## 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.

## A Climate Aware Citizen

A person decides in 2020 that they want to completely eradicate their carbon footprint in 20 months.

Following this decision, they begin to use multiple technologies that decrease their carbon footprint, such as limiting air travel, carbon offsetting and a solar powered home that contributes energy back to the national grid.

This results in their rate of emissions per month, E, following the equation

$E = 12\ln\left( t + 5 \right) – 2t + 2$

where t is the time, in months, since 2020.

a) Show that they achieve zero emissions between 18 and 22 months after they start.

[2 marks]

b) Using the iteration formula $$t_{n + 1} = 6\ln{(t + 5)} + 1$$ with $$t_{o} = 18$$, find the value of $$t$$ at which they achieve zero emissions to 2 decimal places.

[3 marks]

## Allotment Areas

A citizen wants to reduce their carbon footprint, so decides to grow their own produce in their garden.
The area set out for growing vegetables is shown below, and they want to place a shed in the area marked A.

Given that all dimensions are given in metres, and the total area of the plot is 65㎡ , find the values of $$x$$ and $$y$$ and hence the depth and width of the shed that they need to purchase.

[6 marks]

## What is a Carbon Footprint?

Carbon is one of the building blocks of life. Humans, animals and plants are made up of organic compounds. We burn wood and fossil fuels to produce energy and power transport, inadvertently releasing the greenhouse gas, CO2 into the atmosphere.

We will look at a series of calculations that represent the carbon cycle and how CO2 production is related to energy. You will start to see the energy implications of various fuels and technologies and their CO2 footprint.

The associated information sheet will provide the data you need to answer the questions below.

1. ## How much CO2 is emitted by the following activities? (calculate them in kg of CO2)

• Driving 100 miles?

(Using 13 litres of petrol or 10 litres diesel)

• Using your LED TV for 5 hours a day during a week?

(A 50” LED TV uses 100 watts, to convert to kWh, multiply kW by number of hours)

• Boiling water in the electric kettle for a family for a week?

(A kettle uses 1200 W and it takes 3 minutes to boil water and this is done 10 times a day – or does your household drink more hot drinks?)

• Heating the water with natural gas for a week of daily 5 minute showers?

(Heating 30 litre of water to 40°C uses 1.1 kWh in the form of gas, where emissions  from natural gas are 0.2 kg CO2/ kWh burned)

• Charging mobile phones for the family for a week. With an average of two full charges a day.

(Typical phone charges at 0.015 kWh and takes 2 hours to charge fully)

• Play station for 20 hours a week

(A Playstation 4 Pro uses 139 W)

## 2.  How to quantify CO2 emissions in terms of volume and mass?

• How many cubic metres of CO2 would 5000 kg CO2 occupy?

• A factory states that it releases 10 tons C per year (as greenhouse gas emissions). How many m3 of CO2e is this?

• If UK car emissions released 3 GtC in a year and all the CO2 remained in the atmosphere, by how much would the CO2 concentration increase?

• Go to see last year´s UK Carbon emissions published by the government (Provisional GHG emissions). In 2019 it was 351.5 Mt CO2 Considering the UK population is 63 million and the world population is 8.3 billion, are our carbon emissions representative of global average emissions? ((World emissions in 2017 were 36 Bt)

• Why has CO2 not decreased in 2020 if CO2 emissions have dropped? Is there still last years and the decade before´s emissions in the air or are we still emitting more despite the drop in transport and industry in 2020?

## 3.  Steps towards reaching carbon neutrality

• Do you think the UK is on its way to becoming a low carbon economy? Why do you think some countries like Estonia are way behind the UK and countries like Sweden are way ahead?
• The UK has a goal of reaching Carbon neutrality by 2050- do you think we are on our way to reaching that?
• What percentage of our man-made CO2 emissions are absorbed by the oceans?
• If a fully grown tree absorbs 22 kg of CO2 per year and an acre of forests 2.5 tons of Carbon, if we wanted to neutralize our country-wide annual emissions of 351.5Mt* CO2, how many more trees or acres of forest would we need?**

*The latest government statistics on UK annual CO2 emissions (for 2019) was 351.5 Mt CO2 equivalent

**UK forests absorbed 21 million tonnes CO2 in total in 2020, so they are working away continuously at helping to neutralise our emissions!

