## 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.

## IPCC 2021 – Comparing Arctic and Global temperatures – using Excel

According to the IPCC report for Policymakers “It is very likely that the Arctic has warmed at more than twice the global rate over the past 50 years1.

You are going to test this statement to see if it is true.

### Resources

Arctic and global temperatures data spreadsheet

Tutorial: Using Formula in Excel

Tutorial: Creating Line Graphs in Excel

1. Contrast the results of your averages and the range for global air temperatures and those in the Arctic
2. Using the Change over time value in your table consider oif the statement “It is very likely that the Arctic has warmed at more than twice the global rate over the past 50 years” is true.
1. Complete the graph above which shows data on Global and Arctic temperature change from 1900 to 2020;
• Add a title to the graph
• Draw a curved line of best fit between the data shown for the start of each decade for the Global data
• Draw a curved line of best fit between the data shown for the start of each decade for the Arctic data
• Try to predict the future! Continue your line of best fit for both Global and Arctic lines on until 2100. To do so follow the recent tend and try to project that into the future.
• What could change the future? Think about government policies relating to climate change and the future.

### Why is the Arctic warming faster that the rest of the globe?

Place the following information into a logical sequence to explain why the Arctic is warming faster that the global average:

### Sources

1. IPCC, 2021: Climate Change 2021: The Physical Science Basis. Contribution of Working Group I to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change [Masson-Delmotte, V., P. Zhai, A. Pirani, S.L. Connors, C. Péan, S. Berger, N. Caud, Y. Chen, L. Goldfarb, M.I. Gomis, M. Huang, K. Leitzell, E. Lonnoy, J.B.R. Matthews, T.K. Maycock, T. Waterfield, O. Yelekçi, R. Yu, and B. Zhou (eds.)]. Cambridge University Press. In Press. P.3461. Accessed 28th November 2021 at Sixth Assessment Report (ipcc.ch)
2. Ecochard, K., 2021. NASA – What’s causing the poles to warm faster than the rest of Earth?. [online] Nasa.gov. Available at: https://www.nasa.gov/topics/earth/features/warmingpoles.html Accessed 29 November 2021.
3. The annual mean global and Arctic  temperature time series are provided by Dr. Muyin Wang. Values are the weighted average of all the non-missing, grid-box anomalies plus the absolute temperature. They are based on the monthly  global gridded data (5×5 grid box ) and the absolute temperature,   that has been developed by the Climatic Research Unit (University of East Angliaand NCAS) jointly with the Hadley Centre (UK Met Office).

## IPCC 2021 – Comparing Arctic and Global temperatures

According to the IPCC report for Policymakers “It is very likely that the Arctic has warmed at more than twice the global rate over the past 50 years1.

You are going to test this statement to see if it is true.

1. Contrast the results of your averages and the range for global air temperatures and those in the Arctic
2. Using the Change over time value in your table consider oif the statement “It is very likely that the Arctic has warmed at more than twice the global rate over the past 50 years” is true.
1. Complete the graph above which shows data on Global and Arctic temperature change from 1900 to 2020;
• Add a title to the graph
• Draw a curved line of best fit between the data shown for the start of each decade for the Global data
• Draw a curved line of best fit between the data shown for the start of each decade for the Arctic data
• Try to predict the future! Continue your line of best fit for both Global and Arctic lines on until 2100. To do so follow the recent tend and try to project that into the future.
• What could change the future? Think about government policies relating to climate change and the future.

### Why is the Arctic warming faster that the rest of the globe?

Place the following information into a logical sequence to explain why the Arctic is warming faster that the global average:

