## Secondary Maths Lessons

Developed in Partnership with Dr Frost Learning, these resources are suitable to 11-16 maths teaching (KS3 and KS4 in England), unless otherwise indicated.

Each lesson features a lesson PowerPoint as well as printable exercise and investigation sheets.

## Change the Subject of a Linear Formula Involving Brackets and Fractions

Climate change context

2023 being confirmed as the hottest year on record

Prior Learning:

• Solve simple linear equations.
• Form simple expressions & formulae
• Use and interpret algebraic notation, including:
• 𝑎𝑏 in place of 𝑎×𝑏,
• 3𝑦 in place of 𝑦+𝑦+𝑦 and 3×𝑦,
• 𝑎/𝑏 in place of 𝑎÷𝑏,
• coefficients written as fractions rather than as decimals.
• Brackets
• Distinguish between expressions, equations, inequalities, terms and factors
• Order of operations
• Change the subject of a linear formula requiring a single step.
• Change the subject of a linear formula requiring two steps (including simple divisions).
• Change the subject of a formula where the subject is multiplied or divided by more than one constant or variable.
• Change the subject of a formula where the subject appears on the denominator of a fraction.
• Change the subject of a linear formula where the coefficient of the subject is negative.
• Expanding single brackets.
• Change the subject of a linear formula involving multiplication using brackets.

Lesson ppt

Exercise pdf

## Substitution with the Four Operations and Integers

Substitution is the process of replacing the variables in an algebraic expression, usually with a numerical value. We can then work out the total value of the expression.

Climate change context

Calculating household carbon dioxide emissions

Prior Learning:

• Negative numbers and arithmetic
• Decimals and arithmetic
• Fractions and arithmetic
• Powers and roots
• Basic algebraic notation
• Using function machines & their inverses

Lesson ppt

Exercise pdf

## Integer Substitution with Powers and Roots

Climate change contexts:

Substitution and the Sahara

Rainforest deforestation

Prior Learning

• Substitution with four operations and integers
• Using notation for powers and roots
• Knowing powers and roots with base 2, 3, 4, 5 and 10

Lesson ppt

Exercise pdf

## Substitution with Fractions and Decimals

Prior Learning

• Decimals and arithmetic
• Fractions and arithmetic
• Powers and roots
• Basic algebraic notation
• Substitution using integers with the four operations
• Substitution using integers with powers and roots

Lesson ppt

Exercise pdf

Investigation Sheet 1 – Wind Turbine

Investigation Sheet 2 – Wind Turbine

## Form Simple Expressions

Climate Change Contexts

Arctic warming

Building insulation

Carbon footprint of social media

Emissions reductions

Prior Learning

• Use and interpret algebraic notation, including:

–ab in place of a×b,

–3y in place of y+y+y and 3×y,

–a/b in place of a÷b,

–coefficients written as fractions rather than as decimals.

–brackets

• Simplify expressions with sums, products and powers including index laws
• Distinguish between expressions, equations, inequalities, terms and factors
• Algebraic substitution
• Recognise & create equivalent expressions
• Order of operations

Lesson ppt

Exercise pdf

## Form and Use Simple Formulae

Climate Change Contexts

Tree planting

Vehicle emission reductions

Solar panel output

Prior Knowledge

• Use and interpret algebraic notation, including:

–ab in place of a×b,

–3y in place of y+y+y and 3×y,

–a/b in place of a÷b,

–coefficients written as fractions rather than as decimals.

–brackets

• Simplify expressions with sums, products and powers including index laws
• Distinguish between expressions, equations, inequalities, terms and factors
• Algebraic substitution
• Recognise & create equivalent expressions
• Order of operations
• Form simple expressions

Lesson ppt

Exercise pdf

## Form and Solve Linear Equations from Simple Contexts

Climate Change Context

Emission reductions and net zero

Prior Knowledge

• Solve simple linear equations.
• Solve linear equations with brackets.
• Solve linear equations where the variable appears on both sides of the equation.
• Solve linear equations involving brackets.
• Expanding single brackets.
• Form simple expressions & formulae.
• Use and interpret algebraic notation, including:

–ab in place of a×b,

–3y in place of y+y+y and 3×y,

–a/b in place of a÷b,

–coefficients written as fractions rather than as decimals.

–Brackets.

