Myths about teaching can hold you back
- Year 11•
- Higher
Improving the estimate of the gradient of a curve
I can improve the estimate of the gradient by considering the gradient of the tangent at a fixed point.
- Year 11•
- Higher
Improving the estimate of the gradient of a curve
I can improve the estimate of the gradient by considering the gradient of the tangent at a fixed point.
Lesson details
Key learning points
- Since the gradient is improved by moving the points closer together, you could consider a point.
- By drawing the tangent to the graph at a given point, you can estimate the gradient at that point.
- The gradient at that point is estimated by calculating the gradient of the tangent.
Keywords
Tangent - A tangent to a curve at a given point is a line that intersects the curve at that point. Both the tangent and the curve have the same gradient at the given point.
Common misconception
Pupils may think a tangent to a curve at a given point cannot intersect the curve at another point.
Define the tangent at a point as the line which has the same gradient as the curve at that point. If the graph is cubic then it is possible for the tangent at a point to intersect the graph again. There are examples in the lesson that show this.
To help you plan your year 11 maths lesson on: Improving the estimate of the gradient of a curve, download all teaching resources for free and adapt to suit your pupils' needs...
To help you plan your year 11 maths lesson on: Improving the estimate of the gradient of a curve, download all teaching resources for free and adapt to suit your pupils' needs.
The starter quiz will activate and check your pupils' prior knowledge, with versions available both with and without answers in PDF format.
We use learning cycles to break down learning into key concepts or ideas linked to the learning outcome. Each learning cycle features explanations with checks for understanding and practice tasks with feedback. All of this is found in our slide decks, ready for you to download and edit. The practice tasks are also available as printable worksheets and some lessons have additional materials with extra material you might need for teaching the lesson.
The assessment exit quiz will test your pupils' understanding of the key learning points.
Our video is a tool for planning, showing how other teachers might teach the lesson, offering helpful tips, modelled explanations and inspiration for your own delivery in the classroom. Plus, you can set it as homework or revision for pupils and keep their learning on track by sharing an online pupil version of this lesson.
Explore more key stage 4 maths lessons from the Real-life graphs unit, dive into the full secondary maths curriculum, or learn more about lesson planning.
Licence
Lesson video
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Prior knowledge starter quiz
6 Questions
Q1.$$5$$ is __________ for this curve between $$x=-10$$ and $$x=-8$$
Q2.Estimate the gradient of this curve between $$x=-4$$ and $$x=0$$
Q3.Estimate the gradient of this curve between $$x=-2$$ and $$x=2$$
Q4.Match these intervals to the estimated gradient of the curve.
$$x=-2 \ \text{to} \ x=-1$$ -Â
$$4$$
$$x=-1 \ \text{to} \ x=1$$ -Â
$$1$$
$$x=0 \ \text{to} \ x=2$$ -Â
$$5\over2$$
$$x=2 \ \text{to} \ x=3$$ -Â
$$10$$
$$x=-1 \ \text{to} \ x=2$$ -Â
$$2$$
Q5.Order the estimated gradients of these intervals from highest to lowest.
Q6.Which of these intervals give this curve an estimated gradient of $$0$$?
Assessment exit quiz
6 Questions
Q1.A tangent to a curve at a given point is a line that intersects the curve at that point. Both the tangent and the curve have the same __________ at the given point.
Q2.When using two points on a curve to estimate the gradient we can improve the accuracy of our estimate by __________ the distance between the two points.
Q3.If you wanted to calculate the gradient of the curve at a given point, which diagram is likely to be the most helpful?
Q4.Use this tangent to calculate the gradient of this curve at $$x=0$$
Q5.Use this tangent to calculate the gradient of this curve at $$x=1$$
Q6.This tangent was drawn by hand. The triangle enables us to estimate the gradient at $$x=5$$ to be __________.
