Curriculum Update m e i . o r g . u k I s s u e 5 3 M a y / J u n 2 0 1 6
Resits
2017 Mathematics AS/A level
Specification Changes
Decisions have been made about the availability of resit opportunities as specifications change .
What's changing?
New AS and A levels in Mathematics and Further Mathematics are being introduced in England for first teaching from September 2017. The changes include:
For the GCSE in
Mathematics which year 11 will sit in
Summer 2016, resit opportunities will be available in
November 2016 and
Summer 2017.
 New linear structure: AS will be decoupled from A level, and all assessment will take place at the end of the course. Exam questions may draw on the content of the whole A level.
For the current A level in Mathematics, there will be a resit opportunity in summer 2019.
 New emphasis: There will be more emphasis on problem solving, reasoning and modelling, and a requirement for the use of technology to permeate teaching and learning.
Level of demand of new A level
Mathematics qualifications
The new A levels in
Mathematics for teaching from
September 2017 are intended to be at the same level of demand as the current A levels in
Mathematics. Ofqual is planning to conduct research in July 2016 along similar lines to their GCSE research .
(Cont. on page 2)
 New content: The content of AS and A level Mathematics will be fixed. It will include pure mathematics, mechanics and statistics (including analysis of large data sets) - see our comparison and summary . There will be some choice in content for AS and A level Further Mathematics.
The Further Mathematics Support
Programme has produced a document that highlights the changes to the content for AS/A level Mathematics and
Further Mathematics. It summarises:
“All students will be assessed on their knowledge of Pure Mathematics,
Mechanics and Statistics. Decision mathematics will no longer be an option at A level. The assessment objectives include a greater emphasis on
Click here for the MEI
Maths Item of the Month modelling, problem-solving and reasoning, so some questions are likely to be longer with less scaffolding. This builds on the increase in problemsolving in GCSE so students will be better prepared. The examinations will be taken at the end of the course rather than in modules and it is expected that questions will link different aspects of the course.
“The awarding organisation can decide how to assess the content and they are currently producing specifications and sample assessment materials (SAMS) which will be submitted to OFQUAL for accreditation in June 2016.”
You can find out about the awarding organisations’ specifications and the support they intend to provide at the links below:


In this issue

AQA
Edexcel
 Eduqas
 OCR
 OCR(MEI)
 This half term’s focus: What’s



Curriculum Update changing in the new A levels?
Hugh’s Views: Guest writer Hugh
Hunt writes about linear A levels and joined-up thinking
Site-seeing with... Jo Sibley
KS4/5 Teaching Resource:
Forces
M 4 is edited by Sue Owen, MEI’s Marketing Manager.
We’d love your feedback & suggestions!
Disclaimer: This magazine provides links to other Internet sites for the convenience of users. MEI is not responsible for the availability or content of these external sites, nor does MEI endorse or guarantee the products, services, or information described or offered at these other Internet sites.

Curriculum Update
(Cont. from page 1)
A level Mathematics for teaching from
September 2017
Final conditions and requirements for A level Mathematics for teaching from
September 2017 have been published .
They allow for the first sitting of the new A level to be in summer
2018 (after one year of study); the current
A level will also be available in summer
2018.
Ofqual consultation on award of grades
8 and 9 at GCSE
Ofqual has previously published information that for students taking the new
GCSEs, 20% of those getting grade 7 or above would get grade 9.
Ofqual is consulting about a change to this. Ofqual’s modelling suggests that for Mathematics
GCSE the percentage getting grade 9 would reduce to about 17% of those getting grade
7 and above.
MEI support for schools/colleges
 MEI has recorded a describes what is currently known about the changes for 2017. specifications.
presentation content between current and 2017
 MEI has launched a programme of CPD courses . Each course is
that
 MEI has produced a comparison of appropriate for teaching any of the new mathematics AS/A level specifications.
 The MEI Conference in Bath (30
June-2 July) will have a particular focus on the new A levels. In addition to sessions on various aspects of the changes, there will be plenty of opportunities to talk through any concerns with fellow teachers, FMSP and MEI staff.
 During 2016 and 2017, MEI's
Integral online resources for A level
Mathematics and Further Mathematics are being revised and enhanced to support all of the new AS and A level specifications.
FMSP support for schools/colleges
 The FMSP provides support and guidance for preparing for and teaching the new linear specifications: an overview , information about specifications , Further Mathematics , an implementation timeline , and FAQs .
 The FMSP is organising Spring/
Summer 2016 teacher network regional workshops introducing the major features of the changes.
 From September 2016 the FMSP will be running a series of free CPD sessions focusing on preparing your department for linear A levels in
Mathematics and Further Mathematics.
 The FMSP offers a range of extended professional development courses designed to develop subject knowledge and classroom practice.
 Details of FMSP network and CPD events are on their 2017 A level
Mathematics: Professional
Development and Live Online
Professional Development pages.
(Cont. on page 3)

