## 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.

## Physics – Egypt’s Benban Solar Farm

##### In this resource linked to COP27 in Egypt, physics students explore renewable energy production.

Learning Objectives

• Recognise that solar power is a renewable energy source of great value in Egypt
• Describe the energy transfer in a solar cell
• Evaluate the energy dissipated in the Benban solar farm
• Calculate the cost of the energy produced using the formula cost = power (kW ) x time (hours) x price (per kWh).

## Motivation/Outline

In its acceptance speech at COP26, Egypt celebrated its renewable energy resources:

This is an extract from https://unfccc-cop26.streamworld.de/webcast/closing-plenary-of-the-cop-followed-by-cmp-and-c-2 from 09:20

Egypt transitioned from the traditional energy sources to renewable, more sustainable and planet-friendly energy sources…

One of these resources is the huge Benban solar farm.

## Lesson Introduction

Watch the relevant part of the COP26 plenary video and/ or

• The Benban solar farm was supported by the Green Climate Fund. Contributions to the Green Climate Fund were one of the areas which didn’t make as much progress as was hoped at COP26 in Glasgow, 2021.
• COP27 will be at Sharm El-Sheikh in Egypt in November 2022.

## Discussion points:

• What is a renewable energy source?
• Why is it important to develop renewable energy sources?
• What is a solar cell and how is it different from a solar panel? Where have people seen solar cells/ panels?
• What makes a location suitable for a huge solar energy farm? (space, sunshine, access for bringing the equipment in and getting the electricity out…)
• Could we build such a huge solar park in the UK? (no, we don’t have a big desert, but you could research some UK solar farms)
1. Use https://globalsolaratlas.info/map to compare the global horizontal irradiation where you live with that in Benban. (for Benban the value is given as 2366 kWh/m2).
Global horizontal irradiation is the total amount of solar energy reaching a 1m2 horizontal surface on the ground in a year.

Discussion point: What is a kWh? (if 1 kWh is the electrical energy converted by a 1 kW appliance used for 1 hour rephrase this in terms of electrical energy generation. See https://www.bbc.co.uk/bitesize/guides/z2h4dxs/revision/1 for more detail)

Discussion point: So what is a kWh/ m2?

Extension: Express this answer as a proportion or percentage

2. Discuss: what is the initial store of energy and by what pathways is it transferred? (nuclear store in the Sun, energy is transferred by light from the Sun to the panel and is transferred electrically from the panel to homes and businesses)
3. The size of the Benban solar farm is 37.2 km2. Calculate the total energy carried by the light arriving at the site.

(37.2km2 = 37 200 000m2 so 2366 x 37 200 000 = 88,015,200,000 kWh = 88 015.2 GWh = 88.0TWh)

Discuss: kilo, mega, giga, Tera etc.

4. The estimated output from Benban is 3.8TWh. How much energy is not converted usefully?
88.0-3.8 = 84.2TWh

Extension – write this as a proportion or percentage
Discussion – why so much? Solar panels don’t cover the whole of the ground, solar panels are actually less efficient when they get hot, you can see solar panels, so they must be reflecting some of the Sun’s light, not absorbing it all etc.)

5. What is the current electricity price in your region? (see https://www.ukpower.co.uk/home_energy/tariffs-per-unit-kwh and scroll down for regional breakdown).
What is the value of the energy the Benban solar farm will produce during COP27, which is scheduled to last 2 weeks (assume there are 52 weeks in a year)?

(cost = power (kW ) x time (hours) x price (per kWh).
So value = 3, 800, 000, 000 kWh x 2/52 x 28.34 = £41,420,000.

Discussion – is that surprising?

Why might the quantity of electricity produced actually be different? (We started with an annual value, but the seasons and the weather will actually have an impact on how much is produced in a given week).

## 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 Solar Sine Curve

You are given the equation

$f\left( x \right) = 5\cos\theta – 8\sin\theta$

a) Express f(x) in the form $$R\cos{(\theta + \alpha})$$ where $$R > 0$$ and$$\ 0 < \alpha < \pi$$. Write R in surd form and give the value of α correct to 4 decimal places.

[4 marks]

The temperature of a solar panel, T ˚C, can be modelled by the equation

$T = 20 + 5\cos\frac{4x}{15} – 8\sin\frac{4x}{15}\ ,\ 0 \leq x \leq 72$

Where x is the time in hours since 10pm one evening?

b) Calculate the maximum value of T predicted by this model and the value of x, to 2 decimal places, when this value first occurs.

[4 marks]

c) Calculate the times during the first 24 hours when the temperature is predicted, by this model, to be exactly 17 ˚C

[4 marks]

## Warming up a Solar Cell

The temperature T˚C of a solar cell during a 24 hour period is modelled as

$T = 20 – k\left( 15 – \frac{5t}{4} \right)^{2}\ ,\ \ \ 0 \leq t \leq 24$

Where t is the time in hours after midnight and k is a positive constant

The temperature of the solar cell at midnight is 5˚C.

a) Use this information to find the value of k in the model.

[2 marks]

b) Find, according to the model, the temperature of the solar cell at 8:30 am.

[2 mark]

c) Determine the greatest temperature of the solar cell and the time at which this temperature occurs.

[3 marks]

d) State one limitation of the model

[1 mark]

## Solar Panelling a House

A homeowner wishes to cover their roof with solar panels.

Their roof can be modelled as a prism with volume 24m3

The height of the triangular cross section is h.

If solar panels can only be placed on the 2 rectangular sections of the roof

a) Work out the area of roof that could be covered by solar panels. Give your answer to 2.s.f.

[4 marks]

A company provides solar panels that are 1.5m long and 1.0m high and cost £200 each, including installation.

b) If this is the only size of solar panels available, how much will it cost the homeowner to buy and install them?

[3 marks]