Bringing together two sets of constraints

Focusing on the learners:

Distinguishing–eliciting–connecting. How to:

• encourage the use of both number and units when specifying the speed
• draw on everyday experiences to show how distance and duration are necessarily involved in speed
• show that finding speeds is not simple and straightforward

Teacher Tip: These are all related to findings about children's ideas from research. The teaching activities will provide some suggestions. So will colleagues, near and far.

Focusing on the physics:

Representing–noticing–recording. How to:

• avoid getting bogged down in calculations
• use formulae intelligently
• introduce the idea that relative motion is the only motion, without causing indigestion
• differentiate the ways in which time is used

Teacher Tip: Connecting what is experienced with what is written and drawn is essential to making sense of the connections between the theoretical world of physics and the lived-in world of the children. Don't forget to exemplify this action.

Units for speed

Wrong Track: 60 km/h is faster than 40 mph 'cos it's bigger.

Right Lines: A speed has two parts, a value and a unit. Speeds are much easier to compare if they have the same unit.

Converting units

Thinking about the learning

A car travelling at 60 must be going faster than a car travelling at 40.

This kind of statement is very common and reflects what we see around us. Speed limit signs are 30 or 50 with no attention paid to units.

Of course bigger numbers give a bigger message. A younger child offered the choice between 100 pence or one pound might be seduced by the larger number of pence into thinking that it was somehow worth more. In this example, pence and pounds are examples of two different units of measurement. There is a conversion factor between them (100 p is £1).

And so it is with speed. However, in the 60 kilometre / hour and 40 mph example it is not altogether obvious which of the two speeds mentioned above is greater. In fact they are almost the same speed, quoted in different ways.

Thinking about the teaching

A car travelling at 60 kilometre / hour will be going faster than a car travelling at 40 kilometre / hour.

However, a car travelling at 60 kilometre / hour is not going faster than a car travelling at 40 mph. It is important that pupils realise from the outset the importance of specifying units, so that comparisons can be made. Numbers on their own are meaningless.

To illustrate this in another context, imagine that you were told that John is 1420, Max is 150 and Ian is 1.6 tall. Is John the tallest?

Including the units we see that if John is 1420 millimetre, Max is 150 centimetre and Ian is 1.6 metre tall, then Ian is the tallest. Mathematics often concentrates on numbers and relationships between numbers. Science rarely deals with pure numbers but with quantities. A quantity is a number multiplied by a unit. For example 56 is a number but 56 metre / second is a quantity. An education in science must make this distinction in a formal sense. This will not be the first time that pupils have met the challenge of different units. One approach is to ensure that most examples given to pupils are quoted in metres and seconds in the first instance. Hence try to avoid early questions which have objects moving at speeds such as 22 millimetre / second or 47 kilometre / hour. It is far better to use examples such as people walking at 3 metre / second or cars travelling at 12 metre / second or sound travelling at 330 metre / second.

Thinking about units

Here are some careful thoughts on choosing particular ways of writing the units for speed.

It is common for people (adults and children) to think of speeds as being measured in miles per hour. Pupils may also meet speeds measured in kilometre per hour and metre per second.

It is possible to convert between different units. However, the conversion from millimetre / second to centimetre / second or metre / second should be avoided if it is likely to get in the way of pupils' understanding of the basic concept. There is limited value in expecting pupils at this stage to convert between different unit systems (for example between kilometre / hour and mph). If necessary, a conversion table can be provided.

At this level the most commonly used units for speed will be metres per second, often written as metre / second and sometimes metre second^-1. While these all have the same meaning, they will not be equally accessible to pupils.

The notation itself may cause a problem, and it is useful to liaise with the maths department to establish consistency on this issue. The notation metre / second is probably the most straightforward to use, since it is easy to see how the unit metres divided by seconds comes directly from the formula used to calculate speed and can act as a link to it. You might even choose to write metresecond.

Teacher Tip: Choose notation wisely, and liaise with maths colleagues.

Average speed

Wrong Track: If it takes Bill an hour to travel the 40 miles from Glasgow to Edinburgh he must be travelling at 40 mph.

Right Lines: For any journey, the total time and total distance allow us to calculate an average speed.

Describing real journeys

Thinking about the learning

The child's point of view sounds very sensible. However on a journey of 40 miles it is unlikely that Bill would be moving at the same speed throughout. He might need to take a bus to the train station. He'll probably spend some time waiting for the train. On the train he'll probably keep up a high speed but the train might make several stops during which time Bill's speed will reduce to zero and then pick up again. So in effect, Bill might never have maintained a speed of exactly 40 mph during his journey. The value 40 mph is his average speed.

Thinking about the teaching

The journey described above is a real life story of a real journey. It's a journey travelled daily by hundreds of commuters between two of Scotland's great cities.

