MEI Conference 2016

Get Set for September 2017:

Mechanics & Modelling

Sharon Tripconey sharon.tripconey@mei.org.uk

Simon Clay simon.clay@mei.org.uk

2 of 16 Get Set for September 2017: Mechanics & Modelling June 2016 © MEI

Content extracts for reformed Mathematics A level from Sept 2017

All extracts taken from ‘Mathematics AS and A level content’ (2014), DfE. Note that bold text within

[square brackets] is AS content.

Overarching themes

OT1 Mathematical argument, language and proof

Ref Knowledge/Skill

OT1.1 [Construct and present mathematical arguments through appropriate use of diagrams; sketching graphs; logical deduction; precise statements involving correct use of symbols and connecting language, including: constant, coefficient, expression, equation, function, identity, index, term, variable]

OT1.2 [Understand and use mathematical language and syntax as set out in the content]

OT1.3 [Understand and use language and symbols associated with set theory, as set out in the content]

[Apply to solutions of inequalities] and probability

OT1.4 Understand and use the definition of a function; domain and range of functions

OT1.5 [Comprehend and critique mathematical arguments, proofs and justifications of methods and formulae, including those relating to applications of mathematics]

OT2 Mathematical problem-solving

Ref Knowledge/Skill

OT2.1 [Recognise the underlying mathematical structure in a situation and simplify and abstract appropriately to enable problems to be solved]

OT2.2 [Construct extended arguments to solve problems presented in an unstructured form, including problems in context]

OT2.3 [Interpret and communicate solutions in the context of the original problem]

OT2.4 Understand that many mathematical problems cannot be solved analytically, but numerical methods permit solution to a required level of accuracy

OT2.5 [Evaluate, including by making reasoned estimates, the accuracy or limitations of solutions], including those obtained using numerical methods

OT2.6 [Understand the concept of a mathematical problem solving cycle, including specifying the problem, collecting information, processing and representing information and interpreting results, which may identify the need to repeat the cycle]

OT2.7 [Understand, interpret and extract information from diagrams and construct mathematical diagrams to solve problems, including in mechanics]

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OT3 Mathematical modelling

Ref Knowledge/Skill

OT3.1 [Translate a situation in context into a mathematical model, making simplifying assumptions]

OT3.2 [Use a mathematical model with suitable inputs to engage with and explore situations (for a given model or a model constructed or selected by the student)]

OT3.3 [Interpret the outputs of a mathematical model in the context of the original situation (for a given model or a model constructed or selected by the student)]

OT3.4 [Understand that a mathematical model can be refined by considering its outputs and simplifying assumptions; evaluate whether the model is appropriate]

OT3.5 [Understand and use modelling assumptions]

Mechanics content

P Quantities and units in mechanics

Ref Content description

P1 [Understand and use fundamental quantities and units in the S.I. system: length, time, mass]

[Understand and use derived quantities and units: velocity, acceleration, force, weight], moment

Q: Kinematics

Ref Content description

Q1 [Understand and use the language of kinematics: position; displacement; distance travelled; velocity; speed; acceleration]

Q2 [Understand, use and interpret graphs in kinematics for motion in a straight line: displacement against time and interpretation of gradient; velocity against time and interpretation of gradient and area under the graph]

Q3 [Understand, use and derive the formulae for constant acceleration for motion in a straight line]; extend to 2 dimensions using vectors

Q4 [Use calculus in kinematics for motion in a straight line: 𝒗 =𝒅𝒓

𝒅𝒕,

𝒂 =𝒅𝒗

𝒅𝒕=

𝒅𝟐𝒓

𝒅𝒕𝟐, 𝒓 = ∫𝒗𝒅𝒕, 𝒗 = ∫𝒂𝒅𝒕 ]; extend to 2 dimensions using vectors

Q5 Model motion under gravity in a vertical plane using vectors; projectiles

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R Force and Newton’s laws

Ref Content description

R1 [Understand the concept of a force; understand and use Newton’s first law]

R2 [Understand and use Newton’s second law for motion in a straight line (restricted to forces in two perpendicular directions or simple cases of forces given as 2-D vectors)]; extend to situations where forces need to be resolved (restricted to 2 dimensions)

R3 [Understand and use weight and motion in a straight line under gravity; gravitational acceleration, g, and its value in S.I. units to varying degrees of accuracy]

[(The inverse square law for gravitation is not required and g may be assumed to be constant, but students should be aware that g is not a universal constant but depends on location)]

