Estimating the gradient of a curve | Maths | Quizalize & Oak National Academy

Starter quiz

Gradient is the __________.

measure of how steep a line is ✓

measure of the length of a line

rate of change in $y$ -direction with respect to positive $x$ -direction ✓

rate of change in $y$ -direction with respect to negative $x$ -direction
What is the gradient of this line?

$+2$

$+3$

$-4$ ✓

$-8$
What is the gradient of this line?

$4$

$-6$

$-6/4$

$2\over3$ ✓

$3\over2$
What is the gradient of the line going through coordinate pairs $(18,73)$ and $(50,233)$ ?

$5$ ✓

$1\over5$

$-{1\over5}$

$-5$
Which statements are true of this curve?

The gradient is positive.

The gradient is positive sometimes. ✓

The gradient is negative.

The gradient is negative sometimes. ✓

The gradient is zero sometimes. ✓
What is the gradient of the line going through coordinate pairs $\left(-{{1}\over{4}},{{5}\over{3}}\right)$ and $\left({{7}\over{3}},-{{16}\over{5}}\right)$ ?

${{292}\over{155}}$

$-{{292}\over{155}}$ ✓

${{155}\over{292}}$

$-{{155}\over{292}}$

Exit quiz

The gradient of the hypotenuse of the triangle is not the exact gradient of the curve between $x=2$ and $x=6$ , it is only __________.

an estimate ✓

a calculation

a computation

an approximation ✓
Estimate the gradient of this curve between $x=2$ and $x=6$

'2' ✓
Estimate the gradient of this curve between $x=-6$ and $x=-4$

$2$

$5\over2$

$-{{5\over2}}$ ✓

$-{{2\over5}}$

$-5$
What is wrong with using the coordinate pairs (-7, 12) and (-2, 1) to estimate the gradient of this curve?

The points are too far apart. ✓

Nothing. It is quick and accurate to estimate this gradient this way.

The distance between the hypotenuse and the curve means it lacks accuracy. ✓

The changes in $x$ and $y$ are not both $+11$
Match the intervals to the estimated gradient of the curve in those intervals.

$x=0 \ \text{to} \ x=1$

⇔

$4$ ✓

$x=1 \ \text{to} \ x=2$

⇔

$-4$ ✓

$x=2 \ \text{to} \ x=3$

⇔

$-6$ ✓

$x=3 \ \text{to} \ x=4$

⇔

$-2$ ✓

$x=4 \ \text{to} \ x=5$

⇔

$7$ ✓
The interval $x=0$ to $x=$ __________ would be poor because it would give you an estimated gradient of $0$

'2' ✓

Worksheet

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Lesson Details

Key learning points

The gradient of a curve can be estimated.
You can estimate by drawing a straight line between two points on the graph.
The closer the points, the more accurate the estimate is.

Common misconception

Pupils may estimate negative gradients incorrectly by drawing gradient triangles the wrong way round or ignoring the fact that the $y$ values are decreasing as $x$ increases.

Pupils should look at the shape of the graph between two points and decide if it is positive or negative before calculating. If $y$ is decreasing as $x$ increases then the gradient is negative.

Keywords

Gradient - The gradient is a measure of how steep a line is. It is calculated by finding the rate of change in the $y$ -direction with respect to the positive $x$ -direction.