Let’s Experiment: Discovering Area and Volume Formulas

# 5. Prisms: Establishing Volume Formulas

Suggested Learning Intentions

- To build our understanding of the relationship between surface area and volume
- To develop the formula to find the volume of prisms

Sample Success Criteria

- I can identify the surface area of a prism
- I can describe the connection between surface area and volume
- I can describe the formula for calculating volume of prisms and explain why this is true for all prisms
- I can make connections between the area formula and volume formula
- I can demonstrate and justify my thinking using a range of manipulatives

This stage of the sequence builds on the knowledge students have developed of the areas of two-dimensional shapes. Students apply their understanding of area to find ways to establish the formula for volume of prisms. This stage aims to draw connections between the formulas, not only in that the formulas are related to the concept of base x height, but the process for developing the formulas is similar.

The activities are adapted from the task, ‘From Area To Volume’ published by Sullivan, P. (2018) and is used with permission.

Present students with the following problem:

An open-top box has been cut from a sheet of card. Draw what the net of the box might look like and write in the dimensions.

Ask students: “If I want to find the surface area of this box, what am I finding out? What will be the total surface area of this box?” Invite students to suggest the meaning of surface area and how they can find the total surface area of the box.

Challenge students to draw and label the dimensions of different open-top boxes that have a surface area of 30 square centimetres.

Invite students to share their strategies with their peers, explaining their reasoning.

**Enable** students by providing an example of an open-top box made from a single sheet of paper to demonstrate what it looks like. Allow students to build the box in the question with grid paper.

**Extend** students by asking them to prove that they have found all possibilities.

**1. From area to volume**

This activity supports students to see the connection between a two-dimensional shape and the corresponding three-dimensional object. It aims to consolidate the calculation of volume from the dimensions. The following activity is adapted from the ‘From Area to Volume’ task published by Sullivan, P. (2018).

Present the following problem:

We can make open-top boxes out of a sheet of paper by cutting squares from the corners and then folding up the sides.

What might be the volume of some open-top boxes that are made from a sheet of paper that’s 200mm by 300mm?

Observe students’ approaches as they are solving the problem. Are they able to see a pattern in the solution from the size of the corners and the volume of the box? Can they see a relationship between the surface area and the volume of the box?

**Enable** students by asking them to make an open-top box from a sheet of square paper by cutting 5 cm squares out from each corner.

**Extend** students by asking, “What is the largest volume of a box that you can make? What is a general way to describe how to find the largest box?”

**2. Establishing a formula for the volume of prisms**

Pose the following problem:

Can you draw three different prisms with the volume of 64 cubic centimetres? What are the dimensions of each one?

Guide students to use what they have learnt about developing area formulas to determine a rule, or formula, that would allow them to quickly find the volume of the rectangular prism. Prompt students by asking questions such as:

- How can you use what you know about the area formula for rectangles to find the volume formula for rectangular prisms? The volume of a rectangular prism can be described as the area of the base x height.
- Will your formula work for rectangular prisms of any size?

The relationships between the formulas for volume are analogous to those for area. Note if students can make connections of these formulas to the single concept of base x height.

Students test their formulas for other prisms and adjust if necessary. Ask students:

- Is your formula true for any type of prisms? What about a triangular prism? It may be useful for students to first note the similarities and differences between rectangular and triangular prisms.
- Will your formula also work for a cylinder?
- How can you test your ideas?

**Enable** students by providing concrete materials for them to build the rectangular prism using cubes with the same size. Demonstrate how to draw a diagram to represent the model they have built.

**Extend** students by challenging them to write the dimensions for various prisms with a volume of 64 cubic centimetres, for example, a triangular-based prism or a trapezium-based prism. Is it possible to create a cylinder of the same volume? What could its surface area be?

**Areas for further exploration**

**1. Cuboid Challenge**

The Cuboid Challenge activity from Nrich Maths gives students the opportunity to apply their knowledge of the area to find the largest volume of a box they can make from a square of paper.

Recall that the learning intention of this stage is to build understanding of the relationship between surface area and volume and to develop a formula for the volume of prisms.

To assess students’ understanding of these concepts, ask students to draw and label the dimensions of two different prisms with a volume of 36 cubic metres. What is the surface area for each one?

**Enable** students by providing cubes to build the rectangular prism and asking them to find out how many squares they can see on each face.

**Extend** students by asking them to draw rectangular and triangular prisms. Students can also be challenged to draw and label a cylinder that is approximately 36 cubic metres.

Reys, R. E. et al., 2020. *Helping Children Learn Mathematics. *Milton: John Wiley & Sons Australia.

Siemon, D. et al., 2015. *Teaching Mathematics: Foundations to Middle Years. *Melbourne: Oxford University Press.

Sullivan, P., 2017. *Challenging Mathematical Tasks. *South Melbourne: Oxford University Press.

University of Cambridge, n.d. *NRICH: Cuboid Challenge. *[Online]

Available at: https://nrich.maths.org/cuboidchallenge

[Accessed 15 March 2022].

Other stages

1. Quadrilaterals: Investigating Areas of Trapeziums

EXPLORESuggested Learning Intentions

- To investigate the area formulas of trapeziums

Sample Success Criteria

- I can describe the similarities and differences between a parallelogram and trapezium
- I can develop the formula for calculating the area of trapeziums
- I can demonstrate and justify my thinking using a range of manipulatives

2. Quadrilaterals: Investigating areas of Kites and Rhombuses

EXPLORESuggested Learning Intentions

- To investigate the area formulas of rhombuses and kites

Sample Success Criteria

- I can describe the similarities and differences between rectangles, parallelograms, kites and rhombuses
- I can develop the formula for calculating the area of kites and rhombuses
- I can use a range of manipulatives to model and explain my thinking

3. Circles: Establishing Pi

EXPLORESuggested Learning Intentions

- To investigate the ratio of circumference to diameter

Sample Success Criteria

- I can describe how the radius, diameter and circumference of a circle are related
- I can demonstrate and justify my thinking using a range of manipulatives
- I can interpret the formula for the circumference of circles

4. Circles: Establishing Area Formula

EXPLORESuggested Learning Intentions

- To visualise and evaluate how the area of a circle is determined

Sample Success Criteria

- I can explain the strategies used to develop the area formula for circles
- I can demonstrate my understanding of how the formula for area of circles is derived using a range of manipulatives