R4 [Understand and use Newton’s third law; equilibrium of forces on a particle and motion in a straight line (restricted to forces in two perpendicular directions or simple cases of forces given as 2-D vectors); application to problems involving smooth pulleys and connected particles]; resolving forces in 2 dimensions; equilibrium of a particle under coplanar forces

R5 Understand and use addition of forces; resultant forces; dynamics for motion in a plane

R6 Understand and use the 𝑭 ≤ 𝜇𝑹 model for friction; coefficient of friction; motion of a body on a rough surface; limiting friction and statics

S Moments

Ref Content description

S1 Understand and use moments in simple static contexts

Note the following statements from the pure content where modelling or mechanics is mentioned:

Ref Knowledge/Skill

B11 Use of functions in modelling, including consideration of limitations and refinements of the models

C4 Use parametric equations in modelling in a variety of contexts

D6 Use sequences and series in modelling

E9 Use trigonometric functions to solve problems in context, including problems involving vectors, kinematics and forces

F7 [Understand and use exponential growth and decay; use in modelling (examples may include the use of e in continuous compound interest, radioactive decay, drug concentration decay, exponential growth as a model for population growth); consideration of limitations and refinements of exponential models]

G6 Construct simple differential equations in pure mathematics and in context,

(contexts may include kinematics, population growth and modelling the relationship between price and demand)

H8 Interpret the solution of a differential equation in the context of solving a problem, including identifying limitations of the solution; includes links to kinematics

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The Modelling Cycle

Many different variations of the Modelling Cycle can be found. The one above is taken from ‘OCR Level

3 Advanced GCE in Mathematics B (MEI) H640 Draft Specification’ accessed on 15/06/2016 via

http://www.ocr.org.uk/qualifications/as-a-level-gce-mathematics-b-mei-h630-h640-from-2017/

The references in each box namely, Mp13, Mp14, Mp15, Mp16 & Mp17 refer to competence statements which are

listed within the Draft Specification.

2017 AS/A level Mathematics specifications: How does the content compare?

The information below is intended to provide a helpful summary of the differences between the current and the 2017 specifications for AS and

A level Mathematics. It is based on the document Mathematics AS and A level content published by the DfE in April 2016.

We have made every effort to ensure the accuracy of this information; however we shall not be held responsible for any inaccuracies it may

contain.

Mechanics: The new A levels have a greater emphasis on modelling. This is likely to be particularly prevalent in the Mechanics content.

The table below provides a topic-based comparison between current and 2017 AS/A level Mathematics specifications.

Current specifications 2017

AQA Edexcel OCR OCR(MEI) WJEC Specifications

Modelling

The modelling cycle applied to problems M1 M1 M1 M1 M1 AS

SI units - M1 - M1 - AS

Vectors

The properties of vectors M1 M1 M1 M1 M2 AS1

Scalar product and applications - M5 - - M2 -

Kinematics

Terminology M1 M1 M1 M1 M1 AS

Kinematics graphs M1 M1 M1 M1 M1 AS

Constant acceleration formulae M1 M1 M1 M1 M1 AS2

Use of calculus - variable acceleration M2 M3 M1 M1 M2 AS2

Vertical motion under gravity M1 M2 M1 M1 M1 AS

Forces

Identification of forces and force diagrams M1 M1 M1 M1 M1 AS

Vector treatment of forces - equilibrium, resultants M1 M1 M1 M1 M1 AS2

Modelling friction M1 M1 M1 M2 M1 A level

Moments M2 M1 M2 M2 M1 A level

Rigid bodies in equilibrium - sliding, toppling M2 M2 M2 M2 M1 -

Light frameworks M4 - - M2 M1 -

Centre of mass M2 M2 M2 M2 M1 -

1 Extend to 3 dimensions at A level 2 Extend to resolving forces at A level

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Current specifications 2017

AQA Edexcel OCR OCR(MEI) WJEC Specifications

Newton's laws of motion

Newton's three laws M1 M1 M1 M1 M1 AS2

Connected particles M1 M1 M1 M1 M1 AS

Projectiles

The motion of a projectile M1 M2 M2 M1 M2 A level

Equation of trajectory M3 - M2 M1 - -

Work, energy & power

Kinetic and mechanical energy M2 M2 M2 M2 M2 -

Work-energy principle M2 M2 M2 M2 M2 -

Power M2 M2 M2 M2 M2 -

Hooke's law M2 M3 M3 M3 M2 -

Energy in strings & springs M2 M3 M3 M3 M2 -

Momentum and impulse

Conservation of linear momentum M1 M1 M1 M2 M1 -

Impulse M3 M1 M2 M2 M1 -

Coefficient of restitution M3 M2 M2 M2 M1 -

Impact with fixed surface M3 M2 M2 M2 M1 -

Oblique impact M3 M4 M3 M2 - -

Circular motion

Horizontal M2 M3 M2 M3 M2 -

Vertical M2 M3 M3 M3 M2 -

_________________________

2 Extend to resolving forces at A level

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Departmental activities

The following three tasks are activities which could be used with a department to raise awareness of

some of the issues related to teaching mechanics in the reformed A levels. Below are descriptions of the

resources which follow on pages 10-16. The descriptions include the intended purpose, suggested

approaches to using with departments and potential outcomes of each activity. The descriptions of how

they could be used are offered as suggestions.

