Teacher's notes - Royal Academy of Engineering...2 Teacher's notes The Controlling Motion project box has been developed from a more substantial set of Key Stage 3 curriculum units

Teacher's notes

1 Royal Academy of Engineering

Cornerstone Maths was originally conceived following a charitable donation from the Li Ka Shing Foundation and Hutchison Whampoa Europe Limited.

It was introduced into 124 UK schools under a successful partnership between SRI International and the UCL Knowledge Lab, UCL Institute of Education.

This resource has been developed by Bola Abiloye, Alison Clark-Wilson and Celia Hoyles at UCL Knowledge Lab for the Royal Academy of Engineering with support from the following Connecting STEM Teachers programmes' Teacher Coordinators: Scott Atkinson, Annie Beglin Price, Claudia Clarke, Michael Cronk, Richard Daniel, Richard Davies, Lorraine Drybrough, Arthur Harwood, Stuart Higham, Thomas Lavery, Andrew Leech, Eleanor Lucas, Andrew McVean, Garrod Musto, Michael Nelson, Krissi Pink, Gareth Richards, Gaynor Sharp, Martin Simmons, Rebecca Sowden, Philip Sutton, David Thomas, Peter Thomas, Anthony Vaughan-Evans.

2

Teacher's notes

The Controlling Motion project box has been developed from a more substantial set of Key Stage 3 curriculum units of work, Cornerstone Maths, for use in STEM clubs and activities. Controlling Motion uses selected activities from the Cornerstone Maths unit that addresses the teaching of linear functions at Key Stage 3.

Mathematics colleagues can find out more about the Cornerstone Maths curriculum units (linear functions, geometric similarity and patterns and expressions) at www.cornerstonemaths.co.uk

Access to the software in your school

The software is hosted by London Grid for Learning (LGfL).

Most schools in London are already subscribed to LGfL and can access the software directly from the LGfL website (www.lgfl.net).

If you are not in London, or do not subscribe to LgfL, you will need to organise your pupils’ access to the specific Cornerstone Maths software activities that link with this project box.

Please send an email to [email protected] with the subject line ‘controlling motion’ and the contact name and email of your Head of IT Services to organise access to the software in your school.

Accessing the activities

Go to the LGfL website (www.lgfl.net). On the red tool bar, select learning resources, contents and then maths/numeracy. Scroll down to find Cornerstone Maths.

About this resource

Scroll to choose a teacher and a class - it does not matter what you choose.

Select CONTINUE.

Scroll to select at least one pupil.

Select LOGIN.

Select MODULE 1 – Linear functions to access the software activities that are needed for the tasks in the Controlling Motion project box.


This is a set of 10 Tasks which engage pupils in exploring linear functions by controlling motion.

In each Task, pupils are expected to play the role of a junior programmer/coder for a retro mobile games developer. They are given short tasks to complete.

Pupils design segments of journeys for different characters in the game on behalf of the developer. They use their knowledge of how graphs and equations can be used to control journeys to work out and supply the correct graphs and equations to the senior programmers. The pupils do not design the whole game.

Pupils start by investigating actual motion using a real pull-back mechanical toy supported by a video, which is a simplified version of the hands-on activity. They then investigate speed using the software that includes modelled animations of similar simplified situations.

The software shows four representations of motion with which the pupils engage.

■■ The simulation.

■■ The graph.

■■ The table of values.

■■ The equation.

The Controlling Motion materials are designed to engage pupils in using digital technology to explore multiple representations of mathematical situations to bring about a deep and connected understanding of big ideas in mathematics.

These activities have been designed to use as an enrichment STEM activity for all pupils at Key Stage 3 with support by teachers. Teachers should have a STEM background and some experience/understanding of using the software or be ready to explore the concepts using the software alongside pupils. There is a careful mathematical progression within these tasks so, if you do not have time to do all of the tasks in succession, a shorter sequence is possible using just Tasks B, C, D and G.

Each task takes approximately 30 minutes for pupils to complete.

Where pupils are required to plot graphs, a labelled and scaled set of axes on 5mm squared paper is provided for them to copy onto standard graph paper.

