How to answer mathematical examination questions

So you are about to write the end-of-year examinations!

Some of you are about to write the most important examination of your life, the Grade 12 final examinations.
This is a “high-stakes” examination since the results will determine the course of your life. A good result may open up doors which a poor result will not. And among the subjects, it is Mathemtics and Mathematical Literacy are those which are valued highly for entrance to university and other tertiary studies, and also for entrance to a wide range of further studies. For example, Engineering, Sciences, and Computer Sciences are not available if you do not ave a reaonsable pass in Mathematics.
This article provides a simple approach to answering the questions presented in the mathematical papers to help you improve your marks. By now there is little time left to increase your mathematical knowledge, and so I am focussing rather on how you can present your knowledge to the marker of the examination to show that you know and that you do not lose marks by presenting your knowledge in the wrong way.
In reality there are few right or wrong ways, but for every question in the exam paper there is also a memorandum which helps the markers to provide the marks to you. If a question provides 3 marks, than you are expected to show 3 things, not 2 and not 4, just 3. I will now help you in how to answer, using a method I use with all of my students, and which I call “APEX”.

APEX

I have developed the APEX method based upon the well-known work of Polya (“How to Solve It”) which is one of the most widely used mathematical books of all time. Polya’s work was directed at more complex problem situations, and I have applied this to the most simple of mathematical problems, since all such mathematical problems can be solved in this way.

Polya did not give a name to his 4-step method, and I have restructured this and called is the “APEX” approach, which consists of the following 4 steps.

  • A : Analyse
  • P : Plan
  • E : Execute
  • X : Crosscheck

ANALYSE

For this step you will do the following:

  • Read the Question: do not assume that you know the question by just looking at it quickly. Read it in detail.
  • See what you are given:
  • See what you have to produce:
  • See how many marks there are for this question: For every mark you are expected to do something specific. There will not be a situation in which you get two marks for doing one thing, or one mark for two things. Every question / problem is structured
  • Determine what the marks are for: Knowing how many marks you can get for this question, and also what you are expected to produce as outputs from the inputs which you are given, you need to start to consider what you should be providing in your written answer to be able to get the full marks. However, to continue with this you need to move to step : PLAN.

PLAN

Some questions are presented in a way that you know what type of mathematics you are to use, but in others this is not specified and you need to use your knowledge of mathematics to match the question to the right mathematical approach.

Once you have identified the mathematics you will be using, you can then plan what you will be doing.

In many cases you are also given the formulas, and thus you are not required to remember each and every formula, but you are required to know how to use the right formula and how to apply these to the problem.

For the PLAN step you will do the following:

  • What Mathematics?: select the type of mathematics which you think you will need. If this is not clear then consider a shortlist and see which may be more applicable than another.
  • Plan the attack: determine how you will use the mathematics and what specific actions you will do.
  • Map to the marks: you need to see what you will get marks for, since this may not be clear initially. You need to ensure that you provide enough information to get full marks, and also that you reduce the amount of work which will not contribute to the marks.
  • Predict the Answer: given the nature of the problem, you may be able to predict the nature of the answer expected. This is important in step 4, CROSSCHECK, in which you will check whether your answer is reasonable or not. For this you will check back to the initial question.

If the question has a mark of one (1), then you only need to provide the mark.

If there are two marks (2), then you need to provide two things, generally a first element of your workings / actions, and the answer.

If there are three marks (3), then you need to provide some workings, and one or more answers.

However, you will NEVER get marks from repeating the question! If you do repeat this it may help to provide the solution and will also help if it is not evident which question it is in case you wrote the wrong question number onto your paper.

You now have the following:

  • The actions you will undertake, which may be clear or may not be depending on how much still had to be worked out.
  • Which of these actions you are likely to get marks for, and which should be emphasised.
  • The sequence in which you will carry out the actions to arrive at teh solution.

Now you have to actually do the work, and we move onto the EXECUTE step of the APEX method.

EXECUTE

In this step you must carry out the actions you identified in the PLAN step. In most cases the PLAN is not written down, but will be structured in your head, but for larger problems it may be better to at least write the steps as an aide for your work.