The intention is that the activities could form part of a teaching and learning discussion in team meetings

and be a launch pad into working collaboratively to design shared approaches to teaching the new A

levels. We believe the use of these would be beneficial to most mathematics departments and may be

of particular benefit in departments where there is a spread of experience, expertise and confidence in

the teaching of A level maths.

Activity A. Belief statements (pages 10-12)

Purpose: The purpose of this activity is to allow individuals and the department as a whole to reflect

upon the beliefs they hold regarding the teaching of mechanics. Since mechanics has for so long been a

separately assessed module of A level mathematics, many schools and colleges have taught it as a

standalone and separate area of content.

Suggested approach: Individuals to sort statements into two categories: statements they agree with

and statements they disagree with. In small groups the teachers could then discuss their decisions. For

this to be a constructive exercise there needs to be a safe, non-judgemental space created where

disagreement is ok! The groups then could be asked to come to a consensus on a statement they

strongly agree with and one they strongly disagree with. These statements could form the basis of a

whole group discussion.

Outcomes: Having a department where beliefs about teaching are articulated and discussed can be a

stimulating environment in which to develop one’s own practice. There are specific pedagogical issues

related to the teaching of mechanics and having an open discussion about the opportunities and barriers

to effective mechanics teaching is worthwhile as it forms a basis from which the department can develop.

Activity B: Sometimes, Always, Never True statements (page 13)

Purpose: This activity is designed to (a) exemplify the type of activity that can be used in classrooms to

promote discussions amongst students, (b) highlight common misconceptions held by learners of

mechanics and (c) provide an opportunity for subject knowledge development for individuals.

Suggested approach: Each statement is discussed (pairs or threes works well) and placed in a

category of ‘Always true’, ‘Sometimes true’ or ‘Never true’. These categorisations are then discussed as

a whole team. It would also be worth discussing what responses students would typically give and the

misconceptions their responses reveal. Discussion about why cards are placed in particular categories

is crucial to the exposure of commonly held misconceptions. See ‘Mechanics in Action’ (link below) for

further details and more on the themes of misconceptions and practical work.

Outcomes: An effective discussion-based activity for use with students is tried by the teaching staff in a

‘safe’ environment. Potential pitfalls and misunderstanding are discussed by staff before being used with

students. Many of the statements are ‘sometimes true’ which demonstrates that mechanics is an area of

mathematics which allows rich discussion and debate.

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Activity C: Connections between mechanics and other areas of mathematics (pages 14-16)

Purpose: This activity is designed to highlight the opportunities which now exist for teaching A level

mathematics in a coherent way, given that the new qualifications are linear in nature.

Suggested approach: The graph card sort activity (pages 14-16) could be used in pairs and a

discussion including these points then follow:

- Is this a mechanics or a pure mathematics task?

- Does our language change when teaching mechanics or pure lesson? If so, how?

- Think about the language you used when describing stationary points, changing gradients, etc.

How could these helpfully be expressed in a mechanics context? And in a pure context?

The team could then discuss other links they know of between different topics with links written on Post

its and displayed on the wall for easy grouping into related topic areas. Reference to the DfE Content

document (see link below) might be useful. Here is a (non-exhaustive) list to get you started:

Calculus - motion graphs

- constant and variable acceleration

DEs - modelling motion

Inequalities - friction model

Parametric equations - projectile motion

Partial fractions - to solve modelling DEs

Trigonometry - resolving vectors into components

- finding resultant vectors

Vectors - vector and scalar quantities

- resolving into components

Outcomes: An increased appreciation of the many connections and links between topics across the A

level content. The beginning of the department working together on a shared Scheme of Work

highlighting these connections and giving a fresh approach to teaching A level mathematics. MEI is

developing a SoW (appropriate for all specifications) and this will indicate some of the links with other

topics, contain links to resources, etc. It will be made freely available via this link:

http://www.mei.org.uk/2017-sow

Useful documents

‘Mathematics AS and A level content’ (2014), DfE https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/516949/GCE_AS_and_A_level_subject_content_for_mathematics_with_appendices.pdf ‘A Level Mathematics Working Group Report on Mathematical Problem Solving, Modelling and the Use of Large Data Sets in Statistics in AS/A Level Mathematics and Further Mathematics’ (2015), Ofqual https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/481857/a-level-mathematics-working-group-report.pdf ‘Mechanics in Action’ (1990) A fantastic free resource! This is available from the Stem Centre, although you have to set up a free account or log in to access it. https://www.stem.org.uk/system/files/elibrary-resources/legacy_files_migrated/3635-Mechanics%20in%20action.pdf

