Lesson 2.04: Presentation of Information

In module 5.03 we must now look at how a problem is presented on the written page.
The point at which a problem is presented is the moment at which a real-world situation begins to transition into a real-world solution. Some of the modelling will have been completed, and there will be some left to do. The following flowchart offers a schematic structure of this process, followed by an example:

Phase 1: The raw problem, as it occurs in the world
A problem in the real world is, by its nature, the problem in its 'raw' state. It cannot occur on the page without losing its 'rawness' and any attempt to describe it here will inevitably shift it into Phase 2.

Phase 2: Articulation of the problem verbally
A spoken description of the problem, for example:
Last summer we ran a summer training camp for our football team - the first we've ever done - and it does look like we were scoring more goals this year as a result. In previous years most internal games (between our own teams) involved three goals or fewer being scored, whereas this year there were more goals...

Phase 3: A formal presentation of the problem
All elements of the problem, and data related to the problem, are presented clearly on the page.
The table below records the scores of 100 games in a football league. The entries show the number of games in which the teams have scored the number of goals shown in the headings (for example, there were ten games in which the home team scored two goals and the away teams scored one goal).

Phase 4: Representation of the problem by the problem solver
As a problem solver you take the information presented on the page and represent it in a way that makes the solution self-evident.

Phase 5: Presentation of a solution
As a result of your analysis, you are able to clearly present the solution to the problem.
Last season there were 57 games (100-43) in which three or more goats were scored.
This was significantly better than previous years, and on balance the training camp seems to have been effective. Let's organise another one this year.

This five-phase process is an excellent way of breaking any problem into manageable segments. The problems you face may vary in level of difficulty; for example, within this course you will encounter some questions that are closer to Phase 2 and others that are closer to Phase 4, but, regardless of how difficult they are, you need to develop a refined, directed alertness when engaging with presented information.
Information can be presented in a number of different formats, and each comes with its peculiarities.

Text

Text is capable of supporting, and communicating, the most diverse range of information, and all problems will require some text.

All problems, as presented in exam questions, may contain some surplus information which is not vital for the solution of the problem. This is even more the case in real-life problem solving! Take, for example, the case of a doctor trying to diagnose the cause of a patient's illness. The doctor will begin by questioning the patient about their troubles and, from the information the patient gives, will identify and select the details which are relevant - these are the aspects requiring further investigation.

The first job is to separate the information which is needed to solve the problem from that which is not needed.

While a mantra such as 'redact - highlight - bullet' can be useful, it is worth remembering that this is just a gimmick.

Written text often obscures logical relationships, and particular care must be taken when extracting them. The logical relations of sufficiency and necessity are particularly tricky and need great care.

The grammar of a sentence in English does not always reflect the logic. Try to ensure that you interrogate all sentences written in English if their grammar is unclear, especially if English is not your mother tongue. The more varied the sentence structures you encounter, and interrogate, the more alert you will be to subtleties in grammar.

A famous example of how ambiguous statements can be is:
Let's eat Grandma!
Consider why this is ambiguous. It is normally accompanied by the advice: 'Punctuation saves lives.'

Tables are the classic way of showing how two variables relate to each other. What is true of all such tables is the meaning of any number in a cell can be inferred by linking the row heading and the column heading.

Whenever faced with an unfamiliar table or graph, you should check that you are able to articulate its basic relationship first (unpack it).

When you are sure you can take any individual piece of information from the table and parse it as a natural-sounding sentence, you should practise interrogating it. This can involve seeking out pieces of information, comparing pieces of information and making deeper inferences.

Tables can encode many different types of information. They can be used to show the differences between a group of objects, people or places (normally times, distances or costs); they can be used to show how certain groups have scored according to certain metrics (the classic example being the details of how a sports team has performed over a season); they can show the order and the times that certain events occur (a bus timetable being a standard example).

Three examples of how a table can be unpacked and interrogated are given in the Example question below.

