2 Thinking about capacity and volume and their units of measurement

‘Volume’ is the space that a three-dimensional object occupies or contains. Volume can be quantified in different ways depending on its physical state. For example, the volume of a cuboid solid is calculated by measuring the height, breadth and length as the students were attempting to estimate in Activity 1. In this case the volume will be quantified in cm3, m3 or inches3.

Fluid displacement is another way to determine the volume of solids or gas. Fluid displacement involves immersing the object in a fluid. The volume of the object will displace the fluid. Such displacement of the fluid can be measured. In that case it will be expressed in millilitres, litres, fluid ounces, or cups.

The volume of fluids, or a quantity of small loose objects such as grains of rice, can be measured by pouring them into a measuring tool such as a measuring cup (Figure 1).

Figure 1 A domestic measuring cup or jug

Capacity, on the other hand, is the property of a container. It describes how much a container can hold. Confusion can arise from the fact that the measures used for capacity are usually the same as those used for volume.

The next activity aims to let the students understand the conceptual difference between capacity and volume of three-dimensional objects by using two ‘compare and contrast’ techniques. Parts 1 and 2 of the activity use the question ‘Is this always, sometimes or never true?’ to help students become aware of mathematical properties of volume and capacity. Part 3 poses the question ‘What is the same and what is different?’ to achieve the same awareness.

For the students to be able to focus on the samenesses and differences, and not to be caught up in the minutiae of measuring and calculating with precision, some of the examples used are unusual but real. Using such examples also helps to introduce a sense of playfulness into mathematics, as there are many right answers.

Activity 2: Compare and contrast – capacity and volume

This activity works well for students working in small groups or pairs. Do not make the groups too large because then not all students will be able to contribute to the discussion. There are many right answers to these questions and not all students have to agree. As long as their reasoning is based on mathematical properties and on logic, then accept their arguments.

Part 1: Capacity

List the objects on the blackboard. Add some more unusual ones if you want to.

Object Can this object contain a liquid? Is this always, sometimes or never true?
An elephant’s trunk
A beehive
An orange
A bucket
A water tank
A mosquito’s tummy
A lake
A sea
A glass
A coconut

Instruct your students to:

  • Discuss with your classmates whether it is always, sometimes or never true that these objects [written on the blackboard] can contain a liquid? Tell them you will ask for their reasons in five minutes’ time. Then ask the students to help you complete the table on the blackboard.

Discuss with the whole class the reasons for the students’ decisions. Only then add the word ‘capacity’ to make another heading of the column in your table.

Object

Can this object contain a liquid? Is this always, sometimes or never true?

(CAPACITY)

Ranking order of the capacity of the objects
An elephant’s trunk
A beehive
An orange
A bucket
A water tank
A mosquito’s tummy
A lake
A sea
A glass
A coconut

Ask the students to number these objects on the basis of decreasing capacity: number the object with the biggest capacity 1, the object with the second-biggest capacity 2, and so on.

Then discuss with the whole class the reasons for the students’ decisions. A good question to ask here is ‘How do you know this object has the biggest capacity?’ If the students do not think of the role of measurements which they explored in Activity 1, remind them of it. You may want to have a look at the key resource ‘Using questioning to promote thinking [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] ’ to help you prepare for this part of the activity.

Part 2: Volume

Add another column to the table on the blackboard:

Object

Can this object contain a liquid? Is this always, sometimes or never true?

(CAPACITY)

Ranking order of the capacity of the objects This object, without any liquid in it, occupies space. Is this always, sometimes, or never true?
An elephant’s trunk
A beehive
An orange
A bucket
A water tank
A mosquito’s tummy
A lake
A sea
A glass
A coconut

Ask the students to discuss amongst themselves whether it is always, sometimes or never true that these objects [on the blackboard] occupy space when they are empty? Tell them you will be asking for their reasons in five minutes. Then complete the table on the blackboard.

Discuss with the whole class the reasons for the students’ decision. Lastly, add the word ‘volume’ to make a final column, as shown below.

Object

Can this object contain a liquid? Is this always, sometimes or never true?

(CAPACITY)

Ranking order of the capacity of the objects

This object, without any liquid in it, occupies space. Is this always, sometimes, or never true?

(VOLUME)

Ranking order of the volume of the objects
An elephant’s trunk
A beehive
An orange
A bucket
A water tank
A mosquito’s tummy
A lake
A sea
A glass
A coconut

Ask your students to arrange the objects on the basis of decreasing volume. Number the object with the biggest volume 1, the object with the second-biggest volume 2, and so on.

Discuss the reasons for the students’ decisions together.

Part 3: Comparing and contrasting capacity and volume

Ask the students to look again at the completed table. Ask what is the same and what is different for the objects? Do all objects have capacity and volume? If an object has the biggest capacity, does that mean it also has the biggest volume?

Tell them to discuss in their groups and to prepare to share their thoughts with the whole class in five minutes.

Then discuss with the whole class the reasons for the students’ decisions.

Video: Using questioning to promote thinking

Case Study 2: Mrs Meganathan reflects on using Activity 2

I thought this activity would take a whole lesson and it did. I asked the students again to work in groups of four or five.

They loved the examples! Samir, one of the students who always asks questions, asked whether it was a fully grown elephant or a baby one. So I told him he could take either and then grade them accordingly. I think all the groups ended up working it out for both!

I think the activity worked well because we had done Activity 1 in the previous lesson which had made them think about the concepts of capacity and volume. I also noticed that perhaps because the examples such as mosquito’s tummy are so unusual, and actually a bit absurd, the students who normally never speak out in the mathematics lesson, and never put their hand up, were now making suggestions and they made sense as well.

When the groups were reporting on their findings, for about half of the groups I made a point of asking the ‘weaker’ student to report their group’s ideas. I had never done this before, but was impressed with the explanations that were given. I also noticed that their reasoning had become more convincing and sophisticated as we heard more and more explanations from the different groups.

We normally do not work with mathematical activities that have many right answers – they tend to be questions that you get right or wrong. So this was new, both for me and the students. To help them be more open in that kind of thinking I told them what it said in the activity description:

There are many right answers to these questions and not all students have to agree. As long as their reasoning is based on mathematical properties and on logic, then accept their arguments.

I think this helped both me and the students to indeed focus on what were the mathematical properties, and on flaws in logical thinking. I think it is the first time I actually felt confident in understanding the difference between capacity and volume.

Pause for thought

Mrs Meganathan explained how she had chosen some of the ‘weaker’ students to report their group’s ideas. What do you think are the benefits of doing this, and what strategies might you use to ensure that it is a positive experience for the student giving the report?

Now think about how your class got on with the activity and reflect on the following questions:

  • What questions did you use to probe your students’ understanding?
  • Did you modify the task in any way like Mrs Meganathan did?
  • If so, what was your reasoning for this?

1 ‘Compare and contrast’ tasks to learn about mathematical properties

3 Consolidating knowledge