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Mathematics Teaching Self-evaluation Tools

You are viewing a limited version of the NCETMâ€™s self-evaluation tools. Any answers you save during this session will be removed after seven days. Log in or sign up to view, and use, the full version of the tools.

Here you can see a summary of the areas in which you are confident and those in which you are less confident; there are some ideas and suggestions which may help you in your professional learning.

Showing all next steps for the selected topic.Click on a question to show more information.

you understand the language of kinematics?(show/hide all)

What this might look like in the classroom

The key is getting the difference between a vector quantity (such as velocity and acceleration) and a scalar quantity (such as speed or distance).

Students need to experiment with descriptions of journeys, or describing their location in order to clarify the difference between them.

Like other aspects of mathematics the distinctions become easier as the course develops – so at the start the teacher often has to labour the point rather a lot, but this should become clearer in the student’s mind as more of the course is covered.

Taking this mathematics further

The rest of the mechanics course is going to take these concepts further – at this stage it is probably enough to try to seek clarity.

Making connections

The students have met speed & distance at GCSE and even back in Key Stage 3 – so they should be able to understand these general concepts.

Trying to think about the A-level version, about velocity rather than speed, should help them see the connection with previous work – but helping them see the advanced nature of vectors.

Related information and resources

from the NCETMfrom other sites

This subject support section provides some useful links and explains a range of relevant topics.

you know the difference between position, displacement and distance?(show/hide all)

What this might look like in the classroom

Getting the students to stand up in their seats and describe where they are sitting – can they describe their seat in terms of their position – like the 3rd row and 4th seat along – or do they claim to be 8 seats from the door, or 4m diagonally across the room – these different descriptions should help them use the three terms.

Taking this mathematics further

This could be linked with previous work on loci – saying I’ve travelled a distance of 10 km doesn’t describe the end point of the journey very well, students should realise this only gives the end point as being on a circle – radius 10km – from the starting place – but a displacement of 10km north would be sufficient.

Making connections

Position can easily be related to coordinates, or with i and j vectors – so describing a position as 3i + 2j – could be a way of using position within the vectors topic.

Related information and resources

from the NCETMfrom other sites

This subject support section provides some useful lik s and explains a range of relevant topics.

you know the difference between velocity and speed, and between acceleration and magnitude of acceleration?(show/hide all)

What this might look like in the classroom

Students need to appreciate the preciseness conveyed by the vector descriptions – so a change in velocity of 2ms-2 in a northerly direction – given by acceleration =2j ms-2 is much clearer than just an acceleration of 2ms-2.

A speed of 10ms-1 followed by a speed of 15ms-1 could mean many different things – well worth the students explaining the variety of situations this could describe.

(for example – this is just a car accelerating from 10ms-1 to 15ms-1 as it drives along the road – or perhaps it is a car doing a U-turn, and changing from 10ms-1 in one direction, to 15ms-1 in the opposite direction.

Or it could be more interesting – and be 10ms-1 in a south-east direction and then 15ms-1 in a north east direction.

Many situations can be imagined – which should help the students realise how much better the vector description really is.)

Students can use these descriptions to describe a journey – perhaps from their life, or perhaps from a film (or just their imagination).

This should help them become familiar with the concepts involved.

Taking this mathematics further

Considering how vectors might be used to describe 3-dimensional problems is a relatively easy concept, but one that might help make this situation a little more challenging and more true to life.

Making connections

This topic should help consolidate concepts of addition and subtraction of vectors.

Considering acceleration as a change in velocity should mean that an initial velocity can be changed by repeated acceleration vectors – thus practicing vector addition.

Vectors can seem a very abstract concept:- displacement, velocity and acceleration could be the application that brings vectors to life.

you are able to draw and interpret kinematics graphs, knowing the significance (where appropriate) of their gradients and the areas underneath them?(show/hide all)

What this might look like in the classroom

Students need to realise that (a.) they can read coordinates from a graph and use this to answer questions – so in the example above the acceleration between O and A is found by considering the coordinates of the start and end of the journey. (b.) key features of the graph are related to physical concepts – so the gradient of the velocity time graph gives the acceleration and the area under the graph gives the distance.

Displacement-time graphs can also be drawn, and these have the feature that the gradient gives the velocity for a given section.

Taking this mathematics further

At this level graphs will be drawn with straight lines.

Students should consider the approximations involved in representing a journey described in this way (in particular the problems inherent in the pointed corners when changes occur).