Tables

Examples

Try Unpacking and Interrogating each table on your own before reveling key information

Table 1

Unpack it

To unpack the table it is worth choosing a few random numbers and forcing them into a sentence: for instance, the '54' is explained by the link between 'Shola' and 'Ayesh'. In this case, 'the distance between the islands of Shola and Ayesh is 54 km'.

Interrogate it

Find! Give yourself the task of extracting a piece of information from the table. For example, how far is it from Renaee to Ejiro?
Compare! Give yourself the task of comparing or contrasting two pieces of information. For example, where is furthest from Muna?
Infer! This requires using the information to find a limit, or an optimum. For example, show that it is possible to get round all five islands in 151 km (without returning to the starting point).

Table 2

Unpack it

Again, choose a few random numbers and try forcing them into a sentence. Here, the '4' in the top row is explained by the link between 'First Goal Scored' and 'Avocets'. In this case, 'the team called Avocets scored the first goal in four of their matches'.

Interrogate it

Find! Who won the most matches?
Compare! How many teams lost more than half their matches?
Infer! How many matches were 0-0 draws?

Table 3

Unpack it

Interrogate it

Find! Who won the most matches?
Compare! How many teams lost more than half their matches?
Infer! How many matches were 0-0 draws?

Graphs

On the journey from the real world to an artful problem expressed and solved in an exam hall, raw data is often marshalled into a table, and then presented in a graph. While tables are about ordering the data, graphs are about presenting it, often to support a particular claim.

A good graph is like a fully-formed argument - posing a problem, analysing it and presenting the solution in one go:

This famous graph, created by Hans Rosling (a superstar statistician), demonstrates one of the first laws of graph interpretation: respect the labels, and especially the units, on the axes. They introduce the characters in the story. In this case, the scatter diagram manages to capture four different changing variables for each of the countries: the population (represented by the size of the circle), the income of those in the country (expressed so as to avoid confounding factors such as the relative desirability of currencies, or inflation), the average life expectancy, and the year (the latter visible only when you animate the graph online).

There are four main specialist types of graph:

Those that aim to compare the frequencies of certain characteristics, referred to as frequency charts. These are typically composite bar charts, or cumulative frequency diagrams.

Those that aim to demonstrate changes over time, referred to as time charts, and which include any graph with time as a continuous scale along the horizontal axis.

Those that aim to demonstrate proportions, referred to as proportion charts - the most common being the pie chart.

Those that aim to demonstrate correlation, referred to as correlation charts - the most common being a scatter diagram.

As with the previously discussed tables, the key to understanding is to 'inhabit' the graph and then consider what can be inferred from it. This can be done by converting 'the facts of the matter' into a graph, or by analysing a graph (and establishing what it shows). Looking at the four types of graph in turn, we will now consider how our senses need to be heightened when analysing each one.

Frequency charts

These can be unpacked and interrogated, like the tables. Rather than articulating each piece of data with reference to its row and column heading, you need to take a point and identify its position on the horizontal and vertical axes, before attempting to frame the sentence which it represents.

The graph below shows the cumulative distribution of finishing times in a marathon:

Unpack it
The point indicated by the arrow - what exactly is it telling you? Something like '20 people finished the marathon in 210 minutes or less'.

Interrogate it

Find! Flow many people finished in 4 hours or less?
Compare! Which group finished closer together - the first 10 competitors or the last 10 competitors?
Infer! My cousin and I finished exactly 100 places apart, and exactly 50 minutes apart - is it possible to tell where we finished?

Time charts

Time graphs tell a story, and tell them so well that extremists claim that some stories are actually better when graphed. The key to time graphs, as with all graphs, is to reduce them to sets of key points if the 'story' is unclear.

Speed and gradient

The rates at which the variables change will correspond to the gradient of the line. At a descriptive level this just means that the steeper a line is, the quicker it changes over time. At a more precise level, this involves defining the rate of change, and the gradient, as:

Change in vertical variable

Change in horizontal variable

This results in gradient (of a line), rate of change and speed (on a time graph) all being the same. You do not need to meditate too hard on why or how these are the same, but you do need to remember that the speed (on a distance or time graph) can be found by calculating its gradient.

