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The importance of notation and the equals sign

Watch this to unpack the details behind mathematical notation for Higher Maths (listen to the music of numbers)

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As I child I loved to build things from Lego. It was so exciting when my siblings or I got one of those Lego sets. We would unpack the pieces and lay them out in front of us, planning our own little cities.

We would use the little booklet of pictures to help us go from step to step until we had completed the job. We would have been lost without those diagrams in the little booklet.

Surprisingly, the same principles apply if you are building a lego tower or a new billion-dollar football stadium – you need plans and diagrams that communicate. The diagrams must be clear, logical and written in a universal language that has no ambiguity – they must make sense to everyone.

Using diagrams are essential in Higher Maths, both to problem-solve and to communicate.

Let me show you what I mean.

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There are many questions in the exam where the diagram is already given and your job is to use its features to set up the correct method and working to solve the problem. In this example you are going to use the diagram to decide how to find the shaded area. First it is vital that you notice the points of intersection of the two curves, as they become your limits. You then look to see which of the curves is the upper curve and which is the lower. Then you can integrate the upper curve minus the lower curve. Without the diagram this question would be very challenging.

Sometimes there is no diagram given. Often this is in vectors questions and here a diagram can be very important. In this next example you are given three coordinates and told to find an angle. To do this you will need to find two vectors. The diagram will help you determine which two vectors are needed. Students lose marks in questions like this because they do not consider which two vectors are necessary. The diagram would look like this and you can see that the vectors needed point away from the angle shown. I must use vectors that both point away or both towards, otherwise I need to use the version of the scalar product with a negative in it.

There are other areas of the course where diagrams will help you.
When answering questions about the angle between a straight line and the x-axis and using m equals tan theta.
Also when solving circles problems like this one where a diagram can help you find a way to start.
In the vectors topic when you are asked to find where a point divides a line in a ratio, a diagram can help you organise your thoughts and plan next steps.
And in integration when you have areas above and below the x-axis you can plan out how to find the areas of each section bounded by the curve and the x-axis.

Visual stimuli are all around us and they are like building blocks for our thoughts when we are problem-solving. They help us step into a problem so we can get started, and they also help us to then plan ahead to find our next steps. You may have heard the phrase “A picture is worth a thousand words”.

In your maths problems you must use the diagram given effectively and if you don’t have one, then you should draw one yourself as it will help you to organise your thoughts and select the correct method.

Whether you are building your first lego tower or the new national football stadium, learn how to use diagrams to their greatest effect.

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