Advanced problem solving with real-life graphs | Maths

Starter quiz

This speed-time graph shows __________ rate of change between time and speed.

a constant ✓

an increasing

a decreasing

a fast
What speed is this vehicle travelling at after $10$ seconds? ______ m/s

'50' ✓
What distance has this vehicle travelled after $10$ seconds? ______ m/s

'250' ✓
Which equation generalises how far this vehicle has travelled after $x$ seconds?

$A={{x^2}\over{2}}$

$A=5x$

$A=5x^2$

$A={{5x}\over{2}}$

$A={{5x^2}\over{2}}$ ✓
What is the $y$ -coordinate at the point of intersection of the two linear equations $y=2x-10$ and $y=-{1\over2}x+5$ ? ______

'2' ✓
What is the equation of the tangent to the circle at point A?

$y=-{2\over5}x+{34\over5}$

$y=-{5\over2}x+1$

$y={5\over2}x+1$ ✓

$y={5\over2}x+11$

$y={5\over2}x-{27\over2}$

Exit quiz

The __________ to a circle from an external point are equal in length.

radii

tangents ✓

perpendiculars
What is the tangent to the circle at A?

$y=4x-14$

$y={1\over4}x+{29\over4}$

$y=-{1\over4}x+{29\over4}$ ✓

$y=-{1\over4}x-{29\over4}$

$y=-{1\over4}x+{13\over2}$
Find the point of intersection of the tangents to the circle from points A and B.

$(21,2)$ ✓

$(2,21)$

$(19,2)$

$(23,2)$
Tangents from A and B meet at point C. To show that AC $=$ BC we would use __________.

a ruler

Pythagoras theorem ✓

circle theorem

$A=\pi \times r^2$
How far have these cars travelled between $0$ and $6$ seconds?

Both have travelled $30$ metres each

Both have travelled $90$ metres each

'a' has travelled $90$ metres and 'b' has travelled $120$ metres

'a' has travelled $90$ metres and 'b' has travelled $135$ metres ✓

'a' has travelled $250$ metres and 'b' has travelled $275$ metres
After how many seconds will car 'a' overtake car 'b'? ______ seconds.

'12' ✓

Worksheet

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Lesson Details

Key learning points

Different speed/time graphs can be interpreted and compared.
The area under the graph can also be interpreted in context.
Circle theorems can be demonstrated using coordinate geometry.

Common misconception

The intersection of two different graphs on a speed-time graph is the point where the faster particle overtakes the slower moving particle.

Remind pupils that this is a speed-time graph and ask, "How do we find the distance travelled on a speed-time graph?". Then, ask pupils to calculate how far each particle has travelled at the point of intersection.

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.
Tangent - A tangent of a circle is a line that intersects the circle exactly once.