Lesson ppt

Exercise pdf

## Form and Solve Linear Equations for Problems Involving Perimeter and Area

Climate Change Context

Carbon footprint/ growing food

Tree planting

Prior Knowledge

• Find the area and perimeter of simple shapes.
• Solve simple linear equations.
• Solve linear equations where the variable appears on both sides of the equation.
• Expanding single brackets.
• Form simple expressions & formulae
• Use and interpret algebraic notation, including:

–ab in place of a×b,

–3y in place of y+y+y and 3×y,

–a/b in place of a÷b,

–coefficients written as fractions rather than as decimals.

–Brackets

• Simplify expressions with sums, products and powers including index laws
• Distinguish between expressions, equations, inequalities, terms and factors
• Algebraic substitution
• Recognise & create equivalent expressions
• Order of operations

Lesson

Exercise 1

Exercise 2

Mixed Exercise

## Changing the Subject – One Step

Climate Change Context

Ocean Warming

Prior Knowledge

• Solve simple linear equations.
• Expanding single brackets.
• Form simple expressions & formulae
• Use and interpret algebraic notation, including:

–ab in place of a×b,

–3y in place of y+y+y and 3×y,

–a/b in place of a÷b,

–coefficients written as fractions rather than as decimals.

–Brackets

• Simplify expressions with sums, products and powers including index laws
• Distinguish between expressions, equations, inequalities, terms and factors
• Algebraic substitution
• Order of operations

Lesson

Exercise 1

Dr Frost Learning is a UK registered charity with goal of delivering high quality education for all individuals and institutions regardless of income, centred around the philosophy that education is a fundamental right of all and central to addressing social inequality on a global level. The charity was founded by Dr Jamie Frost and he received the Covid Hero Award in the Global Teacher Prize 2020.

## Introduction/Motivation

The 2022 United Nations climate change conference (27th session of the Conference of Parties – COP27) will be held in Sharm El-Sheikh in Egypt, starting on the 7th of November.

In the introduction video screened at the end of COP26 in Glasgow, Egypt celebrated its adaptations and mitigations to climate change. In this resource, students will explore population growth, urbanisation and the greenhouse gas emissions from the construction industry in Egypt.

Resources:

PowerPoint

Learning Outcomes

• To understand what COP27 is
• To describe how the population of Egypt has grown and is projected to grow in the future
• To be able to interpret a population pyramid for Egypt and use that to explain Egypt’s changing population
• To explain how the construction industry has an impact on the climate and what steps can be taken to reduce that impact.

## Optimising Flight Times

Calculate the best flight time from A to B and reduce greenhouse gas emissions!

The table below represents a cross section through the atmosphere and gives wind speeds (in m/s) in boxes which are 200km long and 1km high.

Your task is to pilot an aircraft, which flies at 230m/s when it is flying in the less dense atmosphere higher than 5km, and 150m/s when it is flying in the more dense atmosphere lower than 5km, from A to B in the shortest time possible.

Remember, flying in the same direction as the wind increases your speed but flying against the wind slows you down.

Map your route on the chart below and then calculate the flight time!

Rules

1. You take off from the ground at A and land on the ground at B.
2. You can only climb, or descend, one box per 200km.

Some students may find the following table useful:

## Introduction/Motivation

The 2022 United Nations climate change conference (27th session of the Conference of Parties – COP27) will be held in Sharm El-Sheikh in Egypt, starting on the 7th of November.

In the introduction video screened at the end of COP26 in Glasgow, Egypt celebrates its road-building project. This resource explores efficient road designs and the climate impacts of car travel.

Resources:

PowerPoint

4 City square template

Created with support from MEI

## Section 1: Lesson Introduction

Show the Egypt introduction video from COP26 and show them the pictures of new Egyptian roads.

Or the full Egypt introductory clip: from 09:21-12:26 in https://unfccc-cop26.streamworld.de/webcast/closing-plenary-of-the-cop-followed-by-cmp-and-c-2

Ask students what comments or questions they have on the video: What do they wonder?

They could discuss in pairs or groups before giving feedback to the class.
You could steer the discussion towards some of the following points:

• What do you thing about building new roads compared to the other climate mitigation and adaptation projects mentioned in the video?
• Roads for sustainable development: connecting cities and industry
• Will a new road reduce traffic?
• Building roads versus building railways/airports
• How will building new roads impact greenhouse gas emissions?
• Should houses be demolished to make way for new roads? https://www.reuters.com/world/middle-east/egypts-road-building-drive-eases-jams-leaves-some-unhappy-2021-05-14/
• How should governments decide which new roads to build? How can we reduce travel time for the most people, reduce the length of the new road or reduce the greenhouse gas emissions from people travelling on the road?