Proportion charts

Pie charts show proportions, not absolute values. This is a simple fact, but one that's easy to be caught out by.

When comparing pie charts (with other pie charts or with data represented in other ways) you need to remind yourself that a bigger slice of the pie does not reflect a big number but a bigger proportion of the whole.

Pie charts have no axes - and often have no numbers on them at all - but there are two ways that you can make inferences from them:

The relative sizes of the slices allow you to rank the different categories.
The 'easy fractions' visible in a pie chart allow you to decide which slices represent more or less than 1/4, 1/2 and 3/4 of the whole.

Correlation charts

Scatter diagrams expose correlations between two variables. They are wheeled in as pieces of 'showcase statistics' when you are young, and then exposed as instruments of deception when you come of age (with the slogan 'correlation does not imply causation').

This graph prompts discussion of how correlation relates to causation. The pattern that can be seen in the graph (the fact that the points lie on a roughly straight line going from bottom left to top right) can betray at least three different relationships: A causes B, B causes A, something causes them both. There is little you can actually infer from a trend without knowing about the possible causal mechanisms, however - that is, having some explanation of how changes in one variable can lead to changes in another. This kind of explanation is normally established under close scrutiny in a laboratory.

What is more likely to invite a higher-level problem-solving approach is the interrogation of the individual data points themselves, by applying the same principles as with the other graph.

Unpack it
The point indicated by the arrow - what exactly is it telling you? Something like 'during one month, between January and June, there were four reported shark attacks and approximately 90000 ice creams sold in Prudosia'.

Interrogate it

Find! What was the largest number of reported shark attacks in a calendar month?
Compare! How many reported shark attacks were there in the last six months?
If the trend shown was to continue, how many ice creams would you expect to be sold during a month in which there were two shark attacks?
Infer! What is the greatest and least number of shark attacks that could have occurred in a 30-day period in the last six months in Prudosia?

Questions

The final element of problem solving is the phrasing of the question itself.

Obviously there are as many different questions as there are problems, but it is worth considering the axes on which these questions lie, so as to be sensitive to their characteristics.

Two key aspects are the generality and justification expected by the question. The following table offers examples of the different combinations.

Problem solving is normally motivated by getting an answer that works, rather than finding the universal certainty, or the modality, of the answer (known as a proof). The questions that you can expect to face in a problem-solving paper will not require anything as formal as a proof.

And, although some problem-solving questions may ask you just to state an answer, you should always consider what an explanation or justification might be. This is a very good way to check that the answer is sensible, and that you have not missed out any solutions.

The superficial similarities between the three columns of the previous table hide fundamental differences in the types of answer required.

All instances

Showing that something is true for all cases normally requires the problem solver to explicitly generalise the situation. Using algebra is one classic tool for generalising to all cases.
In contrast, attempting to list all the options is often practicalLy or theoretically impossible (if there are infinite cases). The key feature to appreciate is that even one tiny, non-conforming case destroys an attempt at a 'complete list'. So an attempted solution must begin with completeness built into its plan - not just by starting with proving that 'some cases conform', aiming for 'more cases conform', and then stopping when no more can be found.
Such total demonstrations are rare in problem-solving exam-paper questions - but are clearly useful in real life.

Optimal case

Showing that a certain case is optimal requires the problem solver to find a boundary value or limit. Identifying a limit is a two-step process: you need to show that one value is possible, and that the next one is not. To do this you must list your cases in an ordered fashion, so that it is obvious what counts as 'coming next'.
The key feature here is to order the cases clearly. Only after doing this is it worth trying to find the first value which qualifies/does not qualify.

When considering problems which require a maximum value, it is often usefule to approach the answer from two different angles:

Find a lower bound. Begin by finding any answer that which works. Often it is a good idea to choose the easiest feasible answer. just to check you have understood the restrictions that you need to abide by.
Find an upper bound. In order to know how big the solution space(The range of possible values which satisfies the fixed constraints of the problem.) is, and to enable you to formulate an efficient strategy, you should then try and find an answer that is definitely too big.

Single instance

Showing that a particular case is possible requires the problem solver to search a 'solution space' creatively to find a case that conforms.