Those with sufficient knowledge of calculus will be able to realise how they can solve more realistic problems with curved lines.

Making connections

Students will have met gradient previously in their coordinate geometry work – in mechanics, on kinematics graphs, the gradient means more than just a measure of steepness – now we use the gradient to calculate the velocity or acceleration of that part of the journey.

Students will be familiar with the speed = distance / time formula and should be able to rearrange this to give distance = speed x time – and hence realise that speed x time will relate to an area on a speed-time graph – so also an area on a velocity-time graph.

This site has a free interactive animation which allows you to investigate the behaviour of a car, incorporating driver reaction times and different braking forces.

you are able to differentiate position and velocity with respect to time and know what measures result?(show/hide all)

What this might look like in the classroom

At first students need to realise that represents the velocity.

Thinking of this as distance divided by time should help – but with the advanced nature of differentials because the distance is constantly changing.

Earlier work looking at the relationship between the graphs of distance-time and velocity-time should be helpful.

Evaluating the to give a numerical value for the velocity at an instant might help ground this topic in reality.

Taking this mathematics further

What comes next is really the next topic – but students should be able to realise that if one can differentiate the position and obtain the velocity, then the reverse process ought to work as well – so integrating the velocity ought to give the position.

This again might help them understand the link between integration and differentiation.

Making connections

Students need to realise this is the same sort of differentiation they have studied in their Core lessons – albeit in terms of and not .

For many the extension to the second differential might prove a little challenging.

This topic might help ground their differentiating in reality and may strengthen their understanding of the Core topic of differentiation.

you are able to integrate acceleration and velocity with respect to time and know what measures result?(show/hide all)

What this might look like in the classroom

Students need to think over the previous topic and realise that since they already know, from their Core studies, that integration is the reverse of differentiation then they will realise that the following diagram must be true:

Kinematics diagram 1 to be added here.

Taking this mathematics further

To progress further in mechanics it is often useful to question some of the assumptions being made, and try to make a more refined model that uses some of the things that are often ignored.

It is usual, at first, to solve mechanics problems by ignoring air resistance.

However, as an extension including air resistance would add an extra element and hopefully a more accurate model.

Air-resistance would best be added as a decelerating force that is dependent on the speed –

so a model like would be appropriate.

Making connections

This topic should help consolidate the students knowledge of differentiation and integration from their core studies.

For many this application of calculus to “the real world” might be the revelation of understanding that they are looking for.

Related information and resources

from the NCETMfrom other sites

The following weblinks offer further explanations:

you are able to recognise when the use of constant acceleration formulae is appropriate?(show/hide all)

What this might look like in the classroom

The distinction between the constant acceleration formulae and the previous section involving calculus is solely made on the basis of a constant acceleration. The best example of this is gravity and many questions will be set in the context of vertical motion. Balls dropping are quite simple to understand and yield good mathematics questions.

It is fairly easy to take the class outside and have them throw balls vertically in the air (and hopefully catch them again). Timing the ball from throw to catch will give TWICE the time from throw to the highest point the ball reaches. Taking a few measurements to smooth errors is worth the trouble.

On returning to the classroom students should be able to use the constant acceleration formulae to find how fast the ball was thrown and what height it reached. Changing the velocity into mph often helps the students grasp how slow their throwing actually is – and comparing the height thrown with a nearby building, or lamppost might prove an interesting point for discussion.

Taking this mathematics further

Students should consider a model that incorporates air resistance, and is thus able to explain the difference between throwing a real tennis ball and a child’s lightweight tennis ball.

Making connections

Carefully deriving the constant acceleration formulae is a profitable exercise and allows the students to check they understand the links between velocity and acceleration.

These formulae should be learned to aid quicker calculations.

Related information and resources

from the NCETMfrom other sites

The following websites include constant acceleration formulae, and examples:

The Student Room - Revision of Kinematics - Equations of Motion for Constant Acceleration.

you are able to solve kinematics problems using constant acceleration formulae and calculus?(show/hide all)

What this might look like in the classroom

Students need to practice this two-part problems, and need to be able to recognise when the situation is based on constant acceleration and when it isn’t. The integration required can be quite challenging.

Taking this mathematics further

These problems become harder as the calculus does – so seeking harder, and more realistic, models will lead to some suitably challenging extensions.

Making connections

The integration can often be quite challenging, but is then a useful practical application of their integration skills.

Related information and resources

from the NCETMfrom other sites

These links from Wikibooks.org cover differential calculus and kinematics:

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