Words and meaningsIn maths, words can be very important and sometimes have a more specific meaning than we first think.For example, if we talk about ‘doing a sum’, many people think this means the same as a ‘doing a calculation’, but ‘sum’ means ‘addition’, so 7-3=4 is not strictly a ‘sum’ at all.You will meet other words that are sometimes misused in maths and science during these activities.

Scalar and vector quantitiesWhen talking about certain measures, there are two types: scalars and vectors.• A scalar quantity has magnitude (size)• A vector quantity has magnitude and

direction

An example: If I start from home and walk at 5km per hour, how far from home am I after an hour?

Scalar and vector quantitiesSurely I must be 5km from home…

…but what if I turned around after half an hour?…what if the road isn’t a perfectly straight one?

Scalar and vector quantitiesIf we assume that the road is perfectly straight: • distance travelled (a scalar

quantity) is how far I’ve walked, regardless of whether I’ve turned around or not

• displacement (a vector quantity) is how far I am along the road from my starting point – in this case, home

Scalar and vector quantitiesVector quantities also have direction, so if I walked in the opposite direction along the road from my house, it would be a negative displacement.

Would the distance travelled also

be negative?

Scalar and vector quantitiesWe also sometimes use just ‘distance’ or ‘distance from xxxxxx’. This is a scalar quantity and shows how far from a certain object something is, but takes no notice of direction.

Can you see that I can be in two different positions and still be the same distance from the house?

DisplacementA displacement - time graph could look like this:

What would the distance-time graph look like?

Displacement & DistanceDoes one of the graphs give you a bit more information than the other?

Displacement & DistanceWhat would the corresponding distance travelled-time graph look like?

Displacement & Distance

Which of these is most informative?

Displacement and distanceWhich of the displacement-time graphs on the next slide match with this distance travelled-time one?

1 2 3

4 5 6

Displacement and distanceWhich other pair of the displacement-time graphs would have the same distance travelled-time graphs as each other?

What would the distance travelled-time graph for the third one look like?

3

4

1

PositionAnother term used is ‘position’, this is always relative to a specific object – in this case the house.

Think about the two scenarios on the next slide. For each one sketch:• a position-time graph• a displacement-time graph• and ‘distance from the house’-time

graph

Position1. I start from the house and walk in the

direction of the arrow at a steady pace for 5 seconds, stop for 2 seconds, turn round and walk back at a quicker pace for 5 seconds

2. I start from the red dot and walk in the direction of the arrow at the same steady pace, stop for 2 seconds, turn round and walk back at the quicker pace for 5 seconds

PositionWhat do you notice about the graphs?

You should have noticed that the displacement-time graphs are identical.

The position-time graphs have the same shape as each other, and the same shape as the displacement-time graphs, but translated a little in each case.

PositionThe ‘distance from the house’-time graphs are similar to the others initially, but when the walker gets to the other side of the house, the graph is different as it stays in the positive section of the page (because it’s a scalar measure).

Speed and velocitySpeed and velocity are another pair of related quantities: • speed is a scalar quantity• velocity is a vector quantity

A car travelling at a constant speed of 30km/h can be going in any direction, whereas a car travelling with a constant velocity of 30km/h means that the

Can speed, velocity or both

be negative?

Speed and velocity

car is moving in a specific direction.

Speed and velocity

Speed and velocity are both ‘rates of change’ which means they refer to how quickly something is changing.• Speed is the rate of change of distance• Velocity is the rate of change of displacement

Speed and velocity

From these definitions we can infer that:• the gradient of a distance travelled-time graph

is speed, • the gradient of a displacement-time graph is

velocity.

Speed and velocityWhy would there be an issue with using the gradient to work out speed from the ‘distance from home’- time graph below?

Average speed and velocity

Are they the same?

Always?

Sketch a few graphs to convince a partner.