Understanding the context

Using the resources

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Teacher's notes

Name/Resource Aims

Task A Investigating speed: pull-back toys

For pupils to learn:- that a variety of factors affect the speed of the toy.

Task B Real-life motion: Yari the Yellow Bus

Investigation 1 (video/snapshots)

For pupils to learn:- how speed can be represented on a position-time graph- that if the speed varies, the graph produced is not a straight line.

Task C Controlling motion using graphs: Rita the Red Bus

Investigation 2

Activity 2.1

For pupils to learn:- why real-life motion is sometimes modelled to simplify the situation- how a graph can be edited to control the motion of the toy- how to use the step, edit and refresh tools.

Task D Controlling races using graphs

Investigation 3

Activity 3.3

For pupils to learn:- that on a position-time graph the steeper the line segment, the faster the

car is travelling- how to sketch graphs to represent constant motion- how to control races by editing the graph.

Task E Controlling races with graphs and checking with tables of values

Activity 3.3

For pupils to learn:- how to sketch graphs fluently- how to find the value of the speed (as a rate) from a Table of Values of

distance (position from start) and time.

Task F Controlling races with cars starting at different positions

Activity 3.3

For pupils to learn:- that linear graphs showing do not always represent proportional

relationships- that a different starting positions is represented by the y – intercept- that a different starting position will affect the corresponding table of

values.

Task G Controlling characters using equations

Activity 4.1

For pupils to learn:- how motion can be represented by equations and how the equations can be

represented by graphs.

Task H More control with equations

Activity 3.3

For pupils to learn:- the general form of the equation of a straight line (y = mx + c)- how to create journeys (and their graphs) using the equation of the line.

Task I Wendella’s journeys – Multi-segment graphs.

Activity 7.1 to 7.4

For pupils to learn:- that graphs tell a story- flat or horizontal lines represent standing still- To calculate the gradient (speed) of line segments that do not start at x=0.

Task J Crab Velocity – Controlling journeys using equations part 2

Activity 10.2

For pupils to learn:- that a velocity is a speed with a direction relative to a given direction- that a negative velocity is not always slower- how to use negative and decimal numbers in this context.

Overview of tasks


Aim

For pupils to learn:

■■ that a variety of factors affect the speed of the toy.

Resources

Pull-back toys (in box)

Meter rule

Stop watch

Guidance

The expectation here is for pupils to:

■■ Pull back the toys and watch them move.

■■ Recognise that a number of factors affect the speed of the toy (such as how far they are pulled back or the type of surface they are on).

■■ Discuss how to conduct a fair test using the factors they have identified.

■■ Record the time taken and the distance travelled by each toy.

Possible support

1. If pupils are stuck encourage them to divide total distance travelled by total time in order to calculate the average speed. Discuss common units of speed like km/h or m/s and explain how these are linked to the ratio of distance and time.

2. If needed, possible suggestions (hints) to pupils could be to use:

■■ The same surface for all tests.

■■ The same distance (for example, if pupils decide to create a race).

■■ To pull each toy back the same distance.

Task A – Investigating speed: pull-back toys

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Teacher's notes

Aim


■■ how speed can be represented on a position-time graph

■■ that if the speed varies, the graph produced is not a straight line.

This is a real scenario that scaffolds the introduction of simulated models for real-life motion that pupils will meet in subsequent activities using the software.

Pupils should be encouraged to write their own written definition of speed at the end of the task.

Resources

Cornerstone Maths Linear functions software: Investigation 1 'Yari the Yellow Bus'

Guidance


1. Predict the motion of the yellow bus and check their predictions using the video of the continuous motion segment and the photo snap shots that play afterwards.

2. Yari's distance and time are given in Q1 but pupils are expected to notice this on the video as well (100cm and 10 sec). Yari's average speed is 10cm/s.

3. It is worth suggesting that pupils join the points using straight line segments or join the points to form an S curve. Pupils should not draw a line of best fit as we are not interested in exploring correlation. Encourage pupils to discuss and then record their emerging written definitions of speed.

