Mathematics Grade 10
Teacher Guide
Volume
Series overview
We aim to give learners a deeper understanding of the concept of volume and how to estimate,
measure and calculate the volume of right prisms. We explain the terminology needed for working
with prisms. Then we explore the concept of volume by looking at the amount of space that is taken
up by different objects. We use one formula, and apply it to different right prisms and to a cylinder.
The first seven lessons consolidate an understanding of volume of right prisms and their formulae that
learners may have worked with before. Then we use these tools to address the Grade 10 Assessment
Standard in the last two lessons. We estimate, calculate and then generalise the changes to the
volume of a right prism when its dimensions are changed.
Note that in the video lessons, we use 3,143 for pi and we round off the answers at different steps
of the calculation. Your learners will get different answers if they use pi on their calculators and only
round off at the end.
Curriculum links
The lessons in this series link to the following Learning Outcomes and Assessment Standards of the
National Curriculum Statement:
Learning Outcome 3: Space, Shape and Measurement
10.3.1 Understand and determine the effect on the volume of right prisms and cylinders, of multiplying
any dimension by a constant factor k.
Learning Outcome 1: Number and number relationships
10.1.3: Investigate number patterns to make conjectures and generalisations.
This means that by the end of the series, learners should be able to take a right prism or cylinder and
work out its volume. They should also be able to change the dimensions of the prism and calculate
the new volume of the prism.
E
Educational approach
There are 9 lessons in this series. In the first three lessons we ensure that your learners have the
basic concepts in place for the rest of the series. We identify the properties of a right prism. We
demonstrate a way to view 2-dimensional representations of 3-dimensional prisms and we discuss
the concept of volume and the units of measurement used.
In lessons 4 and 5, we use a simple approach to establish the formula for the volume of a right
rectangular prism. Then we use the formula in some calculations. In lessons 6 and 7, we find the
volume of a triangular prism, a trapezoidal prism and a cylinder. We use the same formula for all
these solids:
Volume = area of the base of the prism x perpendicular height of the prism.
In the last two lessons, we investigate what happens to the volume of a right prism when we change
the dimensions of the prism.
To make the content more accessible for your learners, we use common objects such as matchboxes,
cans and wooden cubes. We encourage learners to become active in their own learning and do their
own investigations. For example, in the eighth lesson, we use pizza boxes to show how the volume of
a shape can be changed when we double one of its dimensions. In the classroom, learners can place
one textbook on top of other similar textbooks and then discuss what effect the increased height of
the pile of textbooks has on the total volume of the pile of textbooks.
To get the full benefit of the lessons, your learners need to engage actively with the concepts
presented. So, when you preview the videos, think about how to introduce each lesson and what
follow up activities will be useful. Also watch out for places in the video where you can pause to have
a class discussion, or ask learners to complete an activity or solve a problem posed in the video. We
have used this pause icon
to suggest some of these places to you.
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Mathematics Grade 10
Teacher Guide
Volume
Series at a glance
Lesson title
Lesson Outcomes
By the end of this lesson, the learner should be able to:
1. R
ight Rectangular
Prisms
• identify a right rectangular prism
2. R
ight Rectangular
Prisms in 2-D
• identify right rectangular prisms from different positions
represented in two dimensions
• use dotted lines to show the hidden parts of prisms
3. T
he Concept of Volume
• explain volume and its unit of measurement in mathematical
terms
4. T
he Formula for
Volume
• calculate the volume of a right rectangular prism
5. C
alculating Volume
• find the volume of a more complicated structure
6. V
olume of Different
Right Prisms
• apply a formula to work out the volume of:
• a right rectangular prism
• a right triangular prism
• a right trapezoidal prism
7. V
olume of Cylinders
• calculate the volume of a right cylinder
• determine the volume of prisms with cylindrical holes in them.
8. D
ouble the Dimensions
• investigate how doubling the dimensions of a prism changes
the volume of the prism
• estimate and calculate the volume of a rectangular prism in
which all its dimensions are doubled
9. C
hanging the
Dimensions by k
• investigate how increasing the dimensions of a prism changes
the volume of the prism
• find a rule to describe the effect of multiplying the dimensions
of a prism by k
• calculate the changed volumes of right prisms.
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Mathematics Grade 10
Teacher Guide
Volume
G
Teaching guidelines
Preparing for your lessons: You and your learners will need:
• matchboxes
• a range of other right prisms
• cylinders, e.g. cool drink cans
• paper and pencils
• calculators
Lesson 1: Right rectangular prisms
Before you show the first lesson, ask your class if they can define a right rectangular prism. Then they
can compare their answers to the definition given in the video lesson. Ask your learners to identify
the faces, bases and edges of a variety of objects that are right prisms, for example textbooks and
cupboards in the classroom.
Task
Learners need to identify which of some common objects in the kitchen are right rectangular prisms.
They can also identify other examples. This task will show whether learners understand the definition
of a right rectangular prism.
Lesson 2: Right rectangular prisms in 2-D
Some learners might struggle with the concepts that are introduced in this lesson, so try to have
matchboxes or textbooks available for them to try out the activities. To demonstrate the different
views of an object, allow each learner to experiment with a right rectangular prism, such as a book
or matchbox. They need to hold it in one position and see how their view of it changes as they move
their eye position. It might take some time to start to see the different views clearly. Learners need to
realise that the faces of a prism that they see depend on their view point. If they are looking at one
face of a prism directly, they will see only one face. If they move the object to a different position, they
will see more faces of the prism.