You will now perform the actions, and at this time will write these down as part of your response and answer.

The following must be considered:

  • When writing the workings and the answers, be sure to highlight those for which you expect to get marks.
  • This could be done by positioning them differently on the page, such as in the centre of the page.
  • All other workings should be positioned elsewhere, such as on the left of the page.
  • However, some of these workings may also contribute to marks, so do not put them elsewhere.

CROSSCHECK

You now must check your answer to see if it reasonable to what was expected, which you may have already identified in step 1, ANALYSE.

The CROSSCHECK step is perhaps the most difficult for learners, since it is not always obvious what a correct answer should look like, and may learners create answers which are wrong, but for which they cannot see that they are wrong.

In many cases, mathematics exists in a real-world setting, and for these cases it is important to understand the nature of the real world from your own experience.

Practice, practice, practice!

There is a quote attributed to the famous golfer Tiger Woods when someone remarked how lucky he was to always get his low scores, with good drives down the fairway, and good putts to get the golf ball into the hole. His response “yes, but the more I practice, the luckier I get!”.

You need to practice how to write examination questions, and I would like to deal with this in my next blog.

Good luck for the examinations, and be sure to write to me by email about your experiences and whether this is useful to you.

Rounding

I encounter many students who struggle with the concept and practice of rounding.
The general procedure taught in school is that if the last digit is 0-4 we round down, and if it is 5-9 we round up, but what does “rounding up” and “rounding down” mean, and why is this important?

The word “rounding” can be quite confusing, since it sounds like you are doing something round, like a circle, but this is not what this term is concerned with. So let us dispel any notion that rounding has something to do with circles. Rather, the term “round” when used as a verb (to round) is concerned with find a round number, which is the closest number to another number, rather than as an adjective (round) which is used to describe circular objects.

Thus the term “rounding” means to find the closest number of a particular form. If we are asked to round to the nearest 10, then the outcome should be a number like 10, 20, 30, etc… and if we are asked to round to the nearest 2 decimal places, then we are expected to provide a number like 243.23 or 0.17, but not 243.2 or 243.22785.
In some cases we have to found to the nearest 5, and for this the outcome will be a number which is divisble by 5 such as 0, 5, 10, 15, 20, 25, ….

Why do we round?

It is important in all mathematics to understand why we do things and what the different mathematical structures are for. Putting mathematics into context makes it more meaningful and less abstract.

Rounding is used for us to determine an approximate amount since this is often easier to communicate. For example, if I am describing to somehow how far it is for me to drive to work every day I will say “about 20km”, rather than saying “exactly 18.6km”. No-one expects me to tell them exactly how far it is to my work, and to get an idea of a close amount is sufficient. This is rounding to the nearest 5km, since I could have also said 15km or 25km, but neither of these are that good and are not that close. I could have said 18km or 19km and this would also have been informative, but when asked a request such as how far it is to drive to work people do not expect a detailed answer and only an approximation.

I am sure that almost every day you are engaged in conversations in which you are discussion measures and numbers in a way in you use the work “about” to approximate the number when you do not know the exact number or if it is not necessary to be so exact.

For example, if I ask you how much per week do you spend on airtime you will probably not say “exactly R20.45″, but will rather say “about R20″.

Rounding is the mathematical operation which allows us to find the number which is closest according fo a simple set of rules, and which is also appropriate to the context.

Rounding to the nearest unit

The simplest case of rounding is where we have to round to the nearest unit.
As an example, we have a number 452.35 and we want the nearest whole number to this. This can be achieved by simply dropping the decimal fraction .35 leaving us with 453. But what if the the numebr is 452.93. This is clearly closer to 453 than to 452 and this it is better to round UP to 453.
The rule mentioned above will apply. If the first digit is 0-4 we round down and if it is 5-9 we round up. In this case we do not need to look at any other digits in the decimal fraction.
For example, all of the numbers 3.4, 3.49, 3.495, 3.4957 will ALL round down to 4, since the first digit of the decimal fraction is 4, and this indicates we should round down.
This example may be confusing to some, since if we take 3.49 to one decimal place this will become 3.5 and if we then round this it will become 4, but rounding up since the digit 5 will cause a round up. However, this means that when we round we do this in a single operation and not in multiple operations, and in all cases we only need to look at a single digit to make the decision.