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Teaching mechanics is scary There is not enough time to do

practical work in Mechanics lessons

It’s better to teach the pure topics separately from the

mechanics topics

Teaching maths in an interconnected way helps

students to learn

It’s not important that students understand the modelling cycle

– they won’t be tested on it!

Students would benefit from the same person teaching them

both pure and mechanics

Combining pure and mechanics topics in the same lesson can

be confusing for students

Not using textbooks requires more teacher preparation time

Variety is important for keeping students engaged

It will save time if topics common to both pure and

mechanics are taught together

Practical activities are just the latest educational fad

Students don’t like doing practical activities

Able students don’t like doing activities that require them to

discuss

Weaker students find mechanics harder than pure or

statistics

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To succeed in exams the most important thing is to practice by

doing lots of past papers

Activities that promote discussion force the students to

engage with the content

Using activities to teach mechanics requires the teacher

to have a willingness to take risks

For activities to be successful, you need to create an

environment where students feel secure

An experienced Mechanics teacher can teach Pure but an experienced Pure teacher can’t

easily teach mechanics

GCSE Higher now includes kinematics so this will really

help students when they start studying mechanics

Teachers that have never taught mechanics will be able to

teach themselves as they go along

You either get mechanics or you don’t

Practical demonstrations in mechanics can be simple but

effective

Teaching mechanics is fun and interesting

Students who study Physics as well as Maths are better at

mechanics

Special equipment is needed to do mechanics experiments

Discussion helps to highlight misunderstanding in mechanics

Teachers need ideas and resources for incorporating practical work into lessons

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Students must learn the basics of mechanics before they are introduced to problems that

require modelling

Modelling in mechanics is really just about knowing what the simplifying assumptions are

“Real problems" can't be discussed until "the basics"

have been learned

By the time the basics have been "covered," little time is left to use them on "real problems"

There’s no real point in studying mechanics because the real world doesn’t work like that

Experiments never work and so you have to tell the students what they should have got

anyway

There’s no real point in studying mechanics because you have to

simplify things so much

Pre-made demos or applets can help students to visualise what

is happening

Students feel that pre-made demos or applets are ‘fixed’ to

give the right outcome

In Mechanics, the approach of confronting misconceptions is usually more successful than

that of avoiding them for as long as possible!

By using practicals, students quickly see something of the principle even if their depth of

understanding is not great

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Sometimes, Always or Never True cards

For each of the statements decide which category it fits into:

Always true Sometimes true Never True

A negative value of acceleration means that you are slowing down

A graph of distance travelled against time can have a negative gradient

The normal reaction acting on a person standing on a horizontal

surface is always equal and opposite to the person’s weight

At its maximum height a projectile has no overall force acting on it.

Forces come in pairs If the resultant force on a body is

zero then the body is in equilibrium

A distance time graph is the same as a displacement time graph

The coefficient of friction is greater than 1

If the resultant force on a particle is zero then the particle is in

equilibrium

A velocity time graph is the same as a speed time graph

If the resultant force on an object is zero then the object is not moving

The weight of an object acts vertically downwards

If the tow-bar between a car and trailer is in compression then the car

is braking

If an object is moving backwards then there must be a force pushing

it backwards

A frictional force acts in the opposite direction to any motion taking place

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Graphs card sort

Task: Cut up the nine cards from Resource sheet A and place them correctly in the grid below.

Fu

nc

tio

n (

f

)

Gra

die

nt

fun

cti

on

(

f'

)

Gra

die

nt

of

the

gra

die

nt

fun

cti

on

(

f ”)

Reflection questions:

Which cards did you place first? Why?

What strategies and features did you use to decide where to place the cards?

Are there any cards that are not unique? (i.e. another graph could be sketched instead.)

Using the blank axes on Resource Sheet 2, sketch a simple graph of your own and choose to place it in the top, middle or bottom row. What would the two other graphs in the remaining rows look like?