### Sources

1. IPCC, 2021: Climate Change 2021: The Physical Science Basis. Contribution of Working Group I to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change [Masson-Delmotte, V., P. Zhai, A. Pirani, S.L. Connors, C. Péan, S. Berger, N. Caud, Y. Chen, L. Goldfarb, M.I. Gomis, M. Huang, K. Leitzell, E. Lonnoy, J.B.R. Matthews, T.K. Maycock, T. Waterfield, O. Yelekçi, R. Yu, and B. Zhou (eds.)]. Cambridge University Press. In Press. P.3461. Accessed 28th November 2021 at Sixth Assessment Report (ipcc.ch)
2. Ecochard, K., 2021. NASA – What’s causing the poles to warm faster than the rest of Earth?. [online] Nasa.gov. Available at: https://www.nasa.gov/topics/earth/features/warmingpoles.html Accessed 29 November 2021.
3. The annual mean global and Arctic  temperature time series are provided by Dr. Muyin Wang. Values are the weighted average of all the non-missing, grid-box anomalies plus the absolute temperature. They are based on the monthly  global gridded data (5×5 grid box ) and the absolute temperature,   that has been developed by the Climatic Research Unit (University of East Angliaand NCAS) jointly with the Hadley Centre (UK Met Office).

## IPCC 2021 – Impacts of Polar Climate Change

According to the IPCC report for Policymakers “Human influence is very likely the main driver of the global retreat of glaciers since the 1990s and the decrease in Arctic sea ice area between 1979–1988 and 2010–2019 (about 40% in September and about 10% in March). There has been no significant trend in Antarctic sea ice area from 1979 to 2020 due to regionally opposing trends and large internal variability. Human influence very likely contributed to the decrease in Northern Hemisphere spring snow cover since 1950. It is very likely that human influence has contributed to the observed surface melting of the Greenland Ice Sheet over the past two decades, but there is only limited evidence, with medium agreement, of human influence on the Antarctic Ice Sheet mass loss.”1

## The Impacts of climate change in Polar regions

Read page 2 of the polar regions fact sheet from the IPCC2.

Produce a spider diagram of all the suggested impacts that are predicted to occur in the Arctic and the Antarctic. Around those impacts suggest some of the negative consequences for humankind.

Sources

1. IPCC, 2021: Climate Change 2021: The Physical Science Basis. Contribution of Working Group I to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change [Masson-Delmotte, V., P. Zhai, A. Pirani, S.L. Connors, C. Péan, S. Berger, N. Caud, Y. Chen, L. Goldfarb, M.I. Gomis, M. Huang, K. Leitzell, E. Lonnoy, J.B.R. Matthews, T.K. Maycock, T. Waterfield, O. Yelekçi, R. Yu, and B. Zhou (eds.)]. Cambridge University Press. In Press. P.10. Accessed 28th November 2021 at Sixth Assessment Report (ipcc.ch)
2. IPCC.ch. 2021. Regional fact sheet – Polar Regions. [online] Available at: https://www.ipcc.ch/report/ar6/wg1/downloads/factsheets/IPCC_AR6_WGI_Regional_Fact_Sheet_Polar_regions.pdf [Accessed 28 November 2021].

## IPCC 2021 – Changing Polar Climate

According to the IPCC report for Policymakers “Human influence is very likely the main driver of the global retreat of glaciers since the 1990s and the decrease in Arctic sea ice area between 1979–1988 and 2010–2019 (about 40% in September and about 10% in March). There has been no significant trend in Antarctic sea ice area from 1979 to 2020 due to regionally opposing trends and large internal variability. Human influence very likely contributed to the decrease in Northern Hemisphere spring snow cover since 1950. It is very likely that human influence has contributed to the observed surface melting of the Greenland Ice Sheet over the past two decades, but there is only limited evidence, with medium agreement, of human influence on the Antarctic Ice Sheet mass loss.”1

The Arctic is largely sea ice with some parts of countries intruding

Sea ice typically covers about 14 to 16 million km2 in late winter in the Arctic

https://www.nasa.gov/audience/forstudents/k-4/stories/nasa-knows/what-is-antarctica-k4.html

Antarctica is a CONTINENT with land mass under the ice, mountains, and volcanoes.

Antarctica is 14 million km2 PLUS 17 to 20 million km2 of sea ice in the Antarctic Southern Ocean in winter.

1. Using your knowledge of global warming explain why the four changes described above have occurred.

*Losing mass – this means that more ice melts than is gained via snowfall in a given year.

