Purpose
The purpose of this activity is to support students in using an informal unit to measure volumes when comparing two or more cuboids. An informal unit is self-chosen by a student and may or may not be recognized by others. It is important that students come to understand the key characteristics of a unit of measure:
- Same size (identical).
- Align to the attribute being measured, e.g., a unit of volume has volume.
- Fit together with no gaps or overlaps.
Achievement Objectives
GM2-1: Create and use appropriate units and devices to measure length, area, volume and capacity, weight (mass), turn (angle), temperature, and time.
Required Resource Materials
- A variety of cuboid shaped boxes from grocery stores, such as cracker biscuits, toothpaste, tea bags, muesli bars, milo, baking powder, sugar cubes, etc. Try to have some packets of equal volume. Label the packets with letters of the alphabet (e.g. A, B etc.) for easy identification.
- Cubes (Unifix, multi-link, connecta-cubes, or similar). Note that these cubes measure 2 cm x 2 cm x 2 cm. Unit place value blocks measure 1cm3 and can also be used.
- Spherical and irregular units, such as marbles, beans, pasta pieces, pieces of scoria, stones, acorns.
- Calculators
Activity
Note: Volume is the measure of space taken up by a three-dimensional object. The space within a container is known as its capacity. As the thickness of many containers is negligible, it has become acceptable to refer to the space inside as volume too. You might frame the purpose for finding capacity within a context that is relevant to your students' interests, cultural backgrounds, and to learning from other curriculum areas. The māori kupu for volume is rōrahi.
- Choose two packets that have similar volume.
I have two packets here. Which packet has the biggest volume? The packet with the biggest volume will have the most space inside it.
- Discuss strategies for finding this out. Show students a variety of possible units, including cubical, spherical, and irregular units. The units must be sufficiently small so many of them fit in each packet.
I have some things I could put in the packets to compare their volume.
Which units should I use? Why?
How would I use these units?
Encourage students to consider using several different units to find which is most effective.
- Act on students’ suggestions about which units can be used, and the method for using them. An effective method could be to fill both packets up with the chosen units then tip out the contents and count the number of units. If students choose to use cubes, encourage them to think about loose or packed arrangements (e.g. in arrays).
- Record the results (e.g. in a table).
| Packet | Marbles | Pasta pieces | Cubes loose | Cubes packed |
| A | 45 | 51 | 29 | 36 |
| B | 39 | 53 | 32 | 40 |
- Discuss how the use of different units impacted the measurement of volume.
Why is it that different units give us different results? If we trust the marbles, packet A has more volume. If we trust the pasta, packet B has more volume.
Students should comment that the units are different sizes and some units do not fit together well so there is a lot of air space. Some may comment that smaller units leave less air space.
Which units are the most accurate to compare the volume of the two packets?
Cubes fit together without gaps or overlaps. Subsequently, they are the conventional units for volume. Students might note that curved units might work better in curved containers.
- Return to the two packets to consider how best to count the cubes that fit into each packet.
Is there a way to figure out how many cubes would fit in each packet without filling the whole packet? How?
Students usually suggest two methods:
- The Tower Method: Make a tower of cubes the same height as the packet and see how many times that tower would fit into the packet.
- The Layer Method: Make a layer of cubes the same size as the base of the packet and see how many times that layer would fit into the packet.
- Compare the volume of the two packets by writing the measures in cubic units. For example:
Packet A has a volume of 168 cubes.
Packet B has volume of 150 cubes.
- Provide students, or pairs of students, with their own packet and ask them to measure the volume in cubes. If needed, you could work with a small group of students, either initially or throughout the whole task, to support their participation and understanding. Make sure each student uses the same cubic unit. Roam as they work. Look for the following:
- Do they use the tower or layer method?
- Do they calculate the volume using multiplication?
- Do any students find the length of each dimension in cubes and use length x width x height to find the volume?
- Do they record the volume using numbers and units?
- After a suitable time gather the students and use their volume calculations to order the boxes by volume on a continuum. You might modify the continuum to a number line by adding units.
Next steps
- Give students open-ended challenges involving volume like this:
- Create cuboid shaped packets that have a volume of 36 cubic units
- A cubic packet has a volume of 64 cubes. How long are the dimensions of the packet?
- Investigate volumes in real life. Students might look at situations like volumes of containers, storage bins, topsoil, rooms, etc. Look at occupations in which people use volume, such as air conditioning, concreting, building, cooking, engineering, etc.
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Level Two