## Carbon Footprint – Teacher’s Notes

Carbon, fossils fuels and CO2

Carbon is one of the building blocks of life. Humans, animals and plants are made up of organic compounds. We burn wood and fossil fuels to produce energy and power transport, inadvertently releasing the greenhouse gas, CO2 into the atmosphere. Students will become more aware of the facts and figures that link the carbon cycle with CO2 emissions and the jargon that is used in the news and in global climate politics.

## Chemistry curriculum links: AQA GCSE

3.2.1 Use of amount of substance in relation to masses of pure substances (Moles)

7.1 Carbon compounds as fuels and feedstock

9.2 Carbon dioxide and methane as greenhouse gases

9.2.4 The carbon footprint and its reduction

## Chemistry in the activity

Calculating the energy from combustion of different fuels is related to the number of Carbon atoms these hydrocarbons contain. The amount of CO2 produced upon combustion is our way of measuring the Carbon footprint of energy sources. Electricity is generated from various forms of energy in each country´s electricity mix and the more renewables and the fewer inefficient coal power plants there are, the less CO2 is released per kWh electricity used. The UK is trying to go below 100 g of CO2 released per kWh by 2030 and is likely to achieve this before that date.

In the associated worksheet the students will carry out calculations based on a range of information they will find in the corresponding information sheet. They will become familiar with conversions between tons of Carbon and tons of CO2, the volume of CO2 and other factors they may hear in the news or that relate to their personal, a country´s or organisation´s carbon emissions.

They will go to websites that provide current global CO2 levels and a breakdown of the UK´s electricity supply, with the corresponding kg of CO2 this will emit per unit electricity used. Questions 1&2 use numeracy skills to evaluate and compare different forms of energy and different technologies.

Question 3 is best used as a classroom discussion and covers carbon neutrality, achieving the UK´s Carbon neutrality goals and calculate how many trees they would have to plant to neutralise this year´s CO2 emissions.

## 1.  Which fuels or activities produce more CO2?

QUESTIONS

Which of these activities produces more CO2 emissions? (calculate them in kg of CO2)

• Driving 100 miles?

(Using 13 litres of petrol or 10 litres of diesel)

Petrol = 2.3 x 13, Diesel = 2.7 x 10 = 29.9 kg CO2 for petrol and 27 kg for diesel

• Using your LED TV for 5 hours a day during a week?

(A 50” LED TV uses 100 watts, to convert to kWh, multiply kW by number of hours)

5 x 7 hours at 100 watts = 3.5 kWh = 3.5 kg CO2

• Boiling water in the electric kettle for a family for a week?

(A kettle uses 1200 W and it takes 3 minutes to boil water and this is done 10 times a day – or does your family drink more tea?)

1200 x 10 x 3 x 7 = 210 minutes (3.5 hours) or 4.2 kWh  x 0.283 = 1.19 kg CO2

• Heating the water with natural gas for a week of daily 5 minute showers?

(Heating 30 litre of water to 40°C uses 1.1 kWh in the form of gas, where emissions from natural gas are 0.2 kg CO2/ kWh burned)

Heating the water for a week uses 7.7 kWh so 0.2 x 7.7 is 1.54 kg CO2

• Mobile phone usage for the family in a week. Assume the family does an average of two full charges a day.

(Typical phone charges at 0.015 kWh and takes 2 hours to charge fully)

4 x 7 x 0.005 = 0.014 kWh x 0.283 = 0.396 kg CO2

• Play station for 20 hours a week

(A Playstation 4 Pro uses 139 W)

139 x 20 = 2.4 kWh = 7.87 kg CO2

## 2.  How to quantify CO2 emissions in terms of volume and mass?

QUESTIONS

• How many cubic metres of CO2 would 5000 kg CO2 occupy? 2500 m3
• A factory states that it releases 10 tons C per year (for its greenhouse gas emissions). How many m3 of CO2e is this? 10,000 kg x 44/12 = 36,667 kg CO2, so ½ x this is 18,333 m3
• If UK car emissions released 3 GtC in a year and all the CO2 remained in the atmosphere, by how much would the CO2 concentration increase?

