Last year’s panic question in GCSE maths:
“There are n sweets in a bag. 6 of the sweets are orange. The rest of the sweets are yellow. Hannah takes a random sweet from the bag. She eats the sweet. Hannah then takes at random another sweet from the bag. She eats the sweet. The probability that Hannah eats two orange sweets is 1/3. Show that n² – n – 90 = 0.”
The main reason that this question caused a struggle in solving this math question is that students generally are not used to use symbols and letters to make an equation rather than using numbers only. There is a difficulty in combining and extending the idea of using symbols and letters in math; in converting questions into abstract explanations: “How many sweets do I have? I have n amount of sweets.”
Difficulties in solving these problems
Every student, can do the following:
“There are 20 yellow and orange sweets in a bag and 6 of them are orange. How many are yellow?”
Well, 14! But if the students are given that there are n sweets instead of 20, they find it hard to say that the number of yellow sweets are n-6.
Similarly, if there are 20 sweets, they can tell the probability of getting an orange is 6/20. But when the number of sweets is given as an unknown quantity (n), they have found it difficult to recognise that the probability is 6/n. If they are able to do this they would recognise that the probability of getting an orange for second time is 5/n-1 as there are now 5 oranges left out in total of n-1 sweets.
The problem again shows when they are asked to combine the probabilities, using the concept of independent events.
Even the students who are weak in this particular subject, can do a tree diagram and calculate the probability of two independent events happening at the same time. So questions in the following style are very easy and most of the students get them correctly:
” 1. Draw or fill a tree diagram
2. Calculate the probability that both events will happen.”
But when the question is asked without giving the steps and instead asks for the final calculation in a single question the students are unable to split it into steps. The last step of the problem where the students need to multiply the two expressions (which are not numbers – multiplying letters…wait, what?) and equate them with 1/3, again demonstrates that the students didn’t equip the skill of forming equations from given sentences.
So, the classical example is:
“Your dad is 30 years older than you, the sum of your and your dad’s ages is 40 years. How old are you?”
These types of questions show that the students are unable to form equations using unknown variables, although the majority of them, are able to solve them if the equation is given to them (by the way, the answer to the above is 5).
Transform sentences into equations
The example that we usually use with our students to demonstrate the idea is that, if you ask the students to make mashed potatoes, they will fail. But they will be very good if you give them the steps of peeling the potatoes, then boil them, then season them, then mash them. Students need to let their imagination free, dare to express things in an abstract way and have the patience to solve a problem in many independent steps, not necessarily in a single one.
We, at Phi Tuition, recognise this problem and give them the skills they need to develop the independent thinking they need when it comes to solving problems. Although the answer cannot be derived in a single calculation, they become more confident and learn to invent and deploy the steps that will lead them to the answer.
Five key points in maths problem solving:
- Learn how to think and imagine in an abstract way. Use the variables with letters to denote unknown quantities. (“I don’t know how many sweets I have. I have n amount of sweets.”)
- Get used to starting by writing down (“Let n be the number of sweets I have” or “n=total number of sweets”)
- Read the question carefully again and again until it makes sense in and abstract way to form expressions and equations based on the numbers and the letters. It is important to translate/convert the words into mathematical symbols, expressions and equations. (“Hannah ate 2 of my sweets. I now have n-2 number of sweets left.”)
- Don’t hesitate to use the theories and concepts from different chapters in maths. A question in probabilities may involve quadratic equations. Who said that problems in maths are solved using only a single concept? As long as it makes sense, all solutions will lead to the same answer.
- Don’t panic if the question takes you a bit longer than the rest. This is one of the skills you need to have during the exams. There will be questions where you need to write down only one line of calculation and there will also be a bit more challenging ones that will require to write down more than two lines to reach the solution.
The critical point
As a rule of thumb: providing that you understand what the question is asking, and what you write is logical, mathematically correct and follows what you are given in the question, there is a big chance that it will be the right way to reach to the answer.