The remainder of the lesson uses maths to explore the last point.

## Section 2: Scenario motivation for the Steiner problem

This could be introduced as motivation for the Steiner problem, or as a real world application once the problem has been solved (after section 3).

Together with the important cities of Cairo and Suez, this can be simplified and framed as an example of the famous ‘Steiner Problem’:

Which looks a bit like:

## Section 3: Steiner Problem using a Square

To simplify the problem, start by investigating 4 cities in a square. What is the most efficient way to connect all the cities (using the shortest distance of road)? You need to be able to visit all the cities on the road network, but you can go via other cities.

This problem is also described here: https://nrich.maths.org/14937

Students can use the sheet of squares (or squared paper) to draw as many different designs as they can think of, using curves and straight lines, or just straight lines.

Collect some ideas and ask students to calculate the total road distance required. For the square side length, you could use realistic road-distance numbers (eg 100km), simple numbers (eg 10, 1) or a symbol such as x. Students could first measure the distances using a ruler, then calculate them using Pythagoras’ theorem.

Help students to arrive at the optimal solution by considering the two designs below: Is there an intermediate design that would be even better?

Once students have arrived at the optimal solution, this video gives a good demonstration using soap bubbles: https://www.youtube.com/watch?v=dAyDi1aa40E

## Section 4: Context Calculations

At this point you could choose to start using realistic road distances or the fact that the cities are not in a square but are closer to a rectangle (see the first extension point below).

• What is the total distance of the optimal network? How long would it take to travel between each city whilst travelling at the Egyptian motorway speed limit of 100km/hr? (https://www.autoeurope.ie/driving-information-egypt)
• Assume that the roads connecting Beni Suef to Cario, Cairo to Suez and Suez to Zaafarana already exist. Which one new road should be built to reduce the travel time from Beni Suef to Zaafarana? What is the reduction in travel time?
• Cars emit around 120 gCO2/km (https://www.eea.europa.eu/data-and-maps/indicators/average-co2-emissions-from-motor-vehicles/assessment-1)
By how much does the new road reduce the CO2 emissions of a journey from Beni Suef to Zaafarana?
• Due to the shorter travel time, the new road might increases the number of journeys between Beni Suef and Zaafarana. How many extra journeys are needed to outweigh the decrease in emissions from the reduced distance?

## Extension Ideas

• Return to the introduction video at the end of the lesson. What do students think about building roads and climate change after completing the activity?
• The 4 cities in Egypt are not in a perfect square, but are close to being in a rectangle. Does this change the optimal road network? This is discussed at https://thatsmaths.com/2015/01/29/the-steiner-minimal-tree/
• Students could use google maps to look at the real travel time between the 4 cities using different routes.
• What happens when we consider more cities?
• Can students think of other situations where this problem could apply? (gas pipelines, rail networks, broadband cables)
• Think about the real-world practicalities that the Steiner solution doesn’t address. Is it the best solution if most journeys are between Cairo and Suez? Should existing roads be removed in order to build the most efficient network? Which journey times would be increased by this?

## How do CO₂ emissions link to global temperatures?

This resource links to B.4.1, FAQ5.4 and to Figure SPM.10 in the IPCC report of 2021. The aim of this resource is to answer the question how do CO emissions link to global temperatures?

It was written with the Royal Geographical Society with IBG

### The carbon budget

A carbon budget is the cumulative amount of carbon dioxide (CO₂) emissions permitted over a period of time to keep within a certain temperature threshold e.g. a 1.5°C target limit for global temperature rise.

It is tricky to estimate because a budget is influenced by core assumptions, chosen characteristics, and different variables (for example the amount of other greenhouse gases in the atmosphere). Read about the difficulties in estimating a budget on the Carbon Tracker webpage Carbon Budgets Explained. Carbon budgets are particularly tricky, because there is so little left in the budget if we are to stay under a 1.5°C level of warming – there is very little room for error in calculating the budget.

Mark Maslin neatly represents the pressing need to reduce CO₂ emissions in this interactive temporal pie chart Using up the carbon budget.

If global warming is to be held to a 1.5°C temperature rise, the current estimate (from Carbon Brief for the 1.5°C target) is that we have a range of 230-440 billion tonnes of CO₂ left (GtCO₂), from 2020 onwards[1]. Since 1751 the world has emitted over 1.5 trillion tonnes of CO₂.

