For primary students, Volume & Capacity is one of the most misunderstood substrands in Mathematics. As part of the Measurement strand, you might spend in total about 10 hours a year (say 2 hours per term) thinking about these issues. In 7 years of primary school, for a student that’s about 70 hours contact all up. In surveys I have conducted over the last 40 years, teachers freely admit that Volume & Capacity is often bottom of the list when they think about what and how to teach in mathematics. And as a mathematics consultant, one of the most common questions from individual teachers would be ” What’s the difference between volume and capacity?” so any blockages are not limited just to the students.

In fact student blockages can be quite profound. For a start the word volume is commonly linked to sound on a computer, TV or radio. That sort of volume makes sense to students. Here’s what some students write when asked to brainstorm what they already know about volume.

Students also understand volume 1 and 2 in a book such as an encyclopedia. So we need to acknowledge this meaning of volume whenever we start a class discussion about it. But what about volume as a mathematical concept in primary school? Students can go for years without experiencing sufficient volume activities and discussions that help them unravel their misunderstandings. My advice is always to create a block of time rather than ad hoc lessons here and there. If you can only spend 10 hours a year thinking about Volume & Capacity, then two 1-week programs (e.g. Term 1, Term 3) where you can tackle any misunderstandings head on will be the most effective.

Start by asking your students to talk about and/or record what they know. This student thinks volume is all about capacity.

The bowl, jug, can of spray, bath and rubbish bin can all be filled with something. But the desk, computer and book are seen as objects that can’t hold anything else inside them. If you don’t know about this misunderstanding, it would be difficult to unblock this student.

The second student agrees and even records their thinking with words.

Notice how capacity is the dominant concept. There is no consideration that the ruler is made of wood that in itself has a volume. Previous experiences must have focussed on filling empty things up.

This third student tells us what many students think – if there is no capacity then there is no volume.

So, thinking about the large red capsicum at the top of this article, does a capsicum have volume? Many students of course will say DEFINITELY NOT as you don’t put anything in a capsicum, unless you are from a Greek family with a granny who cooks delicious stuffed capsicum of course!

How can you define volume? Volume is a measurement of the amount of space an object or substance takes up. You and I take up a specific amount of space, so does every 3D object around us. Even our earth has volume as it takes up space within space. If you think of that red capsicum, it takes up the same space as say a grapefruit, even though they have different shapes. It takes up more space than an egg, less space than a watermelon. It doesn’t matter if it is hollow or solid. It doesn’t matter whether the outside skin is paper thin or really thick.The volume of the capsicum remains the same. It can be measured in units of volume such as cubes, for example. There are several other ways to measure a capsicum. Mass would measure how heavy the capsicum feels and would be measured in units of mass such as grams. Surface area would tell you how much skin area there is in total and would be measured using units of area such as square centimetres. Length or height or width will tell you different length information measured in units of length such as cm or mm. Time would be a measurement of how long it took to grow from a tiny bud to this size. But, if you squash this capsicum flat, the volume will change. It will become smaller. It will take up less space. That’s because there is also a volume of empty space held inside the capsicum that would not be contained there if it was squashed. So there is actually a specific volume of capsicum flesh that is different to the volume of the capsicum as a whole. Measurement outcomes depend on what it is you want to find out. Do you want to know how much space this whole capsicum takes up, or what volume of flesh there is? If you are packing capsicums into a box, then the whole volume matters. If you are chopping capsicums up to make soup then just the volume of the flesh matters.

Think of a balloon. You know that when it is not blown up it is just a small volume of rubbery substance. This might have the same volume as one cubic centimetre, for example. But if you add air and blow it up quite large and tie it off, the volume of the balloon changes. We now think of the volume as the space taken up by this inflated balloon, not just the small amount of rubbery substance on the outside. The volume of this balloon now includes the volume of the air inside it. So what is it you want to find out? Do you want to know how much space an inflated balloon takes up? For example, you may want to fit 20 inflated balloons in your car, to take to a surprise birthday party. Will they fit? But if you are a balloon manufacturer, you want to know how much volume is in the rubbery stuff that the balloon itself is made from. For example, if it takes 1 cubic centimetre of stuff to make one balloon, you will need 10 cubic centimetres to make 10 balloons and 100 cubic centimetres to make 100 balloons. The answer to “what’s the volume?” depends on which volume you have as your focus.

To help your students understand volume better, you need to brainstorm lots of objects like this until they “get it”. Volume is NOT the same as capacity. I will talk about that later. All 3D objects and substances have volume. And the funny thing is that when the student says a piece of paper has no volume, they need to investigate this further. They need to unblock their thinking. A piece of paper looks like a 2D object but really it is an extremely flat 3D rectangular prism. But it still has height, width and breadth. And that paper 3D rectangular prism definitely has volume. If you are a paper manufacturer you know that you need to order large quantities of pulp and stuff before you can create your thin paper sheets. That stuff is the volume. It may be only a tiny fraction of a cubic centimetre for each sheet but if you scrunch up that one piece of paper you will notice that it creates a more obvious visible 3D object.