It is helpful to try to place physics calculations in real life contexts such as this in order to illustrate the complexity of speed calculations. In a laboratory it might just be possible to consider an object moving at a constant speed but in most scenarios all that is possible is a calculation of an average speed.

We'd suggest that whenever you're after a calculation of an average speed you use data from a journey that has taken place (even if that journey is only in the imagination).

Using formulae

Thinking about the learning

Expressing a relationship in a form such as:

s = dt

(where: s represents speed; d represents distance covered during trip; t represents trip time or duration of trip) is a familiar skill for those of us steeped in an education in science. However, for many learners such a formula is as remote as a piece of ancient Egyptian tomb writing. Using symbols rather than words presents a hurdle for many. This along with a lack of confidence in mathematical skills, which is evident in so many pupils, often results in an I can't do this cry of resignation.

Thinking about the teaching

The key approach here is to demystify the science by keeping the examples simple and by offering lots of positive encouragement through a celebration of success. It is a good idea to use words rather than symbols in the first instance. Give plenty of examples and keep the numbers simple. This will give learners a sense of achievement.

So, more like:

speed = distancetime

Or, perhaps, more helpfully:

speed = distanceduration

A fly past

Wrong Track: Everyone agrees – it's on the aeroplane speedometer. We're doing 500 kilometres per hour.

Right Lines: Alice and Bob are sitting next to each other for the whole flight. Alice does not record any change of distance between her and Bob. But Charlie, in a fighter jet, records Alice and Bob's speed as 300 kilometre / hour as he overtakes, putting an extra 300 kilometre between himself and Alice for each hour of flying time.

Speed is always from someone's point of view

Thinking about the learning

Most pupils are comfortable with the concept of speed, though they may not have a neat definition for it. However, comparisons and calculations of speed may raise difficulties.

So can under-defined questions. Speed is always with respect to a point of view, and sometimes we need to carefully state the point of view to avoid ambiguity.

Thinking about the teaching

Take care to specify a point of view when you ask for a speed. The speed specifies the rate at which one object moves away from another.

So you need to locate yourself – perhaps you're moving along with one of the objects.

If you report something as at rest, all we can work out from that is that you and it are moving along together – the distance between you and the object is not changing. We don't know of absolute point of view against which we can all agree that something is stationary.

Differences and values

Wrong Track: Speed is just distancetime, and that's all there is to it.

Right Lines: Distance is the difference between two locations along your track. To find your speed you also need a time interval, the difference between two times on the clock (when you were at the locations). So speed is calculated from difference in location along the track and difference in time (of day).

Speed is always from someone's point of view

Thinking about the learning

Where am I now? is a question that requires a position for an answer (5 kilometre north east of Worcester).

What time is it now? is a question that requires a time for an answer (12:24 on Thursday).

Repeat these questions after a journey and you'll get another position and a different time.

To find a speed you need to combine the two positions, and maybe some more information to find the distance covered on your journey, as well as combine the two clock readings to find the duration of your journey.

Thinking about the teaching

Trying to cut corners to make things simpler often stores up difficulties for the future. This is a place to take care. We'd suggest that you avoid using just time, unless you mean time of day. Avoid What's the time for that journey?, replacing it with something more natural:

Teacher: How long did the journey take?

This kind of phrasing implies a duration, an interval of time. Distance is less of an issue, as we're less likely to use the word to mean many things, as we do for time (which is often used to mean interval of time as well as time on the clock).

We'd suggest that you guide conversations with the following precisely-worded calculations in mind (although this is private, probably not to be shared explicitly with pupils).

distance = position_end − position_begin

duration = time_end − time_begin

speed_average = distanceduration

Then, and perhaps only then, can you use relationships like these with some confidence that you'll be communicating clearly:

speed = distancetime

Or, perhaps, more helpfully:

speed = distanceduration

How to express the physical quantities

By providing the speeds in the format of, for example, 10 metre in one second, you are laying the foundation for ideas about speed as a measured quantity. The important point here is that you have control over the language used (i.e. distance (metres) moved in 1 second.)

The reinforcement of language is important. The teacher can model the language and encourage pupils to use the same form of words in any discussion that may emerge. This is a good time to make a clear link between fastest and slowest and the measured quantity speed.

There's a good chance you could improve your teaching if you were to:

Try these

• speaking about duration and distance in order to calculate speed
• using full quantity algebra when calculating speeds
• relating calculations on speed to useful examples
• always being careful about specifying differences (in position, or in clock time) in order to calculate speeds

Teacher Tip: Work through the Physics Narrative to find these lines of thinking worked out and then look in the Teaching Approaches for some examples of activities.

Avoid these

• uncritical use of the word time
• achieving fluency at the expense of understanding

Teacher Tip: These difficulties are distilled from: the research findings; the practice of well-connected teachers with expertise; issues intrinsic to representing the physics well.