Curriculum Update
Ofqual consultation on award of grades
8 and 9 at GCSE
(Cont. from page 2)
You can respond online or complete a response form and either email or post it to Ofqual - details are on the Ofqual consultation page .
The consultation closes 17 June 2016.
National Reference
Test
There is an update on
Ofqual’s work on a
National Reference
Test on the Ofqual
Blog ; the update includes some examples of the types of mathematics questions which will be included.
 Statistics:
This is appropriate for any teacher involved with teaching the new AS/A levels, including those who already teach statistics at AS/A level. It will focus on the changes between the current and new A level Mathematics specification, including large data sets.
The ‘Get Set’ programme of four one-day courses run by
MEI is designed to help teachers of maths AS/A levels prepare for the new specifications. It provides an introduction to the changes and will focus on AS/A level Mathematics.
Although some content will be relevant to Further Mathematics, more in-depth professional development on the 2017
AS/A level Further Mathematics will be available through the Further
Mathematics Support Programme .
Each of the courses will be available at regional locations throughout academic year 2016-17. There is no dependency between the courses; however it is recommended that at least one teacher from a school or college attends each of the four courses.
 Mechanics:
This is particularly relevant for those who have not previously taught mechanics. It will cover the main mechanics topics in the new A level
Mathematics.
 Principles of the new mathematics
A levels:
This is appropriate for any teacher who will be teaching the new maths AS/A levels. It will focus on the impact of the changes on teaching, including modelling, problem solving, proof, and a linear scheme of work.
 Effective use of technology in the new maths A levels: This is relevant to all teachers. It will focus on promoting mathematical understanding through the use of technology across A level
Mathematics.

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.
Postcard Problems
MEI has produced a set of postcard size problems, based on the principles of the new 2017 mathematics A levels.
“Modelling in mathematics is about starting with a real life situation, making assumptions to simplify and decide what features are importance and which are not. This allows mathematics to be used to provide information about the real situation. It is then necessary to check whether the answer to the simplified situation is good enough – perhaps simplifying has ignored a feature which turns out to be important.
To receive a copy, please contact MEI
Marketing or download them from this page by clicking on the relevant image.
A link to the solution is provided after each postcard image:
Problem Solving
Solution
Large Data Sets
Modelling
Solution
Reasoning
Solution
Technology
“Modelling is already present in current A levels – particularly in mechanics and statistics. The new A levels will place more emphasis on thinking about modelling assumptions and will include more modelling using pure mathematics.” (From the FMSP website )
Solution
Solution