Average speed =

Total distance travelledTime

Average velocity =

Displacement(from start to finish)

Time

Average speed and velocity• Look at this

displacement-time graph.• What is the displacement

from start to finish?• What is the total distance

travelled?• Work out the average

speed and the average velocity.

Velocity, displacement and distance travelled

A set of cards – copied on the next slide - has 4 displacement-time graphs together with the corresponding: • Velocity-time graphs, • Distance travelled• Average speed

Match the sets and fill in the blanks

Average velocity is _______ Average velocity is 1.33m/s Average velocity is 0m/s Average velocity is -1m/s

Distance travelled is ________ Distance travelled is 12m Distance travelled is 14m Distance travelled is 14m

Average speed is _______ Average speed is 2m/s Average speed is _______ Average speed is_________

Teacher notes: KinematicsThis edition looks at an introduction to 1 dimensional kinematics, making it suitable for those undertaking the new GCSE and for those starting Mechanics at A level.

The concept of scalar and vector quantities is introduced, and whilst it is not a specific requirement within GCSE Mathematics to know the terms and understand the distinction, it is in GCSE Science. Additionally, both speed & velocity and displacement & distance are referred to within the DfE objectives for KS4 mathematics and within several exam board specifications.

Making references to both without explaining the difference to students could cause confusion, particularly if they are encountering them in Science, so parts of this activity address this issue.

Teacher notes: symbols

An opportunity for students to discuss something in pairs and then feed back to the class.

Students to write something down or work something out.

A suggestion in the teacher notes of a way to make an activity more ‘girl friendly’ in order to increase the confidence of girls in class.

Teacher notes: Scalar and vector quantitiesSlides 3 & 4: If I’ve walked 5km, I could

be anything from 0km to 5km from home.

We usually make assumptions in maths to help simplify things, but these assumptions are often not shared with students. With ‘distance-time’ work, our underlying assumption is often that the road or path or train track is a straight one.

Slide 6: Distance travelled is always positive. It is cumulative and so can never ‘drop back down’ (have a negative gradient).

Make it ‘girl friendly’:

Ask students to discuss it with a friend

before giving an answer

Teacher notes: Displacement and distanceSlides 10 & 11: It’s really worth ensuring

that students understand the difference between the 3 terms and appreciate how the graphs correspond to each other.Ensure that students realise that distance travelled is cumulative.

Distance and distance travelled are always positive quantities; displacement can be positive or negative

Slides 12 - 15: An A4 sheet of 9 matching cards (Distance and Displacement graphs) is available as an alternative to using the slides.

Make it ‘girl friendly’:

Print out the matching cards and ask students to

work in pairs.

Teacher notes: Distance and Displacement

Answers:Slides 12 & 13: 2, 5 and 6Slides 14 & 15: 1 & 3 have the same distance time graph (below); the second one is for displacement graph 4

Matching cards answers: A with E, H and JB with D and FC has no match • can you draw one? (Yes) • Is there more than once

answer for this? (Yes, infinite possibilities).

G also has no match: • can you draw one?

(Yes). • Is there more than one

answer for this? (No: this graph only).

Teacher notes: Speed and velocitySlide 21 : Speed is always positive, velocity can be positive or negative.

Slide 24: The gradient of the last part is negative, but since speed is a scalar, it can’t be negative. This is a big issue with ‘distance from’ – time graphs, which are often used.

Slide 25 :It can make a difference, sometimes the graphs will look the same and the values will be the same, often they’re not. • When are they the same?

Make it ‘girl friendly’:

Cut out the cards and work with a partner.

Slide 26: An example of when the values are different.

Slide 28: Either show the slide, use the cards or print out as a worksheet. Cards would be easier for students as they will be able to annotate them. Answers on final slide.

Average velocity is1.33m/s Average velocity is -1m/s Average velocity is -1m/s Average velocity is 0m/s

Distance travelled is 12m Distance travelled is 14m Distance travelled is 14m Distance travelled is 16m

Average speed is 2m/s Average speed is 2.33m/s Average speed is 2.33m/s Average speed is 2.66m/s