## 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.

## Ocean Warming and Kettles – Teachers’ Notes

Resource produced in conjunction with Sustainability Physics.

Students’ worksheet.

## Motivation:

• The world’s oceans are heating. Their temperature is not rising as fast as that of the land or air, but they are the major store of the excess thermal energy resulting from greenhouse gas emissions
• According to the abstract of this study https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2012GL051106#support-information-section the top 700m of the oceans have warmed by 0.18°C on average between 1955 and 2010. This resource investigates how big this store of thermal energy is.

• GCSE physics heat capacity, power calculations and estimation
• GCSE maths standard form: the order of magnitude of the numbers is more important for this question than the numerical values

This could be used as a starter exercise: Can students do the whole question at once given only the radius of the Earth, the temperature rise and the specific heat capacity of sea water?

1. Why does the temperature of the sea rise more slowly than the temperature of the land or air?
Water has a higher heat capacity (4kJ/kg/°C for sea water) than land (2kJ/kg/°C for rock) or air (around 1kJ/kg/°C). For the same input of thermal energy, the increase in temperature is smaller for the ocean than it is for the land.
2. Find the area of the Earth’s oceans using the following information: the radius of the Earth is 6371km and the oceans cover about 70% of the Earth’s surface.
A = 4πR2 = 3.57 x1014 m2 ≈ 3.6 x1014 m2
3. Find the volume of the top 700m of the oceans. Ignore all the coastal sections of the ocean which are shallower than 700m.
V = A*height = 2.5 x17 m3
4. Find the mass of the top 700m of the ocean. Use the density of seawater as ρ = 1025 kg/m3
Mass = V* ρ = 2.56 x1020 kg ≈ 2.6 x1020 kg
5. Find the energy required to give this mass a temperature rise of 0.18°C. The specific heat capacity of sea water is 4 kJ/kg/K
Energy = C*Mass*ΔT = 1.8 x1023 J
6. Find the average power over the 55 year heating period
Power = Energy/time = 1.1 x1014 W
7. How big is that power? Find the power ‘per person’ by dividing the total power by the number of people on Earth today (8 billion people)
1.3 x104 W
8. A kettle has a power of 2.5kW. How many kettles would each person on the Earth have to boil to have the same total power?
1.3 x104 W / 2.5 x103 W = 5.3 ≈ 5
The warming of the upper ocean between 1955 and 2010 is equivalent to the energy used by every person on Earth boiling 5 kettles continuously for 55 years! This question only considers the upper ocean. The lower ocean is also warming and storing energy.

## Ocean Warming and Kettles

Resource produced in conjunction with Sustainability Physics.

Teachers’ Notes

## Motivation

The world’s oceans are heating. Their temperature is not rising as fast as that of the land or air, but they are the major store of the excess thermal energy resulting from greenhouse gas emissions

The top 700m of the oceans have warmed by 0.18°C on average between 1955 and 2010.

This resource investigates how big this store of thermal energy is.

1. Why does the temperature of the sea rise more slowly than the temperature of the land or air?
2. Find the area of the Earth’s oceans using the following information: the radius of the Earth is 6400km and the oceans cover about 70% of the Earth.
3. Find the volume of the top 700m of the oceans. Ignore all the coastal sections of the ocean which are shallower than 700m.
4. Find the mass of the top 700m of the ocean. Use the density of seawater as ρ = 1025 kg/m3
5. Find the energy required to give this mass a temperature rise of 0.18°C. The specific heat capacity of sea water is 4 kJ/kg/K
6. Find the average power over the 55 year heating period
7. How big is that power? Find the power ‘per person’ by dividing the total power by the number of people on Earth today (8 billion people)
8. A kettle has a power of 2.5kW. How many kettles would each person on the Earth have to boil to have the same total power?