15 of 16 Get Set for September 2017: Mechanics & Modelling June 2016 © MEI

Resource sheet A: 9 cards to cut up

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Resource sheet B: Blank cards

Session description From September 2017 all students starting A level Mathematics will learn

both Statistics and Mechanics. This applied content will account for a third

of the qualification. In this session we will explore the features of the

Mechanics content including changes in subject content compared to

current M1 specifications, the increased emphasis on mathematical

modelling and the connections to other topics which can be made while

teaching a linear course.

This session is one of a series of 4 which are designed to inform teachers

of upcoming changes to A level. It is therefore suitable to any teachers of A

level and may be of particular interest to heads of department or KS5

coordinators. The resources accompanying this conference session will be

suitable for using with departmental colleagues in order to help teams

prepare for the changes.

To think about…

How many ways are there of arranging

5 different books on a shelf?

Get set for September 2017:

Mechanics & Modelling

Sharon Tripconey & Simon Clay

To think about…

How many ways are there of arranging

5 different books on a shelf?

Section 1: Modelling

Dan Meyer’s

‘Holey Cow!’

video

Modelling in Mechanics

Is the hole really 1500 feet deep?

How can you calculate the depth?

What mathematics have you used?

What simplifying assumptions have

you made?

…?

Modelling in Mechanics

What happens next?

Extract from ‘Mechanics in Action’

Three Overarching Themes

OT1: Mathematical argument, language

and proof

OT2: Mathematical problem-solving

OT3: Mathematical modelling

Three Overarching Themes

OT1: Mathematical argument, language

and proof

OT2: Mathematical problem-solving

OT3: Mathematical modelling

A. Tasks have little or no scaffolding

B. Tasks provide for multiple

representations

C. The information is not given in

mathematical form

D. Tasks have a variety of

techniques that could be used

E. The solution requires

understanding of the processes

involved

F. The task requires two or more

mathematical processes

Problem-solving help from Ofqual

Modelling in Mechanics

Modelling in Mechanics

Modelling in Mechanics

Modelling in Mechanics

Modelling in Mechanics

The Modelling Cycle

Real-world

problem

Simplifying

assumptions

Mathematical

model

(equations

etc)

Analysis and

solve

Prediction

Interpretation

Validation

Experiment

Section 2:

Connections between

mechanics and other topics

Card sort task 1. Cut up the 9

graph cards

2. Place the cards

correctly in the 3 × 3 grid

Connections: Some considerations

- Is the card sort a pure or mechanics task?

- Does our language change when teaching mechanics or

pure? If so, how?

- Draw out language around turning points, etc. These can be

framed in mechanics context or pure context.

- No longer need to compartmentalise the content like in the

modular A level

- What about other links with pure?

- What opportunities for proof are there in mechanics?

Content: Mechanics topics

P: Quantities and units in mechanics

Q: Kinematics

R: Force and Newton’s laws

S: Moments

Section 3: Your department

Departmental considerations Activity 1: Beliefs about mechanics teaching

Departmental considerations Activity 2: Always, Sometimes, Never True

Departmental considerations Activity 3: Connections between mechanics and other topics

Further support

• Dedicated MEI website for the 2017 developments

mei.org.uk/2017

• MEI provides expert advice on the latest developments,

and supports you and your department as the changes

approach

• Support for all specifications is given

Further support - PD

‘Get Set for 2017’ courses:

mei.org.uk/2017-pd

A series of 4 one-day courses in an area near you coming

during the 2016-17

- Principles of the new mathematics A levels

- Statistics

- Mechanics

- Effective use of technology

Further support - PD Sustained PD courses for teachers looking to develop subject

knowledge & pedagogy for AS/A level Maths or Further Maths.

Teaching Advanced Maths (TAM) course mei.org.uk/tam

Teaching Further Mathematics (TFM) course mei.org.uk/tfm

TAM and TFM cover the pure mathematics content of A level

Maths and Further Maths respectively. For Mechanics and

Statistics PD we run the following:

Teaching Mechanics (TM) furthermaths.org.uk/teaching-mechanics

Teaching Statistics (TS) furthermaths.org.uk/teaching-statistics

Further support - FMSP

• Dedicated FMSP website for the 2017 developments

furthermaths.org.uk/2017

• A wealth of advice, guidance and support form the FMSP

team on the latest developments as the changes approach

• Support for all specifications is given

About MEI

• Registered charity committed to improving

mathematics education

• Independent UK curriculum development body

• We offer continuing professional development

courses, provide specialist tuition for students

and work with industry to enhance mathematical

skills in the workplace

• We also pioneer the development of innovative

teaching and learning resources