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.

Dr Frost Learning
Climate Change Quality Mark Content

Climate Change Concept Association Tool

This tool is best used on a laptop or other larger screen and may not function correctly on a phone.

Carbon Dioxide – Seasonal Cycles

Mauna Loa Observatory NOAA

An exam style question suitable for GCSE science.

Notes for Teachers

The units for the data are in fact ppmv which we have simplified to ‘parts per million’ for this question. 

This is a nice visualisation of what 420ppmv looks like.

The questions explore the fact that there is a seasonal cycle in carbon dioxide in the atmosphere because plants take up carbon dioxide during photosynthesis in the spring and summer, which is then released back into the atmosphere when plants die and leaves rot in the autumn and winter. 

Carbon dioxide is a well mixed gas, meaning that the data recorded at Mauna Loa is representative of the Northern Hemisphere, and that at the South Pole is representative of the Southern Hemisphere. 

The seasons are out of phase with each other – when it is summer in the Southern Hemisphere, it is winter in the Northern Hemisphere. 

As there is far less vegetation in the Southern Hemisphere than in the Northern Hemisphere, the seasonal cycle is much smaller. 

Students may notice that there could also be a human element to the cycle – we burn more fossil fuels in the winter than in the summer (and there are also fewer people in the Southern Hemisphere).

The correct answer for the mean is 416.1 parts per million.

As well as the seasonal cycle, the graph provided shows the increase in atmospheric carbon dioxide since 1958. This increase is because of the emissions of carbon dioxide by human activities including land use change including deforestation, burning fossil fuels and cement production. 

carbon dioxide at mauna loa
Climate Change Quality Mark Content

Storm Surges

Watch this short animation to learn about the causes and impacts of storm surges in the UK, as well as the expected impact of climate change on them. 

At the bottom of the page, you can download a Knowledge Organiser to complement the animation.

With thanks to the students and staff at Boston College for their contribution to the animation. 

Isaac Physics

Isaac Physics logo

A diverse range of questions based on applications of physics in weather and climate, including sea level rise, radar frequencies,  aerosols, oceanic circulation, tidal barrages etc. 

Isaac Physics is an online study tool developed by the University of Cambridge. Isaac Physics questions are self marking practice questions for secondary school and undergraduate scientists. 

Snowflake fall speed

Aerosol attenuation

Barometric formula

Concentration of Oxygen

Cooling Tree

Electric Car Electrics

Electric Charge of Earth

Hadley Cell

How much Rain?

Isotrope Concentrations and Ocean Circulation

Kite Heights

Radar Reflectivity Units

Radar wavelengths and frequencies

Earth radiation balance

Sea Level Rise

Temperature Records and Uncertainties

Tidal Barrage

Urban Heat Island

Wind Turbine

Wind Turbine Power

Mass of the Atmosphere

Maths for Planet Earth

Climate-based questions for students and teachers. A team of students and academics at the University of Oxford developed these Maths for Planet Earth questions.

Physics – Egypt’s Benban Solar Farm

In this resource linked to COP27 in Egypt, physics students explore renewable energy production.

Learning Objectives

  • Recognise that solar power is a renewable energy source of great value in Egypt
  • Describe the energy transfer in a solar cell
  • Evaluate the energy dissipated in the Benban solar farm
  • Calculate the cost of the energy produced using the formula cost = power (kW ) x time (hours) x price (per kWh).

Motivation/Outline

In its acceptance speech at COP26, Egypt celebrated its renewable energy resources:

This is an extract from https://unfccc-cop26.streamworld.de/webcast/closing-plenary-of-the-cop-followed-by-cmp-and-c-2 from 09:20

Egypt transitioned from the traditional energy sources to renewable, more sustainable and planet-friendly energy sources…

One of these resources is the huge Benban solar farm.

