## 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.

## An Offshore Wind Farm

Below is a diagram of 3 offshore wind turbines, A, B and C, in a wind farm, as seen from above.

Given that the bearing from turbine C to turbine B is 90˚, and the distance from turbine A to turbine B is 3 km

Calculate the distances AC and CB.

[4 marks]

## Sankey Diagrams for Physics

Energy and Climate Change

Energy is needed in the form of electricity to power our lives, and to fuel our travel and industry. Since 1990, total world energy consumption has increased by over 55% and is projected to increase by another third by 2040.

Globally, oil accounts for over 30% of total energy use, followed by coal, gas and nuclear at 4%. This mix is different when you look only at electricity production, and different again on a country by country level.

A sustainable energy transition is a shift from an energy intensive society based on fossil fuels to energy efficiency with low carbon and renewable energy sources.

The Paris Agreement is a legally binding global climate change agreement, adopted by 189 nations at the Paris climate conference (COP21) in December 2015. It sets out a global framework to avoid dangerous climate change by limiting global warming to well below 2°C and pursuing efforts to limit it to 1.5°C.

Significant changes in energy production, transmission and use are necessary to achieve these commitments.

This should lead to co-benefits including improved air quality and reductions in energy poverty.

Since 2019, the costs of developing new power plants based on hydroelectric power, onshore wind, solar photovoltaic (PV), biomass and geothermal energy have become comparable to the costs of new oil and gas fuel plants.

Physicists play an essential role in all aspects of climate change research and policy decisions as well as in development of technologies and new ideas for preventing and mitigating the effects of future, damaging climate change.

Energy and Core Physics

Energy is a fundamental concept in physics and a key topic in any physics curriculum. The Earth’s climate system is driven by energy stores and transfers. Development of clean, sustainable energy generation and distribution methods relies on understanding the core physics involved. The climate system and sustainable energy production therefore provide engaging and relevant sources of examples for enhancing the teaching and learning of energy as a topic in Physics. They give teachers an obvious opportunity to engage their students in an appreciation of the importance of the physics already in the school curriculum in solving many of the problems surrounding accelerated climate change, as illustrated in the following, brief summary of potential links.

Energy is transferred by radiation from the Sun, increasing the thermal store in the Earth’s atmosphere and ocean systems. Energy transfers within these systems take place through the physical processes of conduction, convection, radiation and changes of state. Seasonal and longer term, natural variations in heating and cooling of the Earth are a result of the alignment of the Earth in space and its orbital motion around the Sun. Land and ice surfaces are heated differentially according to the absorptive or reflective nature of the surface type and rocks are heated internally due to energy released during radioactive decay and large scale, convective motion of the Earth’s interior.

Successful and sustainable, low carbon generation of electricity to meet current and future demands relies on understanding and exploiting many of these natural, physical processes. Atmospheric convection causes winds to drive wind turbines and also generates the ocean waves exploited in wave power devices. The relative motion of the Earth, Moon and Sun causes the ocean tides exploited in tidal barrages and undersea-current driven turbines. Seasonal changes, weather patterns and latitude can all affect the output of solar energy devices as can reflection and absorption of radiation by the materials they are made from. Geothermal energy relies on energy transfers due to radioactive heating of rocks, local volcanism or simply the heat capacity of the soil acting as a thermal store of energy.

Many large-scale electricity generation methods depend on the basic principle of a turbine turning a generator which relies on understanding the principles of electromagnetic induction and factors affecting potential power output and efficiency. Electricity distribution on a large scale, via the National Grid, involves minimising energy dissipation into the surroundings by transmitting electricity at very high potential difference and low current thus reducing thermal transfers of energy within the cables. Domestic uses of electricity involve devices with varying levels of energy efficiency and informed choice of the most efficient appliances and how long they are used for can lead to reductions in an individual’s energy demands, carbon footprint and household bills.

Energy Efficiency

Improving energy efficiency saves individuals money, reduces waste, conserves resources and cuts emissions of greenhouse gases and other pollutants. Discussing personal, financial savings and more immediately obvious environmental impacts can lead to engagement with climate change by an indirect route with valid applications in the physics curriculum. This is also a good opportunity to reinforce accurate vocabulary using the terms energy stores, transfers and pathways as well as the concept of energy dissipation and avoiding terms such as energy saving (https://spark.iop.org/collections/energy-new-curriculum). Examples can be given of more relevant applications of the Sankey Diagram as a tool for accounting for energy transfers in the atmosphere:

By Cmglee – Own work, CC BY-SA 3.0

This could be used to illustrate a more complex example of a Sankey diagram and lead to a discussion of the possible effects of changes to some of the pathways, reinforcing the concept of energy conservation as both sides must remain balanced.

Wind Turbine Example

In a wind turbine, 20% of the energy from the wind is converted to electricity. Lost wind leads to a loss of 30 % of the energy, friction between the wind and the blades of the turbine and the wind leads to a loss of 25% of the energy, and the rest of the energy is lost due to friction in the electric generator.

1. How much energy is lost due to friction in the generator?

2.   Draw a Sankey diagram for the wind turbine, considering that the output in electrical energy is 20 kJ.