0.47 x 3 = 1.41 ppmv

• Go to see last year´s UK Carbon emissions published by the government (Provisional GHG emissions). In 2019 it was 351.5 Mt CO2 Considering the UK population is 63 million and world population is 8.3 billion, are our carbon emissions representative of global average emissions? ((World emissions in 2017 were 36 Bt)

63m/8.3b =0.81 % of population and CO2 emissions are 351.5Mt/36000Mt = 0.98 %, so the population of the UK creates more CO2 than their population dictates, we produce 0.98/0.81 =1.21 times more CO2 than the average world population

(figures for 2020) 500 ppm; increase of 100 ppm between 1950 and 2020 (in 70 years), that is a 0.7 ppm average increase; it has increased 4 ppm since 2018 (in 2 years), 2 ppm increase per year. The rate of increase of CO2 concentration has increased since the 1950s.

• Why has CO2 concentration not decreased in 2020 if CO2 emissions have dropped?

The lifetime of CO2 means that it stays around in the atmosphere for many years and you will not see a decrease in the CO2 from the year that you stop releasing it, it will gradually level off, that is why we need to reach our CO2 emission peak as early as possible, to see the results a few years later.

## 3. Steps towards reaching carbon neutrality

QUESTIONS to discuss as a class

• Do you think the UK is on its way to becoming a low carbon economy? Why do you think some countries like Estonia are way behind the UK and countries like Sweden are way ahead? (http://www.globalcarbonatlas.org/en/CO2-emissions is a useful information source)

Estonia still burns a lot of coal, hence its high CO2 emissions. Sweden has 80 % of its electricity from nuclear and renewables

• The UK has a goal of reaching Carbon neutrality by 2050- do you think we are on our way to reaching that?
• What percentage of our anthropogenic (human) CO2 emissions are absorbed by the oceans?

31 %

• If a fully grown tree absorbs 22 kg of CO2 per year and an acre of forest, 2.5 tons of Carbon, if we wanted to neutralize our country-wide annual emissions of 351.5Mt* CO2, how many more trees or acres of forest would we need?**

351500/2.5 = 140600 acres. There are 60 million acres in the UK, so actually, only adding 0.234 % of the land as forests would do this!

*The latest government statistics on UK annual CO2 emissions (for 2019) was 351.5 Mt CO2 equivalent

**UK forests absorbed 21 million tonnes CO2 in total in 2020, so they are working away continuously at helping to neutralise our emissions!

## Carbon Footprint – Information Sheet

Carbon is one of the building blocks of life. Humans, animals and plants are made up of organic compounds. We burn wood and fossil fuels to produce energy and power transport, inadvertently releasing the greenhouse gas, CO2 into the atmosphere.

## 1.  Which fuels or activities produce more energy or CO2?

What are fossil fuels made up of?   Hydrocarbons with varying amounts of Carbon:

• Coal contains large complex hydrocarbon molecules (with C:H:O ratios of ~85C:5H:10O)
• Diesel is made up of alkanes containing 12 or more carbon atoms. (e.g. C13H28)
• Petrol contains alkanes and cyclo-alkanes with between 5 and 12 Carbon atoms (with an average composition of C8H12 (octane))
• The mass of one mole of pure Carbon is 12 g and the mass of one mole of CO2 is 12 + (2×16) = 44 g (to convert from CO2e to C multiply by 12/44)

What are the combustion reactions and how much energy and CO2 do they produce?

