## 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 – Trees for Net Zero

Resource produced in collaboration with MEI

This resource comprises several stand-alone activities which may be used separately.

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

## Scotland’s Curriculum Trees for Net Zero

Resource produced in collaboration with MEI

Note that this session is made up of separate activities which may be used independently.

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

# Calculation: Trees as Air Conditioners

Learning objectives: to apply the equations

energy for a change of state = mass × specific latent heat

energy transferred = power × time

to a real world situation.

When water evaporates from the leaves of trees, is it called evapotranspiration. The latent heat required for this comes from the thermal store of energy in the surroundings.  This question compares the cooling power of evapotranspiration from an oak tree to air conditioning.

1. An oak tree can lose up to 400 kg of water through evaporation from its leaves in one day (24 hours). Assuming that the energy for evaporation comes from the tree’s surroundings, calculate the average cooling power of the tree. The specific latent heat of vaporisation of water is 2700 kJ/kg.

energy for a change of state = mass × specific latent heat

400 kg * 2700 kJ/kg

power  =  energy transferred / time

time = 24*3600

power = 10 kW

1. An air conditioning unit  has a power of 2.5 kW. How many air conditioning units would be required to provide the same cooling power as the evapotranspiration of one oak tree?

10 kW / 2.5 kW = 4 units

# Describe: Trees as Reflectors

Aim: Students should be able to describe the effects of reflection, transmission and absorption of waves at material interfaces.

This activity would work well as a think/ pair/ share activity, and could also be displayed as a poster or other presentation.

The chlorophyll in green leaves of a tree absorbs the red and blue light in the visible Electromagnetic radiation from the Sun. The green light is reflected.

A green umbrella would do the same.

Both cast a shadow on the ground, implying that less visible light reaches the ground.

The ground also reflects some of the sunlight and absorbs the rest.

Absorbed sunlight increases the thermal store of energy.

The greater the thermal energy stored in an object, the more infrared radiation, heat, it emits.

Would you feel cooler walking along a road that is shaded by trees than you would if you carried a green umbrella?

Both the umbrella and the trees will reduce the amount of sunlight reaching you and warming you up.

They will also reduce the amount of sunlight reaching the ground. However, the trees are permanent and will have been shading the ground all day, so the ground in the shadow of the trees will be cooler than the ground in the shadow of a moving umbrella.

The heat, infrared radiation, reaching you from the ground will be lower in the shadow of the trees.

Extended ideas

These could include:

• evaporative cooling as described above,
• the fact that the trees will also be a barrier to convection so heat could be trapped near the ground,
• the fact that trees will be a barrier to the wind and so air will be trapped below them. This will also reduce the conduction of heat from the ground,
• in the long term, trees remove carbon from the atmosphere and so reduce the greenhouse effect,
• trees can also reduce air pollution,
• trees can reduce flooding by slowing the flow of water.

## Key Stage 3 – Trees and Carbon Capture

Resource produced in collaboration with MEI

Brief overview of session ‘logic’

• Why trees are good
• How much carbon do trees capture and store?
• How does the amount of carbon captured and stored by a tree change during its lifecycle?

Mathematical opportunities offered

• 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

## 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

## Core Maths – Trees and Carbon Capture

Resource produced in collaboration with MEI

Brief overview of session ‘logic’

• Why trees are good
• How much carbon do trees sequester?
• How does the amount of carbon sequestered by a tree change during its lifecycle?

Mathematical opportunities offered

• 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

## A Carbon Conscious Company

The curve

$y = e^{-0.5x} + 4x – 0.1x^{2} + 2$

Can be used to describe a company’s net emissions, in tons of CO2, x decades after 2000.

a) By first using calculus, show that peak emissions occur at 19 ≤ x ≤ 21

[4 marks]

The company owns an afforestation program. As trees absorb CO2, this means that their net emissions can decrease to negative values.

b) Show that the emissions become negative between 39 and 41 years after the year 2000.

[2 marks]

c) Use the Newton-Raphson method to find, to 4 significant figures, the number of decades it takes for the emissions to become negative. Use x0 = 39

[4 marks]

## Arithmetic Afforestation

A country decides to begin a reforestation program, starting in 2020, gradually increasing the number of trees planted per year by the same amount each year.

The timetable for the first four years is shown below:

 Year: 2020 2021 2022 2023 Trees planted: 6.0×10⁶ 1.4×10⁷ 2.2×10⁷ 3.0×10⁷

a) Find an expression, in terms of n for the number of trees planted in year n.

[2 marks]

b) Calculate how many trees in total will be planted if this program is followed for 10 years.

[2 mark]

The government will declare the program a success once 2.45×10 trees have been planted in total

c) Given that the country plants all of the trees that the model would predict in year k, but reaches the target part way through year (k + 1), show that k satisfies

(2k – 49)(k + 25) < 0.

[5 marks]