Alec McEachran’s
Furbles – a large data set!
Colleen Young has come up with an interesting and fun suggestion for using a large data set -
Furbles !
Furbles started life as an idea for teaching statistics in an interesting way with children from
KS1 to KS3. The original version was published in 2003 online, and its popularity spread.
Colleen suggests:
“Talk to students about summarising this data, perhaps ask for their impressions as to which colour is the most or least common. Data can be presented as a bar chart or a pie chart and you can choose to categorise in various ways. It is also possible to vary the number of
Furbles, the maximum and minimum number of eyes and sides.”
Learning with Real Data
“Starting to teach with real data, rather than starting with teaching techniques and considering applications later, helps students see the relevance of statistics and helps them understand how to interpret data,” write MEI’s Stella
Dudzic, Stephen Lee and Charlie Stripp in this article from the Mathematical
Association’s publication Mathematics in School , November 2015. The authors provide some ideas to help you to start thinking about the statistics in the new
A level Mathematics and how you might teach it. We will provide a classroom resource in a future edition of M 4 magazine.
Students should explore a large data set using technology such as a spreadsheet. The Wired.com article
What Does Big Data Look Like?
Visualization Is Key for Humans is useful reading. You may find it helpful to look at the Statistical Problem
Solving paper of MEI’s Quantitative
Problem Solving Core Maths qualification ; this makes use of a large data set. There is some guidance about the large data set in Ofqual’s A level Mathematics Working Group report .
Where to find data sets

 by Department for Transport

Economics Network

Statistical data sets
Road Safety Data
Economic Data
MEI Data Sets listed on Gov.uk
published listed by the includes information about the data and an indication of statistical techniques that may be useful
A level Statistics CPD
Teaching Statistics
This FMSP course is for teachers wishing to develop their own subject knowledge and classroom practice, including the use of large data sets, hypothesis testing and use of technology. There are two modules:
 TS1 will cover the compulsory statistics content in the new A level
Mathematics.
 TS2 will cover the statistics content that is likely to be included in the new
Further Mathematics.
MEI Conference sessions
 Get set for September 2017:
Statistics : In this session we will look at what the Statistics content will look like, the change of emphasis from the current S1 modules, and the connections which can be made while teaching a linear course.
 An introduction to probability distributions : This session will provide an introduction to the binomial and normal distribution s using a context.
 Hypothesis Testing in the new
Mathematics A level : We will consider practical approaches, real life examples and how technology can enhance the understanding of this concept.
 Resources and Investigations:
Sampling Methods and Variation :
Simulations offer the opportunity to compare the variation between samples selected using the same and different sampling methods.

Mechanics
The new A levels have a greater emphasis on modelling. This is likely to be particularly prevalent in the
Mechanics content.
The table on pages 4-
5 of this MEI document provides a topic-based comparison between current and 2017 AS/
A level Mathematics specifications.
Integral resources will be fully updated for the new A level specifications for first teaching in
September 2017.
Here are some sample Mechanics resources from the current site:
 Speed-time graphs activity
 Projectiles
You’ll find a Forces classroom resource created by Carol
Knights at the end of this magazine; you can download the
PowerPoint from the
M 4 web page.
Teaching Mechanics
This FMSP course has been developed with the new A level specifications in mind. It is designed for teachers wishing to develop their own subject knowledge and classroom practice in teaching
Mechanics at A level Mathematics and
Further Mathematics. from 2017. They will cover some of the basic subject content, links to GCSE and other A level topics, as well as exploring ideas and approaches for teaching:
 Preparing to teach projectiles : The session will demonstrate how simple practical classroom activities can provide a stimulus for students to develop their understanding of projectiles.
The Teaching Mechanics course starts
September 2016, and has two modules:
 TM1 covers the compulsory mechanics content for the new linear A level and the current content of all the
M1 and some of the M2 material of the four English examination specifications.
 TM2 covers topics in Further
Mathematics.
 Preparing to teach forces : The session will demonstrate how simple practical classroom activities can provide a stimulus for students to develop their understanding of Forces and how this permeates the other aspects of mechanics.
 Preparing to teach moments : The session will demonstrate how simple practical classroom activities can provide a stimulus for students to develop their understanding of moments.
MEI Conference sessions
 Get set for September 2017:
Mechanics and modelling : Exploring 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.
 Preparing to teach…
Four sessions designed to help teachers prepare for teaching mechanics topics in the new A level
 Preparing to teach motion graphs :
The session will demonstrate how simple practical classroom activities can provide a stimulus for students to develop their understanding of motion graphs.

