A 'voluminous' matter

Jayati Kaushik and Sindhu Sreedevi

Often when students are taught about volume, they are overwhelmed by formulas and calculations even if the formula is just l × b × h; and the concept of what they are calculating or why they are doing it or whether it has any significance whatsoever is lost to them. Thus the process of calculating volume becomes nothing more than a mundane exercise in multiplication. A successful lesson in volume must leave students with a clear idea of what volume is – it is not capacity; a 100ml bottle usually has higher volume; capacity is the amount of substance that can be contained by the vessel, but volume is the amount of space occupied by the solid. They must have an understanding of the parameters that affect volume and finally be able to calculate volume using the correct units and formulae.

Activity 1
Objective: To understand the difference between capacity and volume.

This is much better done through a discussion of how when we buy a 1L bottle of cold drink it is not filled up to the brim. However it occupies a certain amount of space in the fridge where you cannot put anything else. That space that it takes up is the volume but the amount of the stuff in it is its capacity.

As teachers, it is important for us to understand that children have an intuitive idea of measurement. Building upon this idea will facilitate their learning. Also, the idea of volume is close to real-life experience and therefore the need to direct the learning along the right path or appropriate method is important. Thus, posing a question from their natural routine will facilitate the learning. This specifically helps in making students understand better.

Any good situation that highlights the need for measuring volume, (one of my favourite examples to use is which cup gives you more juice) is good enough to pique interest in the idea of measuring volumes. Once the need for measuring volume is established, we can ask how this concept can be developed.

If students are already familiar with the concepts of units, then this is simple enough. All that needs to be done is to introduce standard units of measuring volume and a way for students to visualize it. A cube can be very helpful to this end.

It will be interesting as a teacher to know the right path for learning volume. Of course, the first thing will be understanding ‘capacity’, for which the conservation of volume is important. Most of us must have noticed that if we pour the same amount of liquid from one container to another which has a smaller base, then pre-school children will choose the second container because the height of the liquid will be higher in it. From this discussion, by elaborating on each point, the ideas of unit and standard units and so on, will develop in a natural manner.

The usage and need of units will be the first step to start with. A simple exercise of comparing the volume of one entity in terms of another can be very instructive and an example of a real life application. For example, it can be shown how to measure the capacity of a bottle using a cup. Then the capacity of the bottle will be expressed in terms of cups. So ‘cup’ is the unit. At this stage it is probably a good idea to let the students come up with units of their own choice not just for volume but whatever they might want to measure.

Using different sizes of cups provides a good building ground for understanding the conversion of units later. Using a cup of twice the size will mean you get half the count. It is best if students can come up with some of these ideas on their own through experimentation.

This leads to a discussion of need and importance of standard units. Not everyone may have the same cups as they did in class. So in order to communicate, it is important that there be a unit that everyone knows or recognizes. Then the SI unit of measuring volume can be introduced and explained visually.

After building the understanding that the amount of substance a vessel can hold is capacity, moving to the idea of volume will be appropriate. The explanation of different parameters that affects volume is important. So, to explain the parameters that affect the volume of an object, we can use different shapes of same the volume with just one thing (either length, or breadth, or height) changing. Through experimentation and observation it can be explained to the students that three things primarily affect the volume of an object:
Its length
Its breadth
Its height

This helps to establish the idea of volume as amount of space occupied by an object. Also, connecting area to volume will be interesting for children. Even pages put together to form a notebook which is cubical/cuboidal in shape will add value to the logical reasoning of the formula and concept of 3D shapes.

Activity 2

Objective: To understand the parameters that affect volume.

Materials: Two containers of different heights but the same base, a cup measure.

Procedure: Using the cup measure, pour the same volume of water into each of the containers. During the discussion, ask students what is same and what is the different between the two containers.

Suggestion: Cuboidal containers may be preferable as the concepts of height, length, and width are easier to visualize with these rather than a container with a circular base.

And finally we can discuss how to calculate volume of simple shapes like a cube or a cuboid. Even though we do not derive the formula for students, it can just be explained that to take into consideration the effect of each parameter when we calculate the volume, we simply multiply the measures of all three parameters.

Activity 3

Objective: To move from measuring volume to calculating volume.

Materials : A cuboidal container whose dimensions are multiples of a unit cube of dimension 1 × 1 × 1.

Procedure: Ask students to see how many cubes will fit in the container. With a little experimentation, and reference to the unitary method, it will become intuitive to the students that the volume will be l × b × h. Such an informal understanding is invaluable for conceptual learning.

If in the end, students are a little excited about volume and feel the desire to communicate about volume in a more scientific manner, then the efforts you put into this topic will have been totally worth it.

References
1. SECTION 3 Applying Fundamental Concepts, Attitudes, and Skills, UNIT 14 Measurement: Volume, Weight, Length, and Temperature.
2. A guide to effective instruction in Mathematics by Ontario.
3. The long and short of ‘measurement’ by Neerja Singh, August 2014, Teacher Plus.

Jayati Kaushik is a BSc student doing a triple major course in mathematics, statistics, and economics. She is doing her term paper on teaching mathematics and has limited teaching experience mostly helping fellow classmates, juniors and cousins. She may be reached at informjaka@gmail.com.

Sindhu Sreedevi has been a mathematics teacher and has specialized in learning assessment for the last nine years during which she has had opportunities to learn the practical aspects of teaching, learning, and assessment processes. She has also been part of the large scale assessment in Azim Premji Foundation. She can be reached at sindhusree77@gmail.com.