Task
Learners put a matchbox in the position described in the video and then sketch it. Let learners work
in pairs and assess each other’s drawings.
Lesson 3: The concept of volume
We investigate the concept of volume as the amount of space that an object takes up. We show how
various substances have the same volume even though they have different masses. In one of the
activities, we show that mealie seeds displace water in a container. If you do this activity in class, make
sure that you use a material that does not dissolve in the water.
Make sure that learners can use the terms ‘mass’ and ‘volume’ correctly. We use a unit cube as a
standard unit for measuring volume and we can work out the volume of an object by working out how
many unit cubes it contains.
Task
Learners look at four objects made of unit cubes and decide which of the objects have the same
volume. Ask your learners to explain why they think the specific shapes have the same volume.
Ensure that they use the correct terminology.
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Mathematics Grade 10
Teacher Guide
Volume
Lesson 4: The formula for volume
We now use the unit cubes that we defined in lesson 3. We establish the formula for the volume of a
right rectangular prism. We do this by counting the number of blocks in one layer and then counting
the total number of layers. This is an important step towards understanding the volume formula and it
is worth spending some time on this.
Task
Learners calculate the volume of an object made of two prisms. They can split the object as shown
on the video and then add the two volumes together. This task leads to the next lesson.
Lesson 5: Calculating volume
We use the volume formula to find the volume of more complicated structures made of combinations
of right rectangular prisms. We use actual measurements and units for the sides of the prisms.
Remind learners about how to substitute values into a formula. We name prisms according to the
two bases, for example, IJFE.SKGP. Calculations with complex prisms need to use clear accurate
labelling of the parts of the prism.
Task
Learners find the volume of a complex structure made of two right rectangular prisms. This task
assesses the skills learnt in the lesson.
Lesson 6: Volume of different right prisms
Before starting this lesson, check that learners can identify right prisms with bases of different shapes.
For example, if the base is a triangle, the shape is a right triangular prism. We are now ready to use
the formula for volume on any right prism. We regard the height of the prism as the number of layers
or rows of cubic units it contains.
Revise the formulae for the area of a triangle and the area of a trapezium with learners.
Using the words ‘base’ and ‘height’ in different contexts could be confusing. Discuss this with the
class: The base of a 3-D triangular prism is a whole 2-D triangle, while the base of the 2-D triangle is
a side or a length of it. Use drawings and physical objects to demonstrate this.
Task
Learners are now ready to calculate the volume of any right prism. They need to work out the volume
of wood in a right rectangular prism with a right triangular prism cut out of it.
Lesson 7: Volume of cylinders
Ask your learners to discuss ways of finding the formula for the volume of a cylinder and then ask
them to write it down in mathematical terms. You may need to remind them of the formula for the area
of a circle.
Once learners can find the volume of a cylinder, see if they can work out the volume of paper on a
toilet roll. If possible, let learners work in groups and measure actual rolls of toilet paper. This will help
them to understand that they cannot include the volume of the cylindrical hole through the middle of
the roll.
Before showing the second example, this might be a good time to stop the video. Discuss the next
task with the class. They need to work out the volume of a right rectangular prism with a cylindrical
hole through it in terms of algebraic symbols. Think about your class’s ability in algebra. A stronger
group will easily work with unknowns, while another group might need to be reminded how to multiply
unknowns and be shown how to factorise.
Task
Learners need to calculate and compare the volume of two cylindrical water tanks, using the formula
for volume. They are given measurements in centimetres and metres, so they need to convert the
units before doing the calculations. Remind learners to use the correct units for volume. As water is
usually measured in litres, their answer should be given in litres.
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Mathematics Grade 10
Teacher Guide
Volume
Lesson 8: Double the dimensions
In lessons 8 and 9, we change the dimensions of prisms. In this lesson, we look at how the volume
can be changed by doubling one or more of the dimensions of a prism. Learners can do this at their
desks, using objects such as textbooks or matchboxes. They can increase the height of a pile of
textbooks, and discuss what happens to the volume of the pile. This activity needs objects that are the
same size and shape as each other.
We then look at doubling ALL the dimensions or lengths of a shape. This is a good time to stop
the video and let learners suggest ways to estimate what the volume of the new shape will be. It is
important for them to be able to estimate and also to describe the changes in words. If we double all
the dimensions of an object, the volume of the new object will be 8 times more than the volume of the
original object.
Task
Learners draw their own right prisms, and investigate the change in the volumes when they double
all the dimensions. Allow the learners to discuss and compare their answers.
Lesson 9: Changing the dimensions by k
We investigate what happens to the volume of a cube when we multiply its dimensions by different
numbers. Before starting the lesson, let each learner set up a table as shown in the video to compare
the values. We look at what happens in general when the dimensions of a prism are changed. You
might choose to show your class this video lesson a second time, as the concepts are challenging. In
general, if the dimensions of a right prism are multiplied by k, the volume is multiplied by k3.
You will need to extend the concepts established here to other right prisms and cylinders. We have
only used cubes, rectangular prisms and triangular prisms. Let your learners make and investigate
conjectures about cylinders, or make conjectures about only one dimension becoming smaller or
larger, etc. There is plenty of possibility for classroom activities in this topic.
Task
This task can serve as summative assessment for the entire series. Ask learners to do the task for
their portfolios.
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