How to Attack Problem-Solving Questions in Junior Maths

Here’s the situation…. You know how to solve a quadratic or simultaneous equation. You’ve done loads and loads of questions… but you are still unsure of how to attack ‘problem-solving’ style questions. What do you do?

Problem-solving questions may seem a little tricky from the outset. Sometimes it may be difficult to see how the Maths concepts you have learnt apply to ‘real world’ situations when Maths might seem a little technical and algebraic. So, how do you solve those harder problem-solving questions which often differentiate between Band 5 and Band 6 students?

In this post, we’ll provide you with four tips you can use to attack problem-solving questions with some examples along the way. In sum, our four tips are as follows:

Link parts of the problem to content you already know.
Understand the intuition behind the concepts you are learning.
Organise different types of problem-solving questions on a given topic into ‘categories’
Practise solving problem-solving questions with a small study group.

Let’s dive in!!

#1: Link parts of the problem to content you already know.

An integral part of solving problems is to dissect the problem in terms of the content you already know. Solving ‘one question’ may require you to apply concepts from more than one area.

Let’s take the following example:

If the area of the rectangle is 54 cm2, find the dimensions of the following rectangle.

Now, we are dealing with area, so we therefore need to know our area formulae for plane shapes. We know that the area of a triangle is given by the following formula:

Area=lb

Even though the dimensions of the triangle are not ‘numbers’ per se, we still know that we need to substitute them into the formula in order to get our area of 54. So:

54=x(x-3)

And then we can continue solving. Immediately, we can see that solving the equation requires an understanding of expanding brackets and solving quadratic equations.

54=x2-3x

x2-3x-54=0

x2+6x-9x-54=0

x(x+6)-9(x+6)=0

(x-9)(x+6)=0

∴ x=9;x=-6

Now, here we have two answers. However, we need to interpret things in light of their context. We are dealing with length and area, neither of which can be negative (i.e. a side cannot be -6 cm long). So therefore x=9 only.

Applying this, our dimensions are:

x=9 cm

x=9-3=6 cm

Dimensions: 96 cm

So, in sum, solving the above question required the following:

Knowing our area formulae.
Knowing how to expand brackets and how to solve quadratic equations.

Now onto tip #2..

#2: Understand the intuition behind the concepts you are learning.

Being successful in Mathematics is not about memorising the process behind completing a question or finding the ‘right formula’ to use. Rather, you need to understand the intuition behind a particular concept in terms of why it works and how it gives you the correct answer. Once you have done this, you can apply this ‘intuition’ to solving problems.

Let’s say we have the following problem:

A car is driving from Sydney to Canberra. The car’s speed is constant throughout the journey with the car travelling at 110 km/h. Find an equation that represents the distance travelled (in kilometres) against the time spent travelling (in hours).

This question is all about using equations to model real-life situations, some examples of different equations being linear equations, quadratic equations and cubic equations. The problem needs us to ‘select’ the correct type of equation or ‘relationship’ for the problem at hand.

Now, we know the car’s speed is constant. This is a crucial piece of information – what it means is that for every unit increase in our independent variable, the increase in the dependent variable will increase by the same amount each time. Let’s look at the table below.

Hours	Distance
0	1100=0 km
1	1101=110 km
2	1102=220 km
3	1103=330 km

You’ll see that every time “1 hour” is added to the total number of hours, the “distance travelled” always increases by 110km.

Now let’s think about which type of relationship can best model this situation. This is where we need to understand the intuition behind each type of relationship. We know that linear relationships represent a situation where for each increase in our x variable we get the same increase in our y variable (as is the case here). Linear relationships take on the form y=mx+b.

We know the following:

b=0 because 0 hours travelled produces 0 km.
m represents the gradient, that is, how much the y variable is changing for every unit increase of our x variable. We know the answer is 110 (see the table above).

Here is what follows:

y=mx

Distance=110Time

D=110T

To wrap this all up, make sure you understand the “why” and “how” behind each concept you are learning.

Now onto tip #3.

#3: Organise different types of problem-solving questions on a given topic into ‘categories’.

There are only a handful of ways examiners can ask a question in an exam. What you’ll begin to notice is that questions start to repeat themselves. So what you should be doing is categorising the problem-solving questions into “themes” and then understanding the intuition behind each theme of question.

For example, if you are looking at solving problems using quadratic equations, you can categorise the types of problem-solving questions into the following (these categories aren’t exhaustive):

Area questions – Example: Finding the dimensions of a paddock when given the area.
Projectile motion questions – Example: Where the trajectory of a projectile is given by quadratic equation.

Categorising questions into themes will also help for revision purposes. If there is a particular style of question that you are repeatedly getting wrong, you know now where you need to target your revision.

And now for our final tip.

#4: Complete problem-solving questions with a small study group.

This last step will help turbocharge your study. Group study is a very effective way to share thoughts and ideas. It is also a great way for you to explain your reasoning to others (remember: the best way to remember something is to teach it!!) in addition to getting any of your own concerns clarified.

Group study is particularly important for problem solving questions because there may be more than one way to solve a problem. Learning multiple methods will give you more tools in your toolbox to use during an exam.

In sum…

So there you have it: Our four tips for mastering problem-solving questions. A final point I’ll leave you with is this: Mastering problem-solving questions takes time. While these four tips will help you make more of an effective use of your time, you need to practice, practice and practice (effectively, of course!!).

Good luck!!