Joined-up thinking builds bridges
Dr Hugh Hunt is a
Senior Lecturer in the
Department of
Engineering at
Cambridge University and a Fellow of Trinity
College.
Hugh was the 2012 winner of Royal
Television Society
Award for the best history programme:
"Dambusters:
Building The
Bouncing Bomb"
In 2015 Hugh was the winner of the
Rooke Award for the public promotion of engineering.
You can view Hugh’s appearances in TV, radio and popular press from the his website .
RAEng
Twitter:
@hughhunt links on
View Hugh’s videos on his YouTube channel: spinfun
Follow Hugh on
I can’t imagine how a pure mathematician would design a bridge.
Nor a physicist. Would you ask a statistician? Or a chemist? Perhaps you’d go to an artist, or maybe it’s just all applied maths.
Would a partly-complete span be safe in a violent storm? Probably not so the planning team analyse the local weather to see if there would be enough spells of calm weather for construction to proceed safely.
Viaduc de Millau (click image to enlarge)
By Stefan Krause, Germany (Self-photographed)
[ CC BY-SA 3.0
] via Wikimedia Commons
My favourite bridge is the Viaduc de
Millau in France. It is beautiful and perhaps an artist had something to do with it. But let’s think something about the process of designing the bridge.
First there is a need. Careful analysis of road -usage statistics, cost benefit, health issues, safety – all these factors and more go into making a case for a new bridge. But it’s not at all simple.
Mathematicians have known for a long time that opening a new road can make things worse. It’s called Braess’ paradox . A new bridge might be a white elephant.
How high can the bridge be? In the early stages of design it was thought impossible to span the deep wide valley of the river Tarn. The bridge pillars would be taller than the Eiffel Tower and would they be able to withstand the high winds in the valley? Perhaps the bridge would be strong when complete but what about during construction?
What materials would the bridge be made of? Reinforced concrete pylons?
Steel deck? High-tensile steel cables?
The choice of these materials is not just by fancy. For example an alternative to steel cables is Kevlar – a light-weight super-strong material that resists corrosion – but for such a large project the cost would be prohibitive. Instead the steel cables are protected by being galvanised (zinc coated) then polythene coated and then the entire cable is encased in a weather-strip. The chemistry of corrosion is pretty extraordinary.
Here are two identical nails in salty water. One wrapped with zinc-coated wire. The difference is noticeable after a couple of hours.

Joined-up thinking builds bridges
Braess Paradox
Maths author Presh
Talwalkar explains that:
“The Braess Paradox is an unexpected result from network theory. It states that adding capacity could actually slow down the speed of the network. Applied to highways, the Braess
Paradox means the existence of some roads slows down traffic, or that closing some roads could speed up traffic…
“The Braess Paradox is a result that individual drivers are seeking the fastest roads, and when all drivers make the same choices the roads are congested
(this is an example of a Nash equilibrium from game theory).
The inefficiency is a result of a lack of coordination, so this is known as the "price of anarchy."
So much for the chemistry of corrosion, the statistics of weather, the maths of traffic networks. Let’s look at the physics. How many spans should we have? Lots of short spans would be easier on the spans but then there’d be lots of pylons. How about one single span? The tension in the cables would be too great. There is an optimum and the designers came up with eight spans. It’s all about balancing the forces of gravity with the cost of construction.
There’s lots of trigonometry here, resolving forces in various directions.
And the loads are distributed so forces are integrated along the length of each span. The maximum stress in the material is found by differentiation. The allowable stress in the steel is found by experiment, but because of windinduced vibration correction is required for allow for the possibility of fatigue cracking. The science of fatigue is all about sinusoidal motion and for this we need differential equations. We then understand about natural frequency, resonance and damping.
One of the beautiful features of the bridge is that the cables run down the centre of each span.
But what does this mean for the balance of forces? Suppose there is a traffic jam on the north-bound side loading the structure asymmetrically.
Does the bridge twist and collapse? Of course not, but the engineers had to calculate the forces that would arise in this situation. It’s all geometry and physics.
What I’m trying to say is that you don’t design a bridge unless you can make connections between all sorts of disciplines. Real life requires a combination of skills and we do our best to teach this joined-up approach.
Modular courses lend themselves to a compartmentalised thinking. I am a great proponent of linear courses. I wonder why they chose white cables.
Perhaps they consulted an artist?
Viaduc de Millau cable alignment
(click image to enlarge)
View Presh
Talwalkar ’s
video How Closing
Roads Could Speed
Up Traffic - The
Braess Paradox .
By Delphine de Andria,
France. [ Creative
Commons and Art
Libre via Freemages]

