Pre-calculus Skills

Topic Overview

If you are sitting the Higher Tier GCSE maths exam paper, you will be required to understand various pre-calculus skills and use them to solve mathematical problems.

'Calculus' is the term used to define the mathematical study of change. Within the field of calculus, there are two main branches of study; 'differential calculus' and 'integral calculus'.

'Differential calculus' is the term used to describe the mathematical study of rates of change and the slopes of curved graphs.

'Integral calculus' is the term used to describe the mathematical study of the accumulation of quantities and the areas underneath or between curved graphs.

If you intend to study AS and A Level maths, you will study these mathematical fields of differentiation and integration in more depth. However, for your GCSE Higher Tier maths paper, you will only be required to understand and solve basic pre-calculus problems concerning these topics.

Key Concepts

In the new linear GCSE Maths Higher Tier paper, you will be required to solve problems which require pre-calculus skills. The specific questions you will be expected to answer will vary depending upon which examination board with which you are registered, but as a rule you will be required to:

  • Estimate the gradient of a straight line
  • Estimate the gradient of a curve
  • Estimate the area under a graph using the trapezium rule

Listed below are a series of summaries and worked examples to help you solidify your pre-calculus skills.

Worked Examples

1-Estimating the gradient of a straight line
During your GCSE maths exam, you may be asked to find the gradient of a straight line graph. The gradient of a straight line graph is calculated by:

Gradient = Change in y ÷ Change in x

By doing so, you are dividing the change in height of your straight line graph by the change in its horizontal distance.

In order to calculate this gradient, follow these steps:

  1. Choose any two points along your straight line
  2. Draw a right angled triangle where your straight line is the hypotenuse
  3. Calculate the change in the y co-ordinates divided by the change in the x- coordinates to find the gradient of the line

Example
(a) - Find the gradient of the following line and the equation of this line

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Solution
(a) - Firstly, choose two points of this line, for example : (5, 9) and (1,1). Now draw a right angled triangle where your straight line is the hypotenuse:

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2 - Estimating the gradient of a curve
You may also be asked to find the gradient of a curve. By finding the gradient of curve, you are able to see the rate at which the curve slopes upwards or downwards.

To calculate the gradient of a curve you must:

  1. Draw an accurate sketch of the curve,
  2. At the point where you need to know the gradient, draw a tangent to the curve (A tangent is a straight line which touches the curve at one point only). By calculating the gradient of this tangent you are able to calculate the gradient of the curve.
  3. Calculate the gradient of the tangent, and therefore the gradient of the curve, using the following method:

(change in y-coordinate) / (change in x-coordinate)

The resulting value will be the gradient of the curve at that particular point. Often this will not be an exact value, so you may need to round up to the nearest value.

Example
(a) - Find the gradient of the curve y = x² at the point (3, 9)

Solution

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(a) -The gradient of the tangent = (change in y)/(change in x),

The tangent touches the curve at (2.3, 5),

(9-5) / (3-2.3)= 5.71

Therefore the gradient of the curve y= x² at the point (3,9) = 6

3 - Estimating the area under a curved graph
If you are sitting the Higher Tier GCSE maths exam paper, you will be required to calculate the area under a graph. In order to do this you must understand how to use 'the trapezium rule'. 'The trapezium rule' calculates the approximate area under a graph by calculating the area of a series of trapeziums underneath the graph. These series of trapezium areas are then added together to give a rough estimate of the area under the graph:

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Therefore the total area underneath the graph can be calculated by following the trapezium rule:

\[\int_b^a {f(x)dx \approx \frac{1}{2}h({y_0} + 2(} {y_1} + {y_2} + .... + {y_{n - 1}}) + {y_n})\]
\[h = (a - b)/n\]

Where 'n' the number of strips

Example
(a) - Use the trapezium rule to find the area under the curve:

\[y = 5x - {x^2} + 2\]

between x=1 and x=5

Solution
(a) - First, you need to create a table of values to find the 'y' values for when x = 1, up to and including when x =5 (i.e. when: 1 ≤ x ≤ 5):

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Now that you have the values for

\[\begin{gathered} {y_0} = 6,{y_1} = 8,{y_2} = 8,{y_3} = 6,{y_4} = 2,h = 1 \\\ \\ \end{gathered} \]

you can use the trapezium rule

\[\frac{1}{2}h({y_0} + 2({y_1} + {y_2} + .... + {y_{n - 1}}) + {y_n})\]

to estimate the area under the graph:

Area = 1/2 (6 + (2 x 8) + (2 x 8) + (2 x 6) + 2)

= 1/2 ( 6 + 16 + 16 + 12 + 2)

= 1/2 (52)

Area = 26

Exam Tips

  1. Remember that the gradient of a straight line graph is calculated by: Gradient = Change in y ÷ Change in x,
  2. In order to calculate the gradient of a curve, find the gradient of the tangent to the curve and the point at which it touches the curve.
  3. When using the trapezium rule to find the area under a graph, always create a table of values and double check that your values are correct.

Topic Summary

Questions which require you to understand and demonstrate pre-calculus skills can often appear difficult at first. However, if you follow the step by step processes mentioned in the worked examples above and display all of your working out, you can comprehensively answer all of your exam questions and identify and correct any possible mistakes. Consequently, by practising these pre-calculus skills you can develop confidence in your mathematical ability. If you intend to study AS and A Level Maths, these pre-calculus skills will be incredibly useful and will give you a solid base of knowledge upon which you can rely. As a result, not only will you be able to score maximum marks in your GCSE exam, but you will also be able to adapt to AS and A Level studies much more easily!

Related Topics

  • Solving Equations
  • Manipulating Expressions
  • Change the Subject of a Formula
  • The Straight Line
  • Graphs of Curves
  • Inequalities
  • Sequences
  • Collecting and Using Algebraic Terms
  • Reasoning and Proof • Functions