## A Regrowing Reef

a) Use the substitution $$u = 4 – \sqrt{s}$$ to show that

$\int_{}^{}\frac{\text{dh}}{4 – \sqrt{s}} = – 8\ln\left| 4 – \sqrt{s} \right| – 2\sqrt{s} + k$

where k is a constant

[6 marks]

A coral reef is growing back after global temperatures are reduced from their peak value.

The rate of change of area covered by the reef is modelled by the differential equation

$\frac{\text{ds}}{\text{dt}} = \frac{t^{0.25}(4 – \sqrt{s})}{20}$

Where s is the surface area of the reef in m2 and t is the time, in years, after the reef begins to regrow.

b) Find, according to the model, the range of areas that could be covered by the coral reef.

[2 marks]

The coral reef has a surface area of 1m2 when it starts to regrow.

According to the model,

c) Calculate the time this reef would take to cover 12 m2, giving your answer to 3 significant figures.

[7 marks]

## Ocean Acidification and CO2 Absorption – Teacher’s Notes

Increased CO2 levels in the atmosphere are buffered by the oceans, as they absorb roughly 30 % of this CO2. The negative consequences of this are that the oceans become more acidic. The CO2 reacts with water and carbonate to form carbonic acid, reducing the available carbonate that shellfish, crabs and corals combine with calcium to make hard shells and skeletons.

## Curriculum Links: Core chemistry AQA GCSE

4.2.4 The pH scale

9.1.2 The Earth´s early atmosphere

9.2.3. Global climate change

## Chemistry in the activity

Na2CO3 + 2 CH3COOH → 2 CH3COONa + CO2 + H2O (Bicarbonate of soda reacts with vinegar to form carbon dioxide)

In this experiment the students will initiate a reaction that produces CO2 in an enclosed water-air environment. The CO2 formed will be absorbed into the water, making it more acidic and changing the colour of the indicator. The experiment can be carried out in pairs and takes about 15 minutes. An additional experiment to test the solubility of CO2 in warm and cold water can be carried out afterwards, explaining how global warming can affect marine CO2 absorption.

## Materials

• Bicarbonate of soda (baking soda)
• White vinegar
• Bromothymol blue Indicator (diluted with water: 8 ml bromothymol blue (0.04% aqueous) to 1 litre of water)
• 2 x 500 ml Beakers
• Small plastic or paper cup (100 ml)
• 2 x Petri dishes or lid for large beakers
• Teaspoon or 5 ml measuring cylinder
• Two sheets of white paper
• Safety glasses and lab coat

See the student worksheets for the detailed preparation: Ocean acidification and CO2 Absorption

### Application to the  World’s Oceans

The beaker is like an enclosed ocean-atmosphere and the CO2 from the reaction will equilibrate between the water and the air. Our oceans absorb more CO2 when the concentration in the atmosphere increases. But how much CO2 can they keep absorbing? Will they reach a saturation point?

Corals and shellfish are affected by ocean acidification, making it harder to create their shells, which will affect other fish up through the food web. Global warming caused by the increased CO2 effects the corals and fish as only slight changes in the temperature of the water can have effects throughout the ocean´s food chain. So there is a knock-on effect or a positive-feedback from the ocean heating and the ocean acidification.

If you want to illustrate more about the feedbacks and this double impact, the next experiment demonstrates the effect of a temperature increase on CO2 absorption, thus limiting the water´s capacity to absorb as much CO2.

### CO2 Absorption in Water class practical

This experiment allows  students to determine how much CO2 dissolves in warm or cold water.

See the student worksheet for the detailed preparation.

## Materials

• Water
• Effervescent fizz tablets
• Ice (optional)
• 2 x 500 ml measuring cylinders
• 2 x Petri dishes that fit over the cylinders
• Bowl or container (at least 5 litres)
• Stand and clamp to hold cylinders
• Water heater
• Funnel

Application to the World’s Oceans:

More CO2 has escaped from the warm water, showing that it cannot absorb as much CO2. Warmer oceans will not be as effective buffers at removing CO2 from the atmosphere. However, this phenomenon does prevent these warmer oceans from being as acidic.