Jo is a Central
Coordinator for the
Further Mathematics
Support Programme, which is managed by
MEI.
The new requirements from the DfE require that students are able to interpret real data presented in summary or graphical form, use data to investigate questions arising in real contexts, understand informal interpretation of correlation and understand that correlation does not imply causation.
Jo has responsibility for the Live Online
Professional
Development programme.
Jo gained a BSc in
Mathematics in 1992 followed by a PGCE in 1993. She then taught Mathematics and Further
Mathematics in a school on the south coast for 21 years, including three years as a SENCo. During the latter part of this role she became a
Further Maths
Network Centre
Manager and then an
FMSP Area
Coordinator before joining MEI in 2014 as a member of the
FMSP Central Team.
The Spurious Correlations website: contains a wealth of ridiculous graphs which appears to show connections between, for example, the per capita consumption of mozzarella and the number of civil engineering doctorates awarded.
I like this site for generating discussion, and would ask these questions:
 These graphs are not what we define as scatter graphs in maths – how are they different?
 How do you think these correlations have been found?
 Why does this fail as a scientific method?
This connects to the reason why we must set up null and alternate hypotheses before carrying out a test for correlation.
Connected with this, in his Battling Bad
Science TED Talk
Ben Goldacre l ooks at science research in the media, misuse of data and poor sampling methods amongst others.
In the mechanics section of the new A level mathematics, students will look in detail at the modelling of kinematics both with constant and variable acceleration.
My favourite resource when looking at constant acceleration, which I always use when the question of ignoring air resistance is raised is this
Hammer vs Feather - Physics on the
Moon video .
Astronaut David Scott confirming
Galileo’s reasoning in practice.
“How ‘bout that!”

Have you ever wondered:
What’s it like to walk on the moon?
or
Why does my stomach lurch in a lift?
The answers are both linked to gravity and the following activities will help to answer these questions and others.

Two words that we need to distinguish between are ‘mass’ and ‘weight’.
In everyday language people talk about:
• weighing ingredients
• their own weight
• the weight of a baby
In each of these cases they are actually talking about ‘mass’, which is measured in kilograms.
Weight?... we’ll get to that...

Take two objects of differing masses, a pen and a book would work well.
One person hold them at exactly the same height.
Everyone watch and listen carefully...
Release the two objects at exactly the same time.
What happened?

How many noises did you hear?
What does this tell you?
There is an equation linking initial velocity ( u ), time
( t ), displacement ( s ), and acceleration ( a ).
s = ut + ½ at 2
What can you say about the acceleration for each object?
This acceleration is called the acceleration due to gravity.

Galileo Galilei (1564-1642) is said to have done essentially this same experiment, dropping balls of the same material but different masses from the Tower of Pisa.
He concluded that they reached the ground at the same time. Until then, it was believed that heavier objects accelerated faster.
Historical note: there is some doubt about whether or not it was Pisa, and even whether or not it was Galileo, but it was around this time that Aristotle’s belief was being questioned.

Isaac Newton (1642-1726) devised a theory that between any two bodies is an attractional force which is related to each of their masses and to the distance between them.
The bigger their masses, the stronger the attraction.
The closer they are, the stronger the attraction.