1. Create a pie chart to illustrate the historic carbon budget and the estimated remaining amount of carbon in the budget for the 1.5°C target. To complete this use the following steps.

a) 1000 kilograms is a tonne. 1 billion metric tonnes equal a gigatonne. 1 trillion tonnes equal 1000 gigatonnes. Standardise the total amount of CO₂ in the carbon budget by converting 440 billion tonnes and 1.5 trillion tonnes into GtCO₂.

b) Calculate the estimated total carbon budget. Take the upper estimate of how much carbon we have left in the budget (to emit) and add it to the amount emitted since 1751.

c) What proportion of your circle will be drawn per GtCO₂ by dividing 360° by your total carbon budget figure?

d) Draw the pie chart.

The idea of a carbon budget and the notion that Earth has a remaining amount before a target is missed is based on the near-linear relationship between cumulative CO₂ emissions (the impact on atmospheric concentrations) and the warming of the planet. In other words, as one increases so does the other. The IPCC report of 2021 confirmed that global temperatures rise in direct relation to cumulative emissions.

### CO₂ emissions and global warming

Scientist have investigated the correlation between CO₂ emissions and global warming. Table 1 in Appendix A contains data for CO₂ emissions and historic annual temperature change for the planet.

1. Draw a line graph to illustrate the relationship between cumulative CO₂ emissions and global temperature.

There are 5 projected ‘pathways’ for future cumulative CO₂ emissions and temperature change; SSP1, SSP2, SSP3, SSP4, and SSP5 (standing for Shared Socio-economic Pathways). Currently the Earth is following the SSP2-4.5 or SSP3-7.0 scenarios. These pathways are sometimes also referred to as RCPs, Representative Concentration Pathways of CO₂. The bullet points below clearly highlight the preferable future path and, according to the Paris Agreement (by which 198 countries agreed to try to keep the rise in mean global temperature to well below 2 °C above pre-industrial levels, and preferably limit the increase to 1.5 °C), the only legal trajectory.

SSP1 the ‘green road’, honouring the Paris Agreement, by limiting global warming to 1.5°C

SSP3 ‘regional rivalry’, food and energy security are prioritised, strong environmental decline

SSP4 inequality ‘a road divided’, environmental policies only focus on high-income areas

SSP5 fossil-fuel development taking ‘the highway’ business-as-usual, no-mitigation

Figure 2 in Appendix B is taken from the IPCC report. It shows cumulative CO₂ emissions since 1850 and °C temperature change with the 5 future SSP (Shared Socio-economic Pathways).

1. Describe the relationship between cumulative CO₂ emissions and global warming. Be careful: emissions don’t necessarily determine the temperature of the Earth, read Carbon Dioxide in the Atmosphere – Balancing the Flow to learn more.
1. Do cumulative CO₂ emissions cause annual mean global land-ocean temperatures to rise? Use data in your answer.

Within your answer for question 2 there is variation in emissions by country. Some countries have historically contributed more than others to global warming. Table 2 in Appendix B gives data on CO₂ emissions in 1750 and 2019 for 6 countries.

1. Create a line graph for cumulative CO₂ emissions for Canada, China, India, Kenya, the US, and the UK.
1. Which country emitted the most CO₂ in 2019?
1. Which country has had the greatest relative change between 1750 and 2019?

### Exam-style question

Open the Global Carbon Atlas.

Using all the work you have completed answer the final question below. The instruction describe means you must give an account of the pattern you see in the world map, and how it changes.

1. Press the play button at the bottom of the screen. Describe how the pattern of CO₂ emissions changes from 1960 to 2020.

[1] Carbon budgets are an estimate of the total quantity of CO₂equivalent emissions that can be allowed in order to maintain a 66% chance of staying within the Paris Agreement target of capping global warming at 1.5°C this century.

### Appendix B

Figure 2 is there a relationship between cumulative CO₂ emissions and the increase in global surface temperature? © The 2021 IPCC Working Group I report

1. The Washington Post explains that a gigaton is equivalent to a billion metric tonnes.

a) Standardise the total amount of CO₂ in the carbon budget into GtCO₂. 440bn tonnes and 1.5 trillion tonnes of CO₂ = 440 GtCO₂ and 1500 GtCO₂

b) 440 GtCO₂ and 1500 GtCO₂ = 1940 GtCO₂.

c) 360 ÷ 1940 = 0.18556701. Each GtCO₂ will be worth 0.18556701°.