Students associate volume with water. For example,we can estimate how many millilitres of water match the volume of our red capsicum and then place it in a tub of water. The amount of space the capsicum takes up will push that water somewhere else. If you place the capsicum into a completely full container, sitting in a larger empty bowl, you can measure the amount of water that overflows. This is the displacement. This displaced water can then be measured informally by marking where it comes up to in a container or formally using a container marked in millilitres. The volume of this displaced water matches the actual volume of the capsicum. You can check this measure against your estimate. The volume of this water looks nothing like the shape of the capsicum even though the volumes match. After repeating this guess and check process with different objects each time, your students will become experts at volume estimation. They will realise that objects with the same volume can differ greatly in their shape.

Or instead of using water to help us measure volume, we can estimate how many one centimetre cubes will match a 3D object. We can build a model that matches the size and shape of the red capsicum. Once we think we have a match we can then count up how many cubic centimetres we needed. This will approximate the volume of the red capsicum too. And we can rearrange these same cubes to create any shape we want. All the shapes we create will have the same volume.

Notice it doesn’t matter whether we measure with water or blocks. Both measures are measures of volume. Volume isn’t just about water or liquids.

But now we need to ask “What is capacity?” and “How does this differ from Volume?”. To answer this we need to go back to our red capsicum. The way it is created (apart from the Greek granny stuffing it with rice …) it doesn’t usually contain anything else, other than air or hollow space inside it. It has a volume, but no capacity to hold another volume inside it.

Now think about this cup.

It is made from plastic. The manufacturer (probably in China) had to order in a whole bag of materials that were melted together perhaps in a mould to create this object. Let’s say the manufacturer uses about 25 cubic centimetres of plastic to make one cup. That means the volume of this cup is 25 cubic centimetres. But because it is a cup, this plastic object now has the capacity to hold the volume of something else inside it. This could be a volume of grapes, a volume of water, a volume of flour if you are making a cake and so on. Even though the cup has a volume of 25 cubic centimetres of plastic, it might hold 250 millilitres of water inside it. In other words, there are now TWO volumes. And I think it is at this point that students get very confused. Again, we need to know what question we really want answered. Do we want to know the exact amount of plastic used to make this cup? Or are we more interested in how much it can hold of another volume? I think for most people, we want to know how much this cup can hold. What is its capacity?

Only things with a shape that can fit other things inside them have capacity. A theatre has the capacity to hold say 1000 people inside for a concert. A bath has the capacity to hold so many litres of water. A box has the capacity to hold so many oranges packed in together to put on a truck to take from a farm to a shop. A supermarket shelf has the capacity to hold a specific number of cereal boxes. A human lung has the capacity to hold a specific amount of air inside it. A jar has the capacity to fit a specific amount of biscuits inside it. Capacity can be measured in many units, including units of water and blocks.

Students seem to understand this idea. In fact it pervades all their thinking.

Most students seem to clearly understand that capacity can relate to measuring with water.

But not only do our students start to think of volume as capacity, they then start to get confused with other measurements too.

This student must have had plenty of experiences measuring how far up a container some water came. They start to confuse this with height itself. It is true you measure the height of the water but only as an indicator of where the total volume reaches to in the container. The whole amount is the volume, not just the height. You need to focus on the whole volume.

And this student confuses volume with mass, or how heavy an object feels. You need regular class discussions to fix this blockage. An object might look really large, such as a gigantic cardboard box that a fridge came in. But just because it is very large does not automatically mean this box will be heavy. Similarly a small metal object might feel really heavy, even though its volume is the same as a ping pong ball which is very light. Are you measuring how heavy something feels (Mass)? Or are you measuring how much space something takes up (Volume)?

Sometimes volume and capacity is invisible. A room has the capacity to hold a specific number of invisible cubic metres, for example. We measure the volume of a room in cubic metres. We learn to estimate this space by imagining stacking invisible cubic metres in rows and columns one on top of another. After many experiments with making models from cubic centimetres, we can start to imagine these as invisible cubes instead of using real objects. This skill uses proportional reasoning to work it all out.

Finally, misunderstandings occur when students are introduced to the formula for calculating the volume of a rectangular prism as L x W x H. As this is relatively easy to explain and model, it becomes the end action. Students then believe that volume only occurs when the criteria of length, width and height are measurable. They come to believe that if something is irregular in shape then it doesn’t have a volume. There is nothing wrong with thinking about the volume of a rectangular prism. but at the same time we need to discuss many other objects that are not rectangular prisms. A bean bag has volume, a cabbage has volume, a car has volume, an elephant has volume. The volumes are not easy to calculate using a formula in the primary school. But they can be estimated using invisible units and proportional thinking. Students need plenty of experiences estimating the volume of weird and irregular shaped objects.

Notice that this Year 6 student clearly believes volume is about what is inside this shape. If this is a model of a room, this is true. If this is a model of a solid block then the whole object has a volume. It s not just about what is inside. This next Year 6 student has probably had experiences calculating the surface area of rectangular prisms. This is a totally different measure. Surface area only concerns the outer 2D surface of this shape, not the volume or amount of space this object takes up.

All 3D objects have volume. Not all 3D objects have volumes that are easy to calculate. We need our students to be experienced imaginers, able to use proportional reasoning to divide up spaces into imaginary cubes. This process is certainly helped by talking about rectangular prisms. We can estimate how many cubes would fit along one side, then lay rows like this side by side to fill the base, then stack layers of these imaginary cubes to reach the top. The end result is that we have a measure of length, width and height. But our concentration when thinking about volume should always be on seeing stacked 3D cubes, not 2D lengths.

Enough for now, I need a nice cup of coffee and a walk around the garden. Perhaps my red capsicums are ready to pick!