Newton observed the planets, whose motion seemed to suggest the following formula, known as Newton’s
Law of Universal Gravitation: 𝑚
1 𝑚
2 F = G 𝑟 2
F is the force between the two objects ( N ) m
1 and m
2 are the masses ( kg ) of the two objects r is the distance ( m ) between the two objects, measured from their centres
G is a constant 6.67 x 10 -11 ( Nm 2 kg -2 )

Imagine a 1kg mass placed on the Earth’s surface.
Using Newton’s Law of Universal Gravitation, work out the force F between the Earth and the mass.
F = G 𝑚 𝑟
1 𝑚
2
2
G is 6.67 x 10 -11 Nm 2 kg -2
Mass of the Earth is 5.97 x 10 24 kg
Radius of the Earth is 6.37 x 10 6 m

You should have found that the force is 9.81N.
If we use Newton’s second law f = ma, a formula linking force ( f ), mass ( m ) and acceleration (a ), we can find out the acceleration due to gravity at the Earth’s surface.
In this case, f = 9.81N and m =1kg so a = 9.81ms
-2 which we often approximate to 9.8ms
-2

Weight is a force and is measured in Newtons ( N ).
To calculate the weight of an object we use Newton’s second law f = ma.
Acceleration due to gravity is 9.8ms
-2
So, to find the weight of a person whose mass is
60kg we use:
Weight = 60 x 9.8
Weight =588N

Find the weights of the following masses:
• A 1kg bag of flour
• A 7kg baby
• A 15kg suitcase

What do you think?
Is the Earth a perfect sphere?

The Earth isn’t a perfect sphere; it’s slightly flatter at the poles, where the radius is 6.36 x 10 6 m
(and 6.38 x 10 6 m at the equator)
F = G 𝑚
1 𝑟 𝑚
2
2
Mass of the Earth is 5.97 x 10 24 kg
G is 6.67 x 10 -11 Nm 2 kg -2
F=ma
What weight would a 1kg mass have at the poles?
What is the acceleration due to gravity here?

At 8 848m, Mount Everest is the highest mountain on Earth.
F = G 𝑚
1 𝑟 𝑚
2
2
Mass of the Earth is 5.97 x 10 24 kg
Radius of the Earth is 6.37 x 10 6 m
G is 6.67 x 10 -11 Nm 2 kg -2 f=ma
What weight would a 1kg mass have at the top of Everest?
What is the acceleration due to gravity here?

As with many ‘real life’ contexts, things aren’t ‘ideal’ so we use a good approximation that will serve our purposes reasonably well.
Acceleration due to gravity isn’t the same everywhere on Earth because it’s not a perfect sphere.
In calculations we use an average value of 9.8ms
-2 or 9.81ms
-2.
A level explanation, click here

Use the data on the next slide to find out how much you would weigh on the moon, or on some of the other bodies in our Solar system.
You will still need to use:
F = G 𝑚
1 𝑟 𝑚
2
2
G is 6.67 x 10 -11 Nm 2 kg -2 f=ma

Planet
Radius at equator
(m)
Mass (kg)
Mercury
2.44 × 10 6 3.30 × 10 23
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Sun
Moon
6.05 × 10 6
6.38 × 10 6
3.39 × 10 6
7.14 × 10 7
6.00 × 10 7
2.61 × 10 7
2.43 × 10 7
6.69 × 10 8
1.74 × 10 6
4.87 × 10 24
5.97 × 10 24
6.42 × 10 23
1.90 × 10 27
5.69 × 10 26
8.70 × 10 25
1.03 × 10 26
1.99 × 10 30
7.35 × 10 22

So what do you think it would feel like to walk on the moon?
On which planet would it feel most similar to walking on Earth?

Think about being in a lift.
If the lift is accelerating upwards, do you feel heavier or lighter?
What about when it’s accelerating downwards?

Push your hand against a desk top.
Push it harder.
What do you feel?

Assuming you didn’t break the desk(!), you should have felt that the harder you pushed, the more the desk seemed to ‘push back’.
You’d have felt a ‘push back’ force if you had pushed against a wall instead.
This is called the normal reaction force.
(‘normal’ as in ‘at 90° to the surface’, not as in ‘usual’).
This is what we feel.

When stood still on the ground, we feel a normal reaction (NR) which, in the case when we are stood still, is equivalent in to our weight (W).
NR
W
The forces acting are equal and opposite so we can say the system is ‘in equilibrium’.