Rounding to Two Decimal Places

It is very common in examinations that answers are required to be entered to two decimal places if this is not specified elsewhere. This means that you need to know how to round to two decimal places since this may be used many times on every examination which you write. If you fail to round the answer you are likely to lose a mark, even if this was not specified in the question.
In this case and answer of 2.456 must be rounded UP to 2.46, since the last digit is 6 which causes you to round up.
So what about 2.45633?
In this case you must look at the first decimal digit AFTER the point you are to stop. In this case, since your rounding to TWO decimal places you must look at the digit after the second decimal position, which is the 6 after the .45 and is not the additional 33 at the end, which are not needed for determining how to round.

Try these

Original number

Rounded number
2.45
0.6
0.798
23.1999
452.5

Rounding to nearest 5

In most cases you are asked to round to a number which is a power of ten, such as to the nearest hundred (100), ten (10), unit (1), tenth (0.1 – one decimal place), or hundredth (0.01 – 2 decimal places).
If you are asked to round to a number other than a power of ten, then you cannot use the rule about that 0-4 rounds down and 5-9 rounds up.
For example consider the problem of rounding 372 to the nearest 5. For this you will need to find the lowest and hight numbers around 372 which are multiple of 5, and then to select which is the closest. You should be able to this by inspection, but to help you understand this better you can find these numbers youself. In this case the numbers are 370, which is the multiple of 5 which is lower than 372 and 375, which is the multipel of 5 larger than 372. It is clear that 372 is closer to 370, which is a difference of 2 (372-370), rather than 375 which is a difference of 3 (375-372).

Try these

Original number

Rounded to nearest 5
245
343
798
2319
452

Rounding to nearest 10, 100, 1000

These are essentially the same process as rounding to the nearest unit. But the answer will be a multipe of 10, 100, or 1000.
For example, rounding 2459 to the nearest 10 is 2460, because the last digit is 9, which means we round up. Rounding this to the nearest 100 give 2500, since the last digit is now 5, being the digit after the 100 position. Rounding to the nearest 1000, then gives us 2000, since the last digit when considering 1000s is 4 which means round down.

Try these

Original number

Rounded to nearest

Answer
2459 10
2459 100
2459 1000
765 10
765 100
765 100
245.595 1
245.595 10
245.595 tenth / 1 dec place
245.595 hundredth / 2 dec place

Integer Addition Challenges

I have noticed that a number of relatively senior learners, in Grades 10-12, still struggle with types of integer addition.

For example, consider the following three simple arithmetic integer sums:

8-3
-3+8
-3-8

Whereas the first and second have the same value they often yield different answers. All I have done is to move the
-3 to the start of the sum.

I am hoping that you did understand that both of these will produce the answer of 8.

For the third sum, the situation is different, and I have also found some confusion with this, in which learners tend to misunderstand whether this means 8-3 or 3-8, and struggle to understand how to subtract a number from a negative numbers. It appears that they see the minus (-) signs used as having different meaning depending upon whether this is at the start of the sum or if this is between two other numbers.

The answer to this last sum is 11.

LaTeX : markup for mathematics

LaTeX is a language developed from the TeX computer typesetting language which was originally created by the computer scientists Donald Knuth in the 1970s.

LaTeX is generally represented by the logo \LaTeX, which you will only see if your browser can render LaTeX correctly. If this logo does not display, then you should look at the Getting Started page of this web site, which will indicate what browsers support LateX and which do not. Using LaTeX we can easily enter mathematical expressions such as {x+2=\dfrac{3}{2x-1}} which are in the language of mathematics itself.

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Welcome to mathematics.co.za

This site is dedicated to helping to improve the standards of mathematics in South Africa. It is primarily aimed at school-level learners and teachers but will be a useful resource for those interested in the more advanced aspects of mathematics, as well as those who are simply interested in mathematics or work with mathematics in their daily lives (as do most of us today!).

At present this sites is being started, and thus does not have much content, but over time more and more mathematics will be added to ensure that this becomes a useful reference for all.
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