1. Change two above suggests that there will be more precipitation (snowfall and rainfall) in Arctic and Antarctic regions. Complete the futures exercise below to consider what this change could mean for those regions.
• What possible futures are there for the Arctic and Antarctic with increased precipitation?
• What probable futures are there for the Arctic and Antarctic with increased precipitation?
• What preferable futures exist for these regions? Here you might want to consider the key role these regions play in regulating our global climate.

Changes to the Greenland and Antarctic ice sheet.

The graphic below shows Greenland Ice Sheet cumulative mass changes in gigatonnes (recently observed and projected by models under SSP1-2.6 and SSP5-8.5 scenarios) and equivalent sea level change (in metres). Maps show recent elevation changes (metres/year).

• SSP1-2.6: Global CO2 emissions are cut severely, but not as fast, reaching net-zero after 2050. Temperatures stabilize around 1.8°C higher by the end of the century.
• SSP5-8.5: Current CO2 emissions levels roughly double by 2050. The global economy grows quickly, but this growth is fuelled by exploiting fossil fuels and energy-intensive lifestyles. By 2100, the average global temperature is a scorching 4.4°C higher.
Source: IPCC2
1. Using the graphs on ice mass change complete the table below.

SIMPLE TABLE

A gigaton is a billion tonnes. It is roughly equivalent to the mass all of the living mammals other than humans on earth.

1. Which of the two areas, Greenland or Antarctica, will be most affected by climate change in 2100? Use data from the two graphs and maps to justify your view.

### Projected climate changes in the Polar regions

Look carefully at the graphic below, it shows projected changes in the future (SSP5-8.5 scenario) in mean annual temperature and total precipitation at 2°C global warming compared to 1850–1900 for the Arctic (left) and Antarctic (right).

Answer the questions below:

#### Temperature

1. What is the projected change for Greenland (G)?
2. What is the projected change for Lapland (L)?
3. What is the projected change for Siberia (S)?
4. Which parts of the Arctic will suffer the most from temperature changes under the IPCC’s most extreme climate change scenario?
5. What is the projected temperature change for the Antarctic Peninsula (P)?
6. What is the projected temperature change at Vostok (V)?

#### Precipitation

1. What is the projected change for Greenland?
2. What is the projected change for Lapland?
3. What is the projected change for Siberia?
4. Which parts of the Arctic will suffer the most from precipitation changes under the IPCC’s most extreme climate change scenario?
5. What is the projected precipitation change for the Antarctic peninsula?
6. What is the projected precipitation change at Vostok?
7. Which areas are likely to suffer the most change, the oceans or the land masses?
8. Conduct research online, why is it significant for ice melting that the oceans warm? (e.g. https://www.worldwildlife.org/pages/why-are-glaciers-and-sea-ice-melting)

### Extension

Geographic Information Systems (GIS) – run the simulation at this website https://sealevel.nasa.gov/vesl/web/sea-level/slr-antarctica/.
Describe the changes that take place to try to explain them.

### Sources

• IPCC, 2021: Climate Change 2021: The Physical Science Basis. Contribution of Working Group I to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change [Masson-Delmotte, V., P. Zhai, A. Pirani, S.L. Connors, C. Péan, S. Berger, N. Caud, Y. Chen, L. Goldfarb, M.I. Gomis, M. Huang, K. Leitzell, E. Lonnoy, J.B.R. Matthews, T.K. Maycock, T. Waterfield, O. Yelekçi, R. Yu, and B. Zhou (eds.)]. Cambridge University Press. In Press. P.10. Accessed 28th November 2021 at Sixth Assessment Report (ipcc.ch)
• ch. 2021. Regional fact sheet – Polar Regions. [online] Available at: https://www.ipcc.ch/report/ar6/wg1/downloads/factsheets/IPCC_AR6_WGI_Regional_Fact_Sheet_Polar_regions.pdf [Accessed 28 November 2021].

## Arctic Ice Melting

The increasing global temperature due to human-induced climate change is causing ice in the Arctic to melt, particularly over the summer season, July to September.

In September 1999, there was 6.2 million km2 of Arctic sea ice.

In September 2015, there was 4.6 million km2 of Arctic sea ice.

Work out the percentage decrease in Arctic sea ice from September 1999 to September 2015.

[3 marks]