Lesson Introduction

Watch the relevant part of the COP26 plenary video and/ or

  • The Benban solar farm was supported by the Green Climate Fund. Contributions to the Green Climate Fund were one of the areas which didn’t make as much progress as was hoped at COP26 in Glasgow, 2021.
  • COP27 will be at Sharm El-Sheikh in Egypt in November 2022.
Benban - map
Benban map

images from google maps

Discussion points:

  • What is a renewable energy source?
  • Why is it important to develop renewable energy sources?
  • What is a solar cell and how is it different from a solar panel? Where have people seen solar cells/ panels?
  • What makes a location suitable for a huge solar energy farm? (space, sunshine, access for bringing the equipment in and getting the electricity out…)
  • Could we build such a huge solar park in the UK? (no, we don’t have a big desert, but you could research some UK solar farms)
  1. Use https://globalsolaratlas.info/map to compare the global horizontal irradiation where you live with that in Benban. (for Benban the value is given as 2366 kWh/m2).
    Global horizontal irradiation is the total amount of solar energy reaching a 1m2 horizontal surface on the ground in a year.

    Discussion point: What is a kWh? (if 1 kWh is the electrical energy converted by a 1 kW appliance used for 1 hour rephrase this in terms of electrical energy generation. See https://www.bbc.co.uk/bitesize/guides/z2h4dxs/revision/1 for more detail)

    Discussion point: So what is a kWh/ m2?

    Extension: Express this answer as a proportion or percentage

  2. Discuss: what is the initial store of energy and by what pathways is it transferred? (nuclear store in the Sun, energy is transferred by light from the Sun to the panel and is transferred electrically from the panel to homes and businesses)
  3. The size of the Benban solar farm is 37.2 km2. Calculate the total energy carried by the light arriving at the site.

    (37.2km2 = 37 200 000m2 so 2366 x 37 200 000 = 88,015,200,000 kWh = 88 015.2 GWh = 88.0TWh)

    Discuss: kilo, mega, giga, Tera etc.

  4. The estimated output from Benban is 3.8TWh. How much energy is not converted usefully?
    88.0-3.8 = 84.2TWh

    Extension – write this as a proportion or percentage
    Discussion – why so much? Solar panels don’t cover the whole of the ground, solar panels are actually less efficient when they get hot, you can see solar panels, so they must be reflecting some of the Sun’s light, not absorbing it all etc.)

  5. What is the current electricity price in your region? (see https://www.ukpower.co.uk/home_energy/tariffs-per-unit-kwh and scroll down for regional breakdown).
    What is the value of the energy the Benban solar farm will produce during COP27, which is scheduled to last 2 weeks (assume there are 52 weeks in a year)?

    (cost = power (kW ) x time (hours) x price (per kWh).
    So value = 3, 800, 000, 000 kWh x 2/52 x 28.34 = £41,420,000.

    Discussion – is that surprising?

    Why might the quantity of electricity produced actually be different? (We started with an annual value, but the seasons and the weather will actually have an impact on how much is produced in a given week).

verified climate education resources

KS3 Geography – Egypt’s Construction Problem

In this resource linked to COP27 in Egypt, geography students explore population growth, urbanisation and climate change. 

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

flight path

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.
  3. Give your final answer in hours and minutes.
flight data

Some students may find the following table useful:

flight time table

UK Energy Mix

In this activity students use current data to investigate  the UK’s energy sources.

Go to gridwatch.co.uk and use the table and the key at the bottom of the page to complete the following table. This website shows you where the UK’s electric power is coming from and what the total demand (use) is and has been over the past year.

(1 GW = 1 000 000 000W)

energy source table
  1. In some of the boxes, you may see a negative number – what does that mean?
  2. What is the total net amount of power we are currently getting from France, the Netherlands, Belgium and Norway?
  3. For the power generated in the UK, highlight all renewable energy sources.
  4. What is the total amount of power we are currently generating from fossil fuels in the UK?
  5. Looking at the graph headed ‘yesterday’, when would have been the best time to charge an electric car, if you wanted to use as much renewable power as possible? Why?
  6. Looking at the graph headed ‘last year’ which season(s) have the most energy generated by solar energy?
  7. Which season(s) have the most energy generated by wind energy?

Extension

By looking at the total energy demand, and the production by wind energy, what can you deduce is the purpose of gas turbines?

Can you see any correlation between wind output and gas turbine output?

Opportunity for Group Work

Make a poster or presentation showing what you have learned.