This may require a sense of the 'geography' of the space, to enable a speedy solution. Although such cases are easy to check when they are successful, they can be harder to achieve than the two previous types of question, because there may be no simple way of organising the data. An open-minded sweep for 'salient features' may be all one can do to narrow down the search.

Explanations and logical relations

Finding an optimum, or finding a solution that satisfies a given requirement, are problems that are focused on the end result. However, some problems ask you to explain why something can (or cannot) be. Problems like this depend on you first understanding the relationship that generates or underpins the solution, and then explicitly explaining it (in relation to the solution you are giving).

One element of these explanations involves what information is logically necessary or sufficient for a solution.

The key distinctions are laid out below.

The mantra that must accompany these refined logical relations is:

if A is sufficient for B - it must be B, if it is A
if A is necessary for B - if A doesn't happen then B doesn't happen.

Sample Problems

Question 1

Every Sunday in the town of Handel, there is a free concert in the town hall. Many people of the town walk past the hall during the morning, and decide whether to stay for the concert.

The doors to the hall open 2 hours before the concert starts.

The organizers know, from experience, that people choosing to listen to the concert arrive at a constant rate for the first hour; the rate doubles in the next 40 minutes and then doubles again for the final 20 minutes. No-one is allowed into the hall after the concert starts.

The hall holds 400 people.

What is the smallest number of people arriving in the first ten minutes that would imply there was not enough space in the hall?

Solution

This could be done by trial and error but is most efficiently done with a function machine or algebra.

If we call the number who come in the first 10 minutes ‘ r ’

then r X 6 = the number who come in the first hour

( r X 2) X 4 = the number who come in the next 40 minutes

( r X 4) X 2 = the number who come in the final 20 minutes

and these must add together to give more than 400, if there is enough space

i.e. 6r + 8r + 8r > 400

or 22r > 400

and so r > 18.18.

r must be a whole number : so we can conclude that 19 people in the first ten minutes would be the smallest value to imply the hall is full. It is worth checking that this gives a reasonable answer: (19 x 6) + (38 X 4) + (76 X 2) = 418 people, which is marginally greater than the capacity of the hall.

Question 2

The concert lasts for an hour: when it is over everybody leaves at a constant rate of 30 people per minute, until the hall is empty.

Draw a graph showing the number of people in the hall, if 30 people came in the first 10 minutes that the doors were open.

Solution

The rates given (30 per 10 minutes, 60 per 10 minutes, 120 per 10 minutes, 30 per minute) can be used to give snapshots of the graph.

The following list gives the time in minutes (in brackets) and the number of people at that time:

(10) 30, (20) 60, (30) 90, (40) 120, (50) 150, (60) 180, (70) 240, (80) 300, (90) 360, (97) 400, (120) 400, (180) 400, (190) 100, (193) 0

This can then be converted into a time graph:

Question 3

Below are the timetables for a Shakespeare Festival which will take place from Wednesday 3 July to Saturday 27 July. A total of 10 plays will be performed at three different venues in the town of Chambet.

Richard wants to see all of the plays, but he will not arrive in Chambet until 18 July, when only 10 evenings of the festival remain. He particularly wants to save Twelfth Night for the final evening, and he wants to avoid going to the same venue on two consecutive evenings.

Show that it is possible for Richard to choose plays that fit his wishes.

Solution

Delete all the dates that are not available.

Allocate Twelfth Night on the 27th and delete all other performances.
Identify the impossibility of seeing Timon of Athens on the 24th because the only two performances of King Lear are adjacent to it, and allocate Timon of Athens to the 18th.
Allocate The Tempest to the 19th, since it is the only alternative after the 18th is Twelfth Night.
Allocate Romeo and Juliet to 20th, since only available performance.
Allocate As You Like It to the 21st, since Measure for Measure is the only play that can be watched on the 24th.
Allocate Cymbeline (22nd), King Lear (23rd), Measure for Measure (24th), Love’s Labour’s Lost (25th) and Othello (26th) as the only way of completing the remaining evenings.

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