Teaching sequence

Work through this resource material in the following sequence:

10 minutes

Tune In: Maths and renewables

25 minutes

Find Out: Hydropower and the volume of prisms

30 minutes

Take Action: Designing a reservoir

5 minutes

Reflection

Learning intentions and success criteria

Students will:

analyse pumped hydropower as a source of renewable energy by solving problems involving the volume and capacity of right prisms, as well as the circumference and area of circles and cylinders, based on real project data
calculate and compare rates of water flow.

Students can:

estimate the area of an irregular shape using a grid
calculate the volume of right prisms, including cylinders
calculate and compare rates of water flow.

Tune In: Maths and renewables

Curriculum Connections

In this lesson, students investigate how the Kidston Pumped Hydro Energy Storage Project operates by solving problems involving volume, capacity and rates. The Science lessons in this Year 8 unit explore the energy transfers and transformations occurring within the project, while the Design and Technologies lessons get students to design their own turbines, learning how they generate electricity. Cool.org recommends that Maths, Science and Design and Technologies teachers collaborate to deliver these content materials across their respective subjects so students can understand the content from multiple STEM perspectives.

Step 1

Ask students to think of the types of energy production they know and record a list of students’ responses on the board.

Suggested responses: solar, natural gas, coal, hydro, wind, oil, geothermal

Step 2

Ask students to identify which types of energy production are renewable and non-renewable, and invite discussion on what these terms mean.

Suggested Responses: renewable energy sources can be replenished and replaced as quickly as they are used, while non-renewable resources are finite and used much more quickly than replaced. Renewable sources include solar, hydro, wind and geothermal, whilst non-renewable sources include natural gas, coal and oil.

Step 3

Using the Student Worksheet, invite students to brainstorm in pairs how mathematical concepts might be used in designing, constructing and maintaining different types of renewable energy such as solar, wind and hydro. Use the images of each type of renewable energy in the attached Visual Explainer to prompt students' thoughts. Encourage students to think simple! Concepts such as length, area, perimeter, volume, angles, depth, rates and circumference are as important as more complex mathematical concepts. Check the Teacher Content Information for some more suggested responses.

Find Out: Hydropower and the volume of prisms

Step 1

Explain to students that mathematical concepts are important for people working in renewable energy, no matter their role. The remainder of this lesson will focus on using concepts such as area and volume to solve problems relating to hydropower, specifically at the Kidston Hydro project in Queensland.

The Kidston Pumped Storage Hydro Project is a hydropower project situated in Far North Queensland. Watch the video explaining how the Kidston Pumped Storage Hydro Project works. As students watch the video, encourage them to label the image below (also on their Student Worksheet) with the different parts of the pumped storage system and how it works. An annotated diagram can be found in the Teacher Content Information to guide your discussion with students after watching the video.

March 2018 - The Kidston Renewable Energy Hub: Renewable Energy on Tap - Genex Power (https://www.youtube.com/watch?v=egBJHR6xmiw)

Step 2

Before getting into the problem-solving part of this lesson, you might want to review how the volume of a prism is calculated with your class. If we assume the 2 reservoirs in the Kidston Pumped Hydro project are prisms, this formula will help us estimate their depth.

The volume of a prism can be calculated using the formula V = Ah, where A is the cross-sectional area (base) of the prism and h is the height.

Follow the example below:

Check your students’ understanding here. You might like to consider scaffolding them through a few more examples to consolidate their understanding of how to calculate volume of irregular shapes.

Step 3

Distribute 1 copy of the Kidston Pumped Hydro Map to each student. Explain that the Kidston Pumped Storage Hydro project is made up of 2 reservoirs. The upper reservoir is called the ‘Wises Pit’ and the lower reservoir is called the ‘Eldridge Pit’.

Using the formula for the area of a square (A = L²), show students that the area of each grid is 200 m x 200 m = 40,000 m². Students should write this next to their maps for later use.

Step 4

Explain to students that the surface area of the Wises Pit can be estimated using the grid. A common technique for estimating areas using a grid is to count partially covered squares as one half and any fully covered squares as 1. As some partially covered squares will cover more than one-half and some less, they often cancel out, making this an appropriate approximation.

Step 5

Encourage students to use their maps to count the number of partially and fully covered squares in the Wises Pit. Instruct students to calculate the pit's surface area by multiplying the number of squares by 40,000 m².