Possible support

You may need to support pupils to come up with a definition of speed in their own words based on their personal experiences. They will have an opportunity to refine this later on in the activity. At this stage it is helpful to establish that speed is a rate.

Task B – Real-life motion: Yari the Yellow Bus


Aim


■■ why real-life motion is modelled to simplify the situation

■■ how a graph can be edited to control the motion of the toy

■■ how to use the step, edit and refresh tools.

Resources

Cornerstone Maths Linear functions software: Investigation 2 Activity 2.1 – 'Rita the Red Bus'

Guidance

The expectation here is for pupils to recognise that

■■ The time taken is controlled by the horizontal controller B on the x-axis.

■■ Both distance and speed can be controlled by the vertical controller A.

■■ The start position of a character is controlled by the position of controller C on the y-axis.

1a. 100cm 1b. 10s 1c. 10cm/s.

2. Straight line graph from (0,0) to (10,100).

3. Yari and Rita travel the same distance in the same time but Rita has a constant speed while Yari does not.

4b. The start position and time taken stay the same.

4c. The start position and speed stay the same.

4d. The time taken and the speed stay the same.

Possible support

Encourage pupils to explore the simulation carefully, ensuring that they observe how the timer relates to the graph and how the colour of the line segment on the graph fills as the character moves.

Pupils will need to watch the original motion first before they edit the graph so they can make sense of how the controllers work.

Show pupils how to refresh the activity if they feel they have ‘spoilt’ the graph or want to see the original motion and its graph.

Task C – Controlling motion using graphs: Rita the Red Bus

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Teacher's notes

Aim


■■ that on a position time-graph the steeper the line segment, the faster the car is travelling

■■ how to sketch graphs to represent constant motion

■■ how to control races by editing the graph.

Resources

Cornerstone Maths Linear functions software: Investigation 3 Activity 3.3 'Controlling Characters with graphs'

Guidance

It is important to ensure that pupils predict first (i.e. answer questions 1 and 2) before checking by playing the simulation in question 3. The same applies to question 5 and 6.


1. Begin to control the graph in a meaningful way and become more confident in doing so.

2. They should recognise that Blue Waters will move faster.

Be able to calculate the speed of Blue Waters as 50 miles per hour (350 ÷7) and Green Grass as 35 miles per hour (350 ÷10) and

3b. and c.

Recognise that the cars have travelled the same distance but taken different times so one is going faster than the other.

3d. Justify that Blue Waters is travelling faster because its graph is steeper.

4a. Recognise that the increase in distance travelled after each and every second in the table represents the speed.

5. Use the table to explain that Blue Waters must be the faster car as it had travelled the complete distance of 350 miles by 7 seconds whereas Green Grass travelled the same distance in 10 seconds.

Possible support

■■ It may be useful for pupils to learn that they can change the scale of each axis on the graph using the zoom icons.

■■ Remind pupils that they should refresh the graph if they get stuck.

■■ Many pupils find it easier to reduce the total journey time first before then changing the speed using controller B.

■■ Sketch means to draw the basic shape (a straight line in this case).

Task D – Controlling races using graphs


Aim


■■ how to sketch graphs fluently

■■ how to find the value of the speed (as a rate) from a table of values of distance (position from start) and time.

Resources

Cornerstone Maths Linear functions software: Investigation 3 Activity 3.3 'Controlling Characters with graphs'.

Guidance

The expectation here is for pupils to

■■ Predict – Sketch the graphs first before checking their predictions using the software.

1. The lines start at the same point on the y-axis and end at points which have the same x-coordinate but different y- coordinates. The green line is steeper.

Pupils may describe the common difference between the distances (for unit increases in time) as they way in which they notice the speed in the table of values.

2. The lines start at the same point on the y-axis and end at points which have the same y-coordinate but different x- coordinates. The x-coordinate of the green line is 5 more than that of the blue line.

3. There is only one solution. Pupils will be creating graphs for the first time using information where the speed is given. Pupils can show this on their graph and using the software.

Possible support

■■ Pupils might need help to see that moving Controller B from left to right increases the journey time whereas moving Controller A upwards increases the speed, and therefore the total distance travelled.

