## 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.

## Changing the Subject with Higher Powers, Roots Including Brackets and Fractions

Climate change context

Rearranging the formula for the power derived from a wind turbine, and substituting values into its rearranged form.

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 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.
• Change the subject of a linear formula requiring two steps resulting in a bracket.
• Change the subject of a formula with brackets and fractions.
• Change the subject of a formula where the subject is squared and with additional steps

Lesson ppt

Mixed Exercise pdf

## 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

## 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

## 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

## 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

## 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

## 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

## 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

## 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

## 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.

## Scotland’s Curriculum – EVolution of vehicle sales

Resource produced in collaboration with MEI

Brief overview of session ‘logic’

• Explore the infographic – what can be worked out from this information and what questions does it raise?
• Look at trends in vehicle registrations
• Look at proportions of types of newly registered vehicles over time – why has the percentage of petrol cars being registered increased from 2015 to 2020?
• Do some calculations to show that the number of petrol cars being registered has decreased from 2015 to 2020.
• Reflect upon the implications for misleading representations of data
• Consider the implications of the ban on new petrol and diesel cars by 2035 – what affect will this ban have on the proportions of car types being registered?
•  What questions does the increase in electric vehicles raise?

Mathematical opportunities offered

• Interpretation of data, statistics, graphs, infographics in context
• Critiquing graphs
• Calculating percentages
• Exploring proportions of quantities over time
• Making conjectures about future proportions given available data
• Analysing and comparing data in order to develop and present a conclusion.

## Scotland’s Curriculum- EVolution of vehicle sales

Resource produced in collaboration with MEI

Brief overview of session ‘logic’

• Explore the infographic – what can be worked out from this information and what questions does it raise?
• Look at trends in vehicle registrations
• Look at proportions of types of newly registered vehicles over time – why has the percentage of petrol cars being registered increased from 2015 to 2020?
• Do some calculations to show that the number of petrol cars being registered has decreased from 2015 to 2020.
• Reflect upon the implications for misleading representations of data
• Consider the implications of the ban on new petrol and diesel cars by 2035 – what affect will this ban have on the proportions of car types being registered?
• What questions does the increase in electric vehicles raise?

Mathematical opportunities offered

• Interpretation of data, statistics, graphs, infographics in context
• Critiquing graphs
• Calculating percentages
• Exploring proportions of quantities over time
• Making conjectures about future proportions given available data
• Analysing and comparing data in order to develop and present a conclusion.

## 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?

## 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?

## Core Maths – EVolution of vehicle sales

Resource produced in collaboration with MEI

Brief overview of session ‘logic’

• Explore the infographic – what can be worked out from this information and what questions does it raise?
• Look at trends in vehicle registrations
• Look at proportions of types of newly registered vehicles over time – why has the percentage of petrol cars being registered increased from 2015 to 2020?
• Do some calculations to show that the number of petrol cars being registered has decreased from 2015 to 2020.
• Reflect upon the implications for misleading representations of data
• Consider the implications of the ban on new petrol and diesel cars by 2030 – what affect will this ban have on the proportions of car types being registered?
• What questions does the increase in electric vehicles raise?

Mathematical opportunities offered

• Interpretation of data, statistics, graphs, infographics in context
• Critiquing graphs
• Calculating percentages
• Exploring proportions of quantities over time
• Making conjectures about future proportions given available data
• Analysing and comparing data in order to develop and present a conclusion.

## Key Stage 3 – EVolution of vehicle sales

Resource produced in collaboration with MEI

Brief overview of session ‘logic’

• Explore the infographic – what can be worked out from this information and what questions does it raise?
• Look at trends in vehicle registrations
• Look at proportions of types of newly registered vehicles over time – why has the percentage of petrol cars being registered increased from 2015 to 2020?
• Do some calculations to show that the number of petrol cars being registered has decreased from 2015 to 2020.
• Reflect upon the implications for misleading representations of data
• Consider the implications of the ban on new petrol and diesel cars by 2030 – what affect will this ban have on the proportions of car types being registered?
•  What questions does the increase in electric vehicles raise?

Mathematical opportunities offered

• Interpretation of data, statistics, graphs, infographics in context
• Critiquing graphs
• Calculating percentages
• Exploring proportions of quantities over time
• Making conjectures about future proportions given available data
• Analysing and comparing data in order to develop and present a conclusion.

## A Changing Transport System

Cars and transport, and the gases and particulates that they emit from their exhausts have a serious effect on the environment and on human health.

In the future the transport system will have to look very different.
For example, in the UK there could be

three times as many journeys using electric buses rather than using bicycles

and two times as many journeys using bicycles than using private electric cars.

For the people in the UK write down the ratio of journeys using electric buses to bicycles to private electric cars.
[3 marks]

## Trucking Company’s Carbon Increase

Jean owns a trucking company (with ten trucks) which produces 1062.8 tonnes of carbon dioxide in the year 2015. The company’s carbon emissions increase over the next two years as the business grows and the trucks spend more time on the road.

From the year 2015 to the year 2016, Jean’s company’s carbon emissions increase by  $$x\%$$.

In 2016, the company emits 1126.5 tonnes of carbon dioxide.

From the year 2016 to the year 2017, Jean’s company’s carbon emissions increase by $$\frac{x}{2}\%$$ .

How many tonnes of carbon does the company emit in the year 2017? Give your answer to 3 significant figures.

[4 marks]