Suggested answer: 34 partially covered squares and 12 completely covered squares should be counted. This results in (34 x 0.5 + 12) = 29 full squares of area. Calculating the area results in 29 x 40,000 = 1,160,000 m².

Step 6

Instruct students that the volume/capacity of the Wises Pit is 8.5 gigalitres (Source: Kidston Pumped Storage Hydro Project IAR Supplementary Information, p33).

Explain to students that 1 gigalitre of water is equivalent to 1,000,000 m³ of volume. Conduct a worked example to show students how to calculate the depth of the Wises Pit. This may also be written underneath their map for later reference.

Extension opportunity: You may like to scaffold this worked example or pose the calculation of the depth of the pit as a problem-solving activity for students to tackle in groups. Provide students with the formula for volume of a prism and the information that 1 gigalitre of water is equivalent to 1,000,000 m³. Students requiring extension can be tasked with converting 8.5gl to m³.

Suggested answer:

Unit conversion: 8.5 gigalitres = 8,500,000 m³
Volume: V = Ah →8,500,000 = 1,160,000 x h → h = 7.3 m

Step 7

Before continuing to the Take Action part of this lesson, show your students the images of the reservoirs on the attached Visual Explainer. Based on what they observe in the images, discuss the accuracy of their calculations.

Prompt a discussion with your students using the following questions

Do you think the estimate of the depth is accurate? Why/why not?
Where do you see water in the Wise's Pit?
Do you think the depth is the same throughout the whole pit?
What must we take into account when making calculations such as this?

Take Action: Designing a reservoir

Step 1

Explain to students that the lower reservoir, the Eldridge Pit, is described as ‘relatively circular and deep’, and therefore we can model it as a cylinder. To calculate a cylinder's volume, a circle's area must be calculated first. You might like to show your students the formulas for the area of a circle and the volume of a cylinder using the images on the attached Visual Explainer.

Area of a circle A = r²

Volume of a cylinder V = r²h

Step 2

Ask students to estimate the radius of the Elridge Pit using their maps. Possible responses should range between 200m and 300 m. Write responses on the board and discuss how students estimated their radius.

Step 3

Explain to students that they will use the questions on their Student Worksheet to create 2 possible designs for the Eldridge Pit.

The Elridge Pit has a total capacity of 28 gigalitres (28,000,000,000 litres or 28,000,000 m³). Knowing this, their task is to create 2 designs for the Eldridge Pit, assuming that it is cylindrical in shape. The radius should be similar to those estimated by your class. You may like to choose the best estimation or let students use their estimations.

Explain to students that their designs for the Eldridge Pit must be large enough that it can hold a minimum of 28 gigalitres of water. Students should list their chosen radius and height, perform the calculations showing that the volume is greater than or equal to 28 gigalitres, and label the cylinder's dimensions for each design. Students requiring support can ‘guess and check’ or use similar calculations to those used above to work out the depth of the pit algebraically.

Some students may have set the volume at exactly 28 gigalitres and worked backward to find the exact height needed. If any students used this method, it may be beneficial to have them explain their thought process to the class.

Step 4

Once students have created their designs, show students the diagram at the top of page 5 of this report from Genex, the company that is in charge of the Kidston Project, and discuss with students how the shape of the Eldridge Pit compares to their assumption of a cylinder.

Ask students: “Would the actual height of the reservoir be lower or higher than their estimate? Why?”

Suggested response: the shape of the Eldridge Pit narrows as it gets deeper and has an actual depth of 320m. Since we estimated the volume of the cylinder based on the surface area of the top of the pit, which is larger than the surface area of the bottom of the pit, its likely students would have a lower value for the height.

Extension opportunity: Students requiring an extension or early finishers can complete extension questions 6, 7, and 8 on the Student Worksheet regarding rates and water flow in the system.

Reflection

Reflect on the lesson with your students using the following question.

One of the most common questions in high school maths is, ‘When will I use this?’

Imagine you are a teacher and your students ask, ‘When am I ever going to use area, volume and rates to make a difference in real life?’ How do you respond?

You might like to provide the link to the Future Energy Jobs Guide for your students to find concrete examples of professions and careers that use this knowledge in renewable energy on a daily basis.

Volume and Capacity in Renewable Hydropower

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Lesson Guides and Printables

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Volume and Capacity in Renewable Hydropower

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