■■ Pupils may incorrectly associate ‘shorter time’ with ‘shorter line’ as an indication of faster motion.

■■ Pupils may have difficulty

■■ Accepting that predictions can be wrong.

■■ Sketching correct speeds especially 80 miles per hour because of the scale of the graph. At this point suggesting that pupils consider how far the car will travel in 10 hours may help.

Task E – Controlling races with graphs and checking with Tables of Values

10

Teacher's notes

Aim:


■■ that linear graphs showing do not always represent proportional relationships

■■ that a different starting positions is represented by the y – intercept

■■ that a different starting position will affect the corresponding table of values.

Resources


Guidance

Here pupils are demonstrating their fluency in sketching graphs with the additional feature of different start positions. The scenarios are slightly harder to interpret on a graph, and this provides additional challenge.

The expectation is for pupils to begin to develop an understanding of how a graph and a table can show a rate of change with even when the start value is not zero (non-proportional lines).

Making a link with sequences here may be a good idea.

1a. See graph below b) 100miles

It ought to be obvious that as the two cars are travelling at the same speed, they don’t get any closer to/further from each other while travelling.

1c. As Green Grass car started further ahead, it would reach the end of the race first.

2. Speeds may vary. Pupils should be able to deduce from the table that the speed is still the common difference between the distances travelled but that there is no common multiplier from time to distance travelled when the table does not start at zero.

3a. See graph and table below. It is easier to read the distance travelled from the table Green Grass 690 miles, Blue Water 810 miles. So they are 120 miles apart.

3b. Green Grass 60mph, Blue Waters 90mph.

3c. Blue Waters by 120miles.

Task F – Controlling races with cars at different starting positions

Possible support

May need help

■■ recognising that parallel lines on graphs indicate journeys with the same speed which have different start positions (never at the same place at the same time) not two cars travelling side by side

■■ interpreting the intersection of two lines in a graph.


Aim


■■ how motion can be represented by equations and how the equations can be represented by graphs.

Resources

Cornerstone Maths Linear functions software: Investigation 4 Activity 4.1 'Controlling Characters with equations'

Guidance

This is an easily accessible activity to introduce the concept of the equation of a straight line. Some pupils will be able to complete both Task G and Task H in the same session.

The expectation here is for pupils:

■■ To conclude from questions 1, 2 and 3, that the speed of the character is equal to the coefficient of x in the corresponding equation. Pupils should write this conclusion in response to question 4.

■■ To write the equation y = 6.0x + 0 in response to question 5d and then edit the equation in the software to check if their prediction is correct.

■■ To predict that the equation changes to y = 6.0x + 8 and infer that the coefficient of x stays the same but the start position is added on as a constant.

Possible support

■■ Some pupils will need support in describing the number multiplying the ‘x’ as always equal to the speed. The earlier they are introduced to the word coefficient the better.

Task G – Using equations to control journeys

12

Teacher's notes

Aim


■■ the general form of the equation of a straight line (y= mx + c)

■■ how to create journeys (and their graphs) using the equation of the line.

Resources


Guidance

This activity enables pupils to reinforce what they deduced in Task G but builds up in challenge quickly.

The expectation is for pupils to:

1. Set up the correct race and deduce:

a) Green Grass wins

b) and c) from the graph and table that it takes less time for Green Grass to travel 500 miles (5 hrs) as Blue Waters takes 7 hrs.

Check they create the race correctly by using the equations:

Y= 50x + 150 (Blue Waters) Y= 100x + 0 (Green Grass)

Both speeds, 50 and 100, should be underlined while both start positions, 150 and 0, are circled.

2. a) Predict equations as y= 25x + 200 (Blue Waters) and y = 100x+0 (Green Grass)

b) They should state the coefficient of x is the speed in mph and the constant is the position in miles with respect to the starting position.

