We know from our conversations with teachers that worded problem solving questions can be a challenge for young people. There are so many messages in a ‘wordy’ problem. Some of these messages are helpful to the problem. Other messages are erroneous. After a read through, it can be easy for a youngster to have forgotten the first part of the question. Often, the problem needs to be read, at least, twice for the language to be understood.
In order to improve our chances in solving a problem, we might consider the use of tips, techniques and tactics to help us retain important information. For example, we might circle key words. We may annotate the text. We might write down key information in the margin. We will try and decode the diagrams and tables. This will help us deconstruct the wordiness of the problem. No doubt, we will all have our unique way of approaching this problem.
With this current problem-solving task, we invite the reader to reflect upon the approach needed to provide a solution. Deciding upon the approach, which is the tasks to be done, can be more exciting than finding a solution. It is also important that the order of the tasks, in the approach taken, is thought about. The order of the approach to the task will help the student think about the ‘maths’ knowledge and skills needed to answer the question. The maths teaching will also help students think about the order of the maths knowledge and skills, which will be retrieved. As with most problems to be solved, the management and marshalling of information provided within the text are also important.
And here, I fear I have already slipped on a ‘banana skin’. On one hand, I argue that the approach to solving a problem is as unique as a fingerprint. And yet, in the above paragraph, I try to impose on the reader a preferred way of solving a problem. There is no generalised, or preferred, way of solving a problem.
In terms of balance, this problem-solving question is here for several reasons. The reader may not be interested in thinking about the approach needed. The reader may simply want to answer the question. Others may want to break the problem down into small parts. Some may wish to use this as a classroom activity at the end of this teaching year.
Problem solving is at the heart of mathematics. Mathematics requires us to make sense of numerical challenges through the recall of relevant mathematical knowledge and skills. Perhaps, the application of that retrieved mathematical knowledge and skills poses the greater contest. Maybe, the reader will have their own reasons for why problem solving is important. From our perspective, we hope that this, and other problem-solving tasks, will help our students engage with their mathematical journey of discovery.
The problem solving task below is appropriate for Year 10 or Year 11 students (excluding the website linked at the end of the article). Year 12 and Year 13 students may also get plenty out of this task, although you might want to ask them some deeper questions (or just refer them to the website linked at the end of the article).
Problem Solving Task: Learning from the Coronavirus
In a country’s regional city, the population is represented as blue circles. Each blue circle symbolizes 25,000 people. Table 1 shows the pre-pandemic population. We will show that uninfected population is also known as the susceptible population, (S).
In one country’s city, the population was divided into three groups during the pandemic.
Group One: all the population were Susceptible (S) to the infection – they could catch the coronavirus. No one was immune from the Coronavirus. Hence, all the population is Susceptible (S). This has a blue circle.
Group Two: those sections of the population who were infected with Coronavirus were no longer Susceptible (S), since they were now Infected, which is represented with a yellow circle and known as Infected, with the letter (I)
Group Three: those who no longer had the Coronavirus, either because of recovery or death, were known as the Removed part of the population and given the letter (R). This is shown with a green circle.
Three months into Lockdown, City Chiefs were able to learn about the effect of the Coronavirus.
Table 2 below shows the effect of the Coronavirus on the city’s population after 8 weeks. From table 2, can you work out
- The probability that a person will currently be Infected with the Coronavirus?
- The probability that a Susceptible person will not have been Infected after 8 weeks?
- The probability that a person is now Removed, given that they have been Infected‘?
Using the same table, write your challenging question, based on probability, for your classmates to solve. Solutions will appear in the next newsletter.
When you look at Table 2 – remember please that Blue is Susceptible (S), Yellow is Infected (I) and Green is Removed (R)
|Total population||20 blue circles each = 25,000 people||500,000 people|
|Susceptible population||This is the total population.||500,000 people|
|Infected population||5 yellow circles = 25,000 x 5||125,000 people|
|Susceptible but not infected||11 blue circles = 25,000 x 11||275,000 people|
|Removed population – no longer infected||4 green circles = 25,000 x 4||100,000 people|
|The population currently not infected||Susceptible plus removed but we don’t know how many of the removed population are dead.||No answer|
|The population who caught the Coronavirus||The yellow circles plus the green circles. This statistic does not require the removed population to be alive. 5 yellow circles plus 4 green circles.||9 x 25,000 = 225,000|
|a) The probability that a person will currently be Infected with the Coronavirus?||Susceptible population = total population|
Infected population = 5 yellow circles
|125,000/500,000 simplified = 1/4|
|b) The probability that a Susceptible person will not have been Infected after 8 weeks?||Those who have been infected includes both Infected and Removed.|
This is the P (Uninfected person from the whole sample).
|P(S) = 500,000|
P (not I or R) = 275,000/500,000 = 275/500 in its irreducible form = 11/20
Or you could simply count the blue circles!
|c) The probability that a person is now Removed, given that they have been Infected?||In simple terms, we could treat the infected and the removed as a subset – that is, as a ringfenced subset of 9.|
However, there are too many ‘ifs and buts’ with this question – we could make some assumptions, which effect the parameters of the data in Table 2. For example, are any of the Susceptible population Infected but asympomatic? Are any of the Susceptible population ‘pre-pandemic infected’ and now Removed? Were those who are Removed, all infected for the same period of time?
And as the questions continue, so does the complexity of the response. Three possible options are proposed for this question
To begin an investigation, where the possible permutations and combinations are considered, and a spreadsheet developed.
To develop a model based on Calculus.
To treat the Infected and the Removed group as a subset of 9. There being 4 Removed and 5 Infected.
The probability of a person, who has now been infected, and who is now removed, would be 4/9
This worded problem is based on an idea created by Chris Robbins who led an online student enrichment session for the AMSP in the North West on the mathematical and statistical inferences and messages from the data available about the Coronavirus on Wednesday 30 June 2021. Chris Robbins said, “I like the use of probability for formulating these kind of ODE (ordinary differential equation) problems. The true starting point is numbers and a probability, which are then recast as population fractions using standard non-dimensionalisation (a change of variable) – you can view details on this website and by clicking ‘Need more detail? Show the maths’.
My thanks too to James Groves for his advice and support with the development of this article.