance covered, but inversely proportional to the time.
It is the gradient of the timeline.
It tells you how many metres are covered in one second (and a billion other little facts about other distances).

It is particularly useful to remember that the unit (metres per second, or miles per hour, or whatever) tells you a specific, intuitively accessible fact about how long it takes to cover a certain distance. This allows speeds to be converted into snapshots, and enables you, the problem solver, to gain control of a slippery situation.

Time and speed

Relationship between distance , speed and time

Units and bases

A second aspect of time which affects how we model it and how we solve problems is the unusual units we use. We work in sexagesimal:

a.k.a. base 60 - a.k.a. minutes and seconds.

We are so used to working in base 10 that we have trouble seeing the choices that have been made in the way we represent numbers and the restrictions that these choices might place on us. Working in 'base 10' means that we use just ten different numerals (the digits 0 to 9) before leaping 'up a level' and introducing a new column (the tens column), and referring to the next number as 1 ten and 0 ones. This introduction of a new column is like the introduction of a new unit of time after 59 seconds - we refer to the next unit as 1 minute and 0 seconds. Another unit is introduced when we reach 60 minutes (we refer to elapsed times, for example, as '1 hour and 12 minutes and 30 seconds later...').

This use of base 60, rather than base 10, is also favoured by global navigators: degrees of latitude are divided up into minutes of latitude, which are further divided into seconds, which are normally known as nautical miles (roughly equal to a regular, statute mile).

Hopefully, by the time you complete this course, working in sexagesimal should come naturally to you, even if you don't think of it as a change of base. As problem solvers you should be aware that the use of standard time notation in a problem (be it the 24-hour clock, or a.m. and p.m.) requires an attitude of'heightened alertness'.

It is easy to read 'add 1.5 hours to 1:50 p.m.' and write down the answer 15:00, if you are not paying attention.(What should the time be here, instead of 15:00?)

There are other cases where our instinctive use of base 10 can Lead to misjudgements, but these are less common in problem-solving papers:

Binary

the use of only two symbols (1 and 0 or on and off) to represent information lies beneath all computer coding. In this case, the column headings become powers of 2, rather than powers of 10.

Students are not expected to be fluent in converting numbers between bases.

Dates

the numbers of days in the different months are not bases, but they can produce similar problems, and require care.

You do need to know how many days there are in each month!

Non-Metric

you will not be expected to remember any of the subdivisions that make up the Imperial unit system (how many inches there are in a foot, feet in a yard, yards in a mile, and so on).

You do need to be able to apply the logic of units to systems where you are given the conversions.

Conversions

You should be particularly sensitive to 'converting the wrong way':

for example, concluding that 35 dm = 350 m (given that 1 decimetre is 1/10 of a metre). The correct conversion is 35 dm = 3.5 m.

Relative measures

When measures are significant in relation to their group, or when values change over time, there are a number of ways of expressing the change.

These are barely part of the modelling of the problem, in that most of the alternatives are logically equivalent (in fact, they relate to the model descriptions rather than the model). But as a developing problem solver you must be well aware of in the different ways of referring to absolute and relative changes, and be able to choose the most appropriate.

For example, if the number of one cent coins in my pocket goes from three (out of twelve coins) on Monday to four (out of twenty coins) on Tuesday, this could be presented in any of the following ways in the table on the right.

This is not an exhaustive List, but it shows how the combination of proportion and change can be presented in diverse ways. As a problem solver, one is often trying to make inferences about the original data from the summaries given.

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