In fact, it’s a bit more complicated than it first seems.
Newton’s third law says that for every force there is an equal and opposite force, sometimes known as an action-reaction pair.
Beware! Just because the two forces acting are equal and opposite, doesn’t mean that W and R are an example of the action-reaction pair mentioned in
Newton’s third law!
Find out more

When a person is stood in a lift which is accelerating downwards, the forces acting on the person are not equal and opposite.
Newton’s second law, f=ma, says that to have an acceleration, there must be a resultant force, f.
R a
W

Supposing a 45kg child is in a lift which is accelerating at 2ms -2 downwards.
Using Newton’s second law, f=ma, we have:
W-R = ma
441 R = 45 x 2
R = 441-90 = 351 N
So the child feels lighter.
R
W
Weight doesn’t change
45 x 9.8 = 441N a

Can you work out what the 45kg child feels in a lift which is accelerating at 2ms -2 upwards?
R a
W

If we consider an object of mass 𝑚 𝑘𝑔 a small distance, ℎ
, from the Earth’s surface using Newton’s Law of Universal
Gravitation, with
𝑀 being the mass of the Earth and
𝑅 being the radius of the Earth:
𝐹 = 𝐺
𝑀𝑚
𝑅 + ℎ 2
= 𝐺
𝑅 2
𝑀𝑚
1 + ℎ
𝑅
2 ℎ
= 𝑔 1 +
𝑅
−2
Using the binomial approximation we obtain:
−2 ℎ 2ℎ ℎ
2 ℎ
3
1 + ≈ 1 − + 3 − 4
𝑅 𝑅 𝑅 𝑅
+ ⋯

Since ℎ is very small and especially considering the ratio ℎ
𝑅 we have:
−2 ℎ 2ℎ
1 + ≈ 1 −
𝑅 𝑅
Substituting our expression back into our first equation we can express the force of gravitational attraction as:
𝐹 ≈ 𝑔 1 −
2ℎ
𝑅 and using values discussed earlier we can see how little the change is, even on top of Everest!
Back to the activity

Newton’s Law of Universal
Gravitation tells us that two masses (e.g. a person and the
Earth) attract each other with the same force.
W
This is an action-reaction pair.
W’
And in Australia........

Newton’s Law of Universal
Gravitation tells us that two masses (e.g. a person and the
Earth) attract each other with the same force.
This is an action-reaction pair.
W
Read on...
W’

If the person is in contact with the Earth’s surface then, in addition to their weight W, they will experience a Normal
Reaction force, NR.
The person exerts an equal and opposite Normal Reaction force on the Earth . The forces on the
Earth are W’ and NR’
NR and NR’ are another action-reaction pair.
W’
W
NR
NR’

The W and NR are NOT an action-reaction pair.
The fact that they are equal and opposite, such as the case where the person is stood still, often leads people to think they are.
W
W’
NR
NR’

This issue looks at forces, particularly gravity.
This content will be compulsory in the new Mathematics A level, and is currently studied within A level Mechanics modules.
In GCSE Mathematics, students don’t need to know about gravity and other forces, however, in GCSE Science, they do. In GCSE Science, students also learn about Newton’s laws and the equations of constant acceleration, so using them in maths may help students connect their learning in different subjects.
The activities simply require the use of familiar and unfamiliar equations, so should be accessible to both GCSE and A level students, and provide opportunities for practising efficient calculator use as well as consolidating understanding of algebraic rules.

» Students should have the opportunity to discuss this with a partner or in a small group
» Students should sketch or calculate (as appropriate)

Slide 4
If the items are dropped at the same time, from the same height, they should hit the ground at the same time.
• There may be some air resistance which will slow down the object with the larger surface area, but over this distance, the effect should be negligible.
Slide 5
If the distance is the same, and the time is the same (one sound), and they both start from rest, then the acceleration must be the same for both.