3. a) y=80x + 0 (Green Grass), Y = 80x+ 20 (Blue Waters)

b) Answers vary

c) Answers vary but

80 x 2 + 20= 180

By substituting the time in place of x

4. Here pupils are selecting the best representation to use for different problems

a) y= 30x +150 b) y = 60x + 0 c) After 5 hours.

d) The graph – but is also confirmed by the table.

e) Green Grass 420 miles Blue Waters 360 miles.

f) The table. g) 1050 miles and 1800 miles. The equation bysubstituting.

5. Pupils make general statements about how they will use each of the different representations if they had to.

Task H – More control with equations

Possible support

■■ Pupils may find it difficult to understand why they should substitute time into an equation to find the distance. Using an equation of y= mx +0 and its table may support this.


Aim


■■ that graphs tell a story

■■ flat or horizontal lines represent standing still

■■ to calculate the gradient (speed) of line segments that do not start at x=0

Resources

Cornerstone Maths Linear functions software: Investigation 7 Activities 7.1 to 7.4 'Wendella’s Journey'

Guidance

Pupils will describe journeys in which the dog Wendella changes her speed and describe contexts that match the changing speeds.

The expectation is for pupils to:

1. Recognise and match the different line segments the different terrain path – swamp - stop bark for help –path. Most pupils really enjoy creating a story here.

2.Segment Where Minutes Metres Speed

A Path 1 300 300m/min

B Quicksand 2 0 0 m/min

C Swamp 4 100 25m/min

D Path 2 600 300m/min

E Quicksand 4 100 25m/min

3. Answers may vary.

Acceptable answers include

Wendella walks at a speed of 25m/min for 4mins before speeding up and running at a speed of 300m/min for 2 mins. She then turns round and runs 600m back towards her start position in 2 mins (300m/min) before turning around again and walking away at a speed of 25m/min for 4 mins again.

3e. This question supports pupils to calculate the speed in the third segment of this journey. Distance travelled is 600m in the 2 mins.

Task I – Wendella’s Journeys – Multi-segment graphs

Possible support

Pupils may discuss Wendella’s journey with others in order to reveal misconceptions such as:

■■ Wendella’s journey involving climbing a hill and coming down again.

■■ Sketching impossible journeys that show Wendella travelling back in time or covering some distance in no time (a vertical line).

14

Teacher's notes

Aim


■■ that a velocity is a speed with a direction relative to a given direction

■■ that a negative velocity is not always slower

■■ how to use negative and decimal numbers in this context.

Resources

Cornerstone Maths Linear functions software: Investigation 10 Activity 10.2 'Crab Velocity'

Guidance

This activity draws knowledge from all of the previous activities together. Pupils use the links between equations and graphs in addition to journeys that move towards the starting point. The task has a few challenges but is also easily accessible. It also provides a different context for our junior programmers to work in.

The expectation is that:

1–3 Pupils recognise that the speed is the same as the velocity except that the velocity is also described with a positive sign (+) for upward motion and, in this case and a negative sign (-) for downward motion. The additional challenge is working with negative numbers and decimal numbers.

Rory’s speed: 1m/s. Rory’s velocity: +1m/s

Galileo’s speed: 0.4 m/s Galileo’s velocity: - 0.4m/s

4. It may be necessary to show those struggling why the equation for Rory’s journey is

y = 1x - 12 or y = x – 12.

5.Speed (m/s)

Velocity Start position

Equation

Bettina 1 -1 12 y =-1x +12

Galileo25

or 0.4 –

25

or - 0.4 -5 y = -0.4x -5

Penelope25

or 0.425

or 0.4 0 y =

25

x

Orlando25

or 0.425

or 0.4 +2 y = 0.4x +2

Task J – Controlling journeys using equations Part 2 – Crab Velocity

Possible support

■■ Pupils will be hesitant to write down decimal or fractional answers – encourage them to use them.

■■ Pupils may repeatedly omit the 'x' from the equation. It is important to highlight that the speed is the distance travelled in each second. This has to be multiplied by the number of seconds which is the value of 'x' in order to calculate the total distance.

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Teacher's notes - Royal Academy of Engineering...2 Teacher's notes The Controlling Motion project box has been developed from a more substantial set of Key Stage 3 curriculum units