Expected level of development
Australian Curriculum Mathematics V9: AC9M9M01
Numeracy Progression: Understanding units of measurement: P9
At this level, students solve problems involving the surface area and volume of right prisms.
Students should identify the ‘base’ of a prism and recognise that a prism is named according to the shape of its base.
When calculating the surface area, use nets to help students visualise the 2D components of the 3D prism. Although some prisms have specific formulas for calculating the surface area, it is more important for students to understand that surface area is found by finding the area of each side of the prism, then adding these.
Students should understand that the volume of an object is the measure of the space inside a 3D object. Recognising the base of a prism is critical when calculating its volume. The general rule for the volume of prisms is Volume = Area of the base × height. Students must know that the height of a prism will always be perpendicular to its base regardless of the orientation of the prism. Use concrete examples to demonstrate.
This topic connects with other mathematics topics at this year level, such as Pythagoras, ratio and proportion, and circles and cylinders. Finding the surface area and volume of composite solids will be the next stage of exploration and development.
Identifying, understanding and using the correct units of measurement for volume and area allows students to communicate with specificity and with context. Students consolidate their understanding of using and providing units to show that their calculations, formulas and comparisons are correct for different contexts.
Teaching and learning summary:
- Build on students’ understanding of area and volume.
- Introduce the concept of surface area.
- Introduce the formula for a prism – cross-sectional area × height.
- Make connections to composite area.
- Make connections to circles, cylinders and cones.
- Demonstrate the difference between a ‘right’ and an ‘oblique’ prism. Show real examples of both to give context.
- Use concrete examples to demonstrate the uniformity of the cross-section of a prism.
- Ensure students identify, understand and use correct units of measurement for volume and area.
- identify a prism
- use the formula to find the volume of a prism
- use the correct units for volume
- use knowledge to solve problems involving prisms and volume
- use correct units of measurement for volume and area.
Some students may:
- confuse area and volume.
- give the solution with incorrect units.
- find the product of all measurements given, not just those that they need.
- believe that for all prisms you multiply three dimensions, without appreciating that volume is the cross-sectional area × length.
- get confused with the terminology 'length', 'height' and 'width' when shown the same object in different orientations.
- be unable to visualise the unobserved sides of a solid object.
- not partition the solids correctly.
- have difficulty identifying the ‘base’ of a prism; so it is important that students recognise that a prism is named according to the shape of its base.
The Learning from home activities are designed to be used flexibly by teachers, parents and carers, as well as the students themselves. They can be used in a number of ways including to consolidate and extend learning done at school or for home schooling.
- We are learning about the volume and the surface area of a prism.
Why are we learning about this?
Working on this problem will give you a deeper understanding of the relationship between volume and surface area, and how they change as the dimensions of a cuboid (rectangular prism) are altered.
What to do
Surface area of a cuboid
This activity allows you to vary the lengths of the sides of a cuboid and calculate the surface area.
- Open GeoGebra interactive resource Surface area of a cuboid self assessment drill.
- Change the dimensions of the cuboid, varying length, breadth and height. Calculate the volume using the volume formula and compare this to the surface area calculation provided in the interactive resource.
- For three different cuboids, record your results in a table, including dimensions, volume and surface area.
- Analyse your results and write down any generalisations about the relationship between volume and surface area.
Cuboids: changing areas, changing volumes
- Open the resource Changing areas, changing volumes.
- Complete the task that uses your knowledge of area and volume to order cuboids.
- Draw your completed grid that shows the cuboids in the correct order according to the criteria listed. Explain your reasoning for deciding where each cuboid is positioned in the grid.
- Optional: you could use the Surface area of a cuboid self assessment drill interactive resource to help you with this task.
- I can describe a prism using relevant dimensions.
- I can use a formula to find the volume of a prism.
- I can calculate the surface area of a cuboid.
- I can describe the relationship between volume and surface area of a cuboid.
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A collection of evidence-based teaching strategies applicable to this topic. Note we have not included an exhaustive list and acknowledge that some strategies such as differentiation apply to all topics. The selected teaching strategies are suggested as particularly relevant, however you may decide to include other strategies as well.
Concrete, Representational, Abstract (CRA)
The CRA model is a three-phased approach where students move from concrete or virtual manipulatives, to making visual representations and on to using symbolic notation.Go to resource
Explicit teaching is about making the learning intentions and success criteria clear, with the teacher using examples and working though problems, setting relevant learning tasks and checking student understanding and providing feedback.Go to resource
A range of resources to support you to build your student's understanding of these concepts, their skills and procedures. The resources incorporate a variety of teaching strategies.
Under the surface
In this resource, students investigate the difference between surface area and volume using dynamic geometry. There are teacher notes and links to activities.Resource details
Surface area and volume of prisms and cylinders
This resource, part of the SAM-MY project, discusses the volume of a rectangular prism and how the units of volume and capacity are related.Resource details
Volume of a pyramid and a cone
A proof of the formula for the volume of a pyramid and a cone using the ancient Chinese yangma.Resource details
Volume of right prisms
This lesson explores the volume of solids by looking at layering cubic units.Resource details
By the end of Year 9, students can apply formulas to solve problems involving the surface area and volume of right prisms.
This task allows students to demonstrate their knowledge of rectangular and triangular prisms in an engaging context.Resource details
Assessment: Three-dimensional shapes
Use this two-page quiz to assess student understanding of volume and surface area of prisms and cylinders.Resource details