## 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.

## Key Stage 3 – Trees for Net Zero (Extended Resource)

Resource produced in collaboration with MEI

Brief overview of session ‘logic’

• Why trees are good
• People are planting trees – estimates around what the numbers look like in terms of land use
• Some companies encourage you to offset flights by planting trees – how many trees for one flight?
• How much carbon do trees capture and store?
• How does the amount of carbon captured and stored by a tree change during its lifecycle?
• What happens to that carbon when a tree dies?
• Can you plant a tree to offset a flight?
• What is Net Zero?

Mathematical opportunities offered

• Estimation and proportional reasoning
• Developing a sense of scale of large numbers
• Interpretation of data, statistics, graphs, infographics in context
• Critiquing graphs
• Analysing and comparing data in order to develop and present a conclusion
• Making assumptions
• Making predictions

## Trees for Net Zero (Extended Resource)

Resource produced in collaboration with MEI

Brief overview of session ‘logic’

• Why trees are good
• People are planting trees – estimates around what the numbers look like in terms of land use
• Some companies encourage you to offset flights by planting trees – how many trees for one flight?
• How much carbon do trees sequester?
• How does the amount of carbon sequestered by a tree change during its lifecycle?
• What happens to that carbon when a tree dies?
• Can you plant a tree to offset a flight?
• What is Net Zero?
• Can trees be used to achieve Net Zero?

Mathematical opportunities offered

• Estimation and proportional reasoning
• Developing a sense of scale of large numbers
• Converting between m2 and km2
• Interpretation of data, statistics, graphs, infographics in context
• Critiquing graphs
• Analysing and comparing data in order to develop and present a conclusion
• Making assumptions
• Making predictions

## Wiring up a Net-Zero Home

A homeowner decides to make their house carbon-neutral. They place solar panels on the roof, which then connect o their mains circuit via a wire.

The wire can be modelled as leaving the solar panels at A = 2i + 3j + 4k, and connecting to the mains at the point B = -3i + j – 3k, with the distances measured in metres and both points measured relative to the same fixed origin.

a) Show that $$\overrightarrow{\text{AB}} = – 5\mathbf{i -}2\mathbf{j -}7\mathbf{k}$$ and hence find the length of wire needed to 2 decimal places.

[2 marks]

In many homes powered by solar energy, when excess power is generated, it can be put onto the national grid, so that more renewables power the grid rather than fossil fuels.

The wire leading to the grid is on an automatic switch system M, which divides the wire $$\overrightarrow{\text{AB}}$$ in the ratio 2:1.

b) Calculate the distance of the automatic switch system M from the origin.

[4 marks]

## 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]

## 50 Years to Net-Zero

A country wishes to achieve net-zero CO2 emissions in 50 years.

At the start of the program their emissions are 800MtCO2 year-1.

They decide that they will be able to reduce their emissions at a stable rate so that each subsequent year they emit 12MtCO2 less than the previous year.

a) Calculate the total emissions that the country had produced over the 50 years, giving your answer in MtCO2.

[2 marks]

b) Show that a graph of MtCO2 produced per year against the year follows a straight line with equation:

$y = 800 – 12x$

[1 mark]

At the same time as reducing their emissions, the country decides to start a carbon dioxide removal program, whereby a certain amount of carbon dioxide is captured from the atmosphere and sequestered underground each year.

The program begins in the tenth year.

When the graph of MtCO2 removed per year is plotted against the year, it follows the curve with equation

$y = 0.1x^{2} – x$

c) Determine whether the country achieves their goal by finding the year in which the emissions removed are equal to the emissions produced, and thus the net emissions from the country are zero.

[3 marks]

After the 50 year program, the countries emissions stabilise at the final value.

The MtCO2 absorbed per year follows the same trend as before.

The country wishes to have not contributed to global warming at all since the start of the program. To achieve this, their net total CO2 emissions over the entire program would have to be zero.

d) Given the above information, by using calculus show that it takes 109 years for the country to have had a net zero effect on global warming since the start of the study.

[5 marks]