Slide 7 Note:
Newton’s law of gravitation has been superseded by Einstein’s theory of general relativity, but we still use Newton’s law as it gives a very good approximation to the force of attraction in most cases and is much simpler to use. Where more precision is required, or where the masses are particularly large or the distances (relatively) close, Einstein’s theory should be used.
This would make a good discussion point: using something that we know is wrong, just because it’s easier and is good enough, for now.
Similarly, in mathematics we sometimes overlook things, for now. For example, KS3 or KS4 students might be told “you can’t find the square root of a negative number”.

Slide 8
It’s worth spending some time understanding the formula and knowing
‘which bits you do first’.
Note: the value of the Gravitational constant G varies according to the source.
Slide 9
The calculation should result in 9.81N
Slide 12
Flour: 9.8N Baby: 68.6N Suitcase: 147N

Slide 13
No, it’s not a perfect sphere. There are mountains and valleys as well as deep trenches in the ocean. The main shape is not spherical, it’s slightly flattened at the poles and slightly bulgy around the equator.
Slide 14
At the poles acceleration is 9.84ms
-2
At the equator acceleration is 9.78ms
-2
Slide 15
Students will need to add Everest’s height to the radius of the Earth to obtain r.
At the top of Everest, acceleration is 9.79ms
-2

Slide 18
It all depends on the mass of the person. Values given below are for a
1kg mass, so simply multiply by the number of kg required.
Planet
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Sun
Moon
Radius at equator (m)
2.44 × 10
6
6.05 × 10
6
6.38 × 10
6
3.39 × 10
6
7.14 × 10
7
6.00 × 10
7
2.61 × 10 7
2.43 × 10 7
6.69 × 10 8
1.74 × 10 6
Mass (kg)
3.30 × 10
23
4.87 × 10
24
5.97 × 10
24
6.42 × 10
23
1.90 × 10
27
5.69 × 10
26
8.70 × 10 25
1.03 × 10 26
1.99 × 10 30
7.35 × 10 22
Weight of 1kg (N)
3.70
8.87
9.78
2.69
24.86
10.54
8.52
11.63
296.57
1.62

Slide 19
What’s it like to walk on the moon? Your guess is as good as anyone’s. On the moon, gravity is about 1/6 of Earth’s gravity. This means that objects will accelerate to the surface more slowly.
When walking on Earth, gravity helps pulls our feet back down to the surface, this force would be greatly reduced on the moon, perhaps resulting in a slower pace and a lighter feeling.
The planet with gravity most similar to ours is Saturn, however, since this is a Gas Giant, probably consisting of liquids and gases, there is likely to be no solid surface to walk on!
In terms of gravity, Venus is next closest to our own – and has a solid surface. However, with a mean temperature of 462°C, humans are unlikely to walk there in the near future.

Slide 20
It’s important to emphasise that this is about when a lift is accelerating
(positive or negative acceleration). When the lift is travelling at a constant velocity, the weight and reaction forces acting on the person are balanced, as they are when it is at rest.
Slide 24
The ‘find out more’ content is recommended for A level students, but is equally accessible to others.
Slide 25
The idea of a resultant force is important. In this case the two forces are in opposite directions, and there is an excess in one direction.
Refer to directed numbers:
2+2=0 this is balanced, the numbers are ‘equal and opposite’
2+3= + 1 this has a result in a specific direction and is different to
+ 2+ 3= 1

Slide 27
Accelerating upwards: using Newton’s second law, f=ma, we have:
R-W = ma
R - 441 = 45 x 2
R = 441+90 = 531 N
So the child feels heavier.
In the lift activities, action-reaction pairs are mentioned. These are a very common source of confusion for students and will be explored in more details in an M 4 classroom resource next academic year.

https://en.wikipedia.org/wiki/Earth_radius#Polar_radius https://en.wikipedia.org/wiki/Isaac_Newton https://en.wikipedia.org/wiki/Galileo_Galilei
Thank you to Simon Clay and Sharon Tripconey for providing the ideas for these activities.
If you would like to explore these themes further you may be interested in the Get Set for Mechanics course which will be run by MEI in preparation for the new A level. See the webpage for details.