TES Maths: Inspect the spec - precalculus

Craig Barton
15th December 2016
TES Maths, inspect the spec, GCSE, new specification, precalculus, gradient of curve, area under graph, secondary, KS4, Year 10, Year 11

Top resources to help you to explore the changes to teaching precalculus as part of the new GCSE specification

Everyone is talking about functions and frequency trees, but what else has changed with the advent of the new specification? And what resources are available to help? Throughout this series, TES Maths aims to find out.

What does the specification say?

The expectation is that:

  • All students will develop confidence and competence with the content identified by standard type
  • All students will be assessed on the content identified by the standard and the underlined type; more highly attaining students will develop confidence and competence with all of this content
  • Only the more highly attaining students will be assessed on the content identified by bold type. The highest attaining students will develop confidence and competence with the bold content.

A15h: Calculate or estimate gradients of graphs and areas under graphs (including quadratic and other non-linear graphs), and interpret results in cases such as distance-time graphs, velocity-time graphs and graphs in financial contexts.

R15h: Interpret the gradient at a point on a curve as the instantaneous rate of change; apply the concepts of average and instantaneous rate of change (gradients of chords and tangents) in numerical, algebraic and graphical contexts.

What's the same?

While the concepts of estimating the gradient of a curve and the area under a curve are technically new to the specification, the skills that underlie them are in fact old classics.

To estimate the gradient of a curve, students need to be able to draw a tangent and work out its gradient. They should know the former from their work with circle theorems; the latter essentially involves calculating the gradient of a straight line.

To estimate the area under a curve, students need to break it up into trapeziums and then calculate their area. This, of course, is something that they have been doing since Year 7.

What has changed?

Obviously it would be a bit risky to assume that students will piece together the separate skills themselves, so they will need plenty of practice in drawing tangents and estimating areas.

Likewise, the context in which these questions could be asked may be unfamiliar. Hopefully, they should already know that the gradient represents speed on a distance-time graph or acceleration on a velocity-time graph. But they will also need to know that the area under a velocity-time graph represents the distance travelled. Students may be tested in financial and other contexts and will need to adapt their skills accordingly.

Additionally, students should be able to say whether their trapezium-based estimate is an under- or an overestimation of the true area and illustrate this using a diagram.

Finally, it is worth pointing out that knowledge of calculus, namely differentiation and integration, is not required as part of this specification.

How can TES Maths can help?

As ever, the wonderfully talented authors of the TES Maths community have stepped up to the mark to lend a hand. Here is a selection of my favourite resources to help support the teaching of this topic:

  1. Gradients of curves and areas under graphs lesson sequence
    Introduce your students to the new concepts using this fully resourced series of lessons, including exam-style questions based on sample assessment materials.

  2. Gradients on a curved graph lesson
    Designed for the IGCSE but easily adapted to suit any exam board, this complete lesson offers a step-by-step approach to estimating the gradient at points on a curve.

  3. Finding the gradient of a curve worksheet
    Get students practising the skill of finding gradients using tangents before moving onto application tasks using this worksheet, including answers.

  4. Speed and acceleration graphs revision
    This colourful worksheet is ideal for recapping existing learning of all aspects of speed-distance-time graphs.


Craig Barton, TES Maths adviser

Craig is a secondary maths teacher in the North of England.


Find more resources to support the changes to the GCSE maths specification by taking a look at the rest of the series.

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