Solving Difficult Gcse Problems

In my time teaching, I have noticed that one of the things students find difficult is knowing how to solve complex problems, which involve multiple steps. There are numerous reasons why this may be the case: do they have the necessary key skills to tackle the problem? Have they seen a similar problem before? Do they know where to get started? Most importantly, are they even sure what the problem is asking?!

Problem solving is a one of the things that draws mathematicians to mathematics in the first place. It gives them that "a-ha!" moment, which everyone enjoys! G. Polya`s how to solve it is an excellent book on how to approach mathematical problems, and one I would highly recommend.

There is a huge amount to cover to become an expert problem solver, and not all of this can fit into one article. What I wish to cover for now is how to apply some basic principles to difficult problems. I will show how, if you ask yourself two very important questions, you will be able to process the problem, and work your way through more effectively. (And at the very least, gain some vital marks)

As an example, I have picked a problem from the AQA GCSE Higher 2018 Paper 1 exam. You can have a go first if you wish:

Beth and Mia translate documents from Spanish into English.

A set of documents that would take Beth 8 days would take Mia 10 days.

Beth starts to translate the documents.

After 2 days Beth and Mia both work on translating the documents.

How many more days will it take to complete the work? You must show your working. (4 marks)

This is tough. And most people will doubtless get stuck after reading the information. So where do you start?

For this, there are two questions you should always, always ask yourself. Every mathematician does this:

1) What do I definitely know?

2) What is the problem actually asking?

The first question is crucial to making a start, and hence will help when you are stuck. Just writing down what you know, and getting some working done from this, will likely take you closer to the answer, and at the very least give you a feeling of getting something done. In the case of an exam, it will likely get you some marks as well. So let`s look at how question 1) helps for this problem:

We definitely know that Beth takes 8 days to complete the work, and Mia 10 days. From this, you can work out how much they get done in a single day.

In Beth`s case, she`ll get 1/8 done. In Mia`s she`ll get 1/10 done.

(At this point, you may not know how you will use this information. But you have gotten started, and it is likely you will use some of it.)

Next, we definitely know that Beth works on the documents for 2 days. This can help us work out how much has been done. Then we can know how much there is left to do!

2 days is 1/4 of 8 days. So Beth will get 1/4 of the work done. Therefore, there is 3/4 of the documents left to translate.

All this working has come before even worry about answering the question. In the exam mark scheme, this will already get you 1 mark.

Finally, we definitely know that after 2 days, Beth and Mia both work on the documents.

This is usually where students become unsure on what to do. This is because we don`t know the actual number of documents they need to get done, and most students would feel more comfortable with a number (a problem solving technique is to make up an easy number, but that is for another article...) But the important thing is: we don`t need that information. We can bring back what we wrote down earlier! This is why writing down what you definitely know is so helpful:

In 1 day, Beth gets 1/8 done, and Mia gets 1/10 done. If they work together, then we know they get more done as a result. Beth gets her 1/8 done, plus Mia gets her 1/10 done. So in total, they get 1/8 + 1/10 = 9/ 40 done.

You would now have another mark.

What needs to be clear from this working is that all of this has come from asking 'what do we definitely know?" The question has helped us formulate in our heads what information we have, and what must follow from this. We then have working, and gained marks before even worrying about our second question.

Why is it so important to ask 'what is the problem actually asking"? There are two reasons: firstly, you will be surprised how many students will initially struggle to say what a problem is asking, and it helps clarify things. Secondly, it will help you understand where you need to get to, and think about how to get there.

With these big problems, the first question tells you how to get started, and the second question tells you how to get to the finish line. It will then set up the path for you, and make clear what working needs to be done to answer the problem. A problem is usually defined as something where you do not know how to get to the answer. These two questions you ask yourself will help deal with this.

In this problem, and with our working, we can now ask 'what is the problem actually asking?". In this case, it is asking how long it will take Beth and Mia to complete the remaining documents. We can now combine this with our earlier working, that has come from asking ourselves what we definitely know, and say:

There are 3/4 of the documents left to complete, and in 1 day Beth and Mia will complete 9/40 of the documents.

So we can ask ourselves: how many lots of 9/40 go into 3/4?

This second bit may not seem clear initially, but it`s because we need to know what we must multiply 9/40 by to get 3/4, as this will tell us how many days it will take to complete the documents. Nonetheless, it has arisen from asking ourselves these two important questions.

To work this out, we calculate 3/4 divided by 9/40, which gives 10/3. So it will take 10/3 days ( 3 and 1/3 days)

You get 1 mark from noticing you have to divide, and clearly the final mark for the correct answer.

So there you have it. A problem which appeared to lack information, and gave no guidance on where to start, has been solved. And a lot of this has come down to asking those two key questions.

It is important to stress at this stage that the ultimate way to improve your problem solving skills is by....solving problems (practice practice practice!) UKMT and Nrich are an excellent resources for this. Practice is the secret to becoming good at anything, and problem solving is no exception. It is also because another useful question to ask is: "have I seen this problem before?' , and drawing from previous experiences on familiar problems is incredibly useful (Once again, that is for another article...)

However, now when you are revising, or building your problem solving skills, always ask yourself those two questions. They are incredibly helpful, and the more you do it, the better you will get at using these two questions effectively.

I hope you find this article helpful. I know a lot of actual working was left out for the problem, however I wanted the focus to be on the thought process, rather than the actual mathematics. Have a go yourself if you are not convinced by the answer!

If you have any questions, get in touch.