2. As instructed.

1. There is a strong relationship between CO₂ emissions and global warming. Both historical and future emission pathways show that as CO₂ increases as a gas in the atmosphere, global temperatures rise. When analysing the paleoclimate record this strong correspondence between temperature and the concentration of carbon dioxide in the atmosphere is equally evident over the past the past several hundred thousand years.
2. Yes, cumulative CO₂ emissions cause annual mean global land-ocean temperature change. Figure 2 clearly shows the near linear relationship. If SSP1-1.9 (with a temperature increase under 2°C) is to be achieved, then world population will have to be held at 8.24 billion with CO₂ emissions being cut to net zero by 2050.
3. As instructed.
4. Kenya emitted 449.09 million t in 2019.
5. Column 5 from Table 2 is complete below.

8. As instructed.

## City Emission Levels

The emissions of a city from 2000 to 2012 are modelled by the equation

$$p\left( t \right) = \frac{1}{10}\ln\left( t + 1 \right) – \cos\frac{t}{2} + \frac{1}{10}t^{\frac{3}{2}} + 199.3$$

$0 \leq t \leq 12$

a) Show that the emissions reach a local maximum in the interval $$8.5 \leq t \leq 8.6$$

[5 marks]

The emissions reach a local minimum between 9 and 11 years after the measurements began.

b) Using the Newton-Raphson procedure once and taking $$t_{0} = 9.9$$ as a first approximation, find a second approximation of when the emissions reach a local minimum.

[6 marks]

## UK Carbon Dioxide Emissions

The rate of CO2 emissions for the UK was measured every 5 years, from 1990 to 2015.

The results are given in the table below with the rate of CO2 emissions measured in x109 kg year-1

 Year 1990 1995 2000 2005 2010 2015 Rate of CO2 emissions (x109 kg year-1) 595.7 557.5 555.7 555.2 496.7 403.8

Using all of this information,

a) Estimate the total CO2 emissions from the UK between 1990 and 2015, giving your answer in standard form.

[3 marks]

Given that the curve produced by plotting a graph of the rate of CO2 emissions against the year is concave,

b) Explain whether your answer to part (a) is an underestimate or an overestimate of the total CO2 emissions between 1990 and 2015.

[1 mark]

## Finding the Total Carbon Dioxide Emissions from 1850

The graph below shows how the rate of CO2 emissions varies from 1800 to 2017.

This curve can be approximated as $$E = 1.5e^{0.02t}$$ where E is the rate of CO₂ emissions per year (GtCO₂ year⁻¹) and t is the number of years since 1800.

a) Using the above equation, calculate the total mass of CO₂ emitted between the years 1800 and 2017. Give your answer to 3.s.f.

[3 marks]

b) By comparing the exponential model with the best fit line on the graph shown at $$t = 150$$, evaluate whether your answer to (a) is an overestimate of the true emissions or an underestimate.

[1 mark]

## A Total Emissions Goal

The warming caused by carbon dioxide (CO₂) emissions over any given period is proportional to the total amount of CO₂ emitted over that period. Recognising this, a country decides to limit its carbon dioxide emissions to less than 12×10¹²kg in total emitted across 20 years.

For the first 4 years, the countries emissions are stable at 800×10⁹kg year⁻¹

The country decides that they will be able to reduce their emissions so that each subsequent year produces 5% less emissions than the previous year.

Using the model,

a) Show that the country’s total CO₂ emissions from the
first 6 years are estimated to be 4682×10⁹kg CO₂

[2 marks]

Show that the estimated total emissions per year in the rᵗʰ year, with units
x10⁹kg year⁻¹, for 5 ≤ r ≤ 20, is

800 x 0.95ʳ⁻⁴

[1 mark]

Determine whether the country will meet their 20 year emissions goal.

[4 marks]

## A Carbon Negative Company

A small company’s carbon dioxide (CO₂) emissions since 2000 can be modelled using the parametric equations

$12x = t,\ y = 8t – 4.9t^{2} + 10,\ t \geq 0$

Where x is the number of years since 2000 and y the yearly CO₂ emissions, in tonnes.

Due to an afforestation program, the company’s emissions can go negative, as the trees planted absorb CO₂ from the atmosphere.

a) Find the year in which the company’s CO₂ emissions go negative.

[4 marks]

b) Find the greatest amount of CO₂ emissions from that company in one year

[5 marks]