• 1 kg of petrol burned yields about 47 MJ of energy (1litre, 34.2MJ)
• 1 kg of diesel burned yields about 46 MJ of energy (1litre, 38.6MJ) (diesel is denser than petrol and has more energy per litre)
• 1 kg of coal burned yields about 30 MJ of energy
• 1 kg of wood burned yields about 19 MJ of energy
• 1 kg of coal (containing 0.78 kg Carbon) will produce 2.4 kg of CO2
• 1 litre of petrol (containing 0.63 kg of carbon) will produce 2.3 kg of CO2
• 1 litre of diesel (containing 0.72 kg of carbon) will produce 2.7 kg of CO2

The most up-to-date information on the make-up of the UK electricity grid (which is a mix of sources) can be found at RENSmart and the value in February 2021 was that 1 kWh produces 0.23314 kg CO2. (kWh are calculated by multiplying kW by the number of hours). If you live in another country you could compare its CO2 emissions per kWh electricity factor. Here is a good comparison site for many countries but with older data.

## 2. How to quantify CO2 emissions in terms of volume and mass?

Volume and mass of CO2

You will often hear about kg of CO2 emitted, relating to the energy usage of different forms of transport, of a household, of a company, of a particular industry (like the cement industry) or of a country or a person.

From what we know about the combustion processes, their efficiency and our energy needs, we can use emission factors to calculate carbon footprints. We also know that a mole of any gas occupies 22.4 dm3 at ambient temperature. So we can express the emissions as a volume of CO2. If we know how much of a gas is emitted and what the original concentration of that gas was in the atmosphere, we can see whether the emissions will change the concentration.

• 1 kg pure CO2 occupies a volume of half a cubic metre (500 dm3 (or litres))
• CO2 emissions are often stated in GtC (109 tonnes (or Gigatonnes) of Carbon)
• Concentrations of CO2 in the atmosphere are expressed in parts per million by volume (ppmv). 1 ppmv takes up 0.0001% of the volume of the atmosphere. Check out the Mauna Loa CO2 measurement station in Hawai for today´s level.
• A release of CO2containing 1 GtC would increase the atmospheric CO2 concentration by 0.47 ppmv if all the CO2 remained in the atmosphere, BUT carbon sinks nearly balance out the sources
• There was a CO2 increase of 2.5 ± 0.1 ppmv between 2017 and 2018
• The lifetime of CO2 is 5 to 100 years

Don’t forget:

• The mass of one mole of pure Carbon is 12 g and the mass of one mole of CO2 is 12 + (2×16) = 44 g (to convert from CO2eq to C multiply by 12/44)

Effects of the Covid-19 on the economy and thus CO2 emissions

• Between 2019 and 2020 global CO2 emissions decreased due to the COVID-19 Pandemic (in the region of 4 Gt CO2 and the CO2 emissions fell by 7 % in 2020, the largest ever decrease since the Second World War!)
• A Carbon brief article suggests that in 2020 we reduced the annual increase in CO2 concentrations by 0.32ppmv, putting it at 2.48ppm.
• Note the difference between emissions of CO2 and actual concentrations. The CO2 already in the atmosphere from previous year´s emissions (it lasts up to one hundred years).

## 3. Steps towards carbon neutrality

This Figure shows the latest (calculated every 3 months) fuel source mix for the UK electricity supply. Go to: OFGEM. Note the elimination of coal and the increase in wind and solar energy.

The UK electricity supply now has well over 20 % from renewables. The UK is trying to get to below 100 g CO2/ kWh by 2030 and we might achieve 5 % renewables by 2025. In late 2019 the electricity from British windfarms, solar panels and renewable biomass plants surpassed fossil fuels for the first time since the UK’s first power plant fired up in 1882.

We saw in section 1 that at RENSmart you can get the latest value for how many kg CO2 are produced per kWh of electricity. Let´s compare other countries from the table at the bottom of this website. Sweden currently has an emission of 0.013 kg CO2/ kWh (21 times lower CO2 emissions per kWh!). In Sweden 80 % of electricity comes from nuclear and renewables (with 66 % from renewables). By the way, renewables do have an embedded energy (of up to 50 g CO2/ kWh).

And look what these natural Carbon Sinks can do:

• Between 1994 and 2007, the oceans absorbed 34 Gt CO2 (31 % of what humans put into the atmosphere during that time)
• One acre of new forest can sequester about 2.5 tons of carbon annually. Young trees absorb CO2 at a rate of 6 kg per tree each year and after 10 years they absorb 22 kg of CO2 per year. At that rate, they release enough oxygen back into the atmosphere to support two human beings.