Working with volume and surface area together
The purpose of this activity is to support students to distinguish between surface area and volume of cuboids, use multiplicative methods for finding the measurements, and record the measurements using appropriate units: square centimetres (cm2) for surface area and cubic centimetres (cm3) for volume. Students also need to develop a sense of size for important multiples of these units, such as 1000 cm3 and 100 cm2.
About this resource
New Zealand Curriculum: Level 3 to early Level 4
Learning Progression Frameworks: Measurement sense, Signpost 5 to Signpost 7
These activities are intended for students who understand how to use units of measure to find length and areas of rectangles. They should understand the following:
- Units relate to the attribute being measured (for example, area is measured with iterations of area).
- Identical units need to be used when measuring.
- Units should be tiled (or iterated) with no gaps or overlaps to create a measure.
- Units can be equally partitioned into smaller units when greater accuracy is needed.
Students should also know how to use a measurement scale, such as a ruler or tape measure. They should be familiar with the common metric units of length, metres, centimetres, and possibly millimetres, though they may not be able to convert measures (e.g., 45 cm = 450 mm). Students should have a partial or full grasp of their basic multiplication facts and the division equivalents.
Working with volume and surface area together
Achievement objectives
GM4-3: Use side or edge lengths to find the perimeters and areas of rectangles, parallelograms, and triangles and the volumes of cuboids.
Required materials
- connecting cubes 2 cm x 2 cm x 2 cm
- calculators (if needed)
1.
Present students with a group of 60 connecting cubes and ask them to count them, or tell them the total number of cubes. This might be a good opportunity to consolidate your students' use of additive and/or multiplicative strategies.
2.
Tell students a definition of volume (rōrahi), for example,
- Volume is the measure of space taken up by a three-dimensional object.
- What is the total volume of all 60 cubes in cubic centimetres?
First, establish that each cube measures 2 cm x 2 cm x 2 cm, meaning that each cube has a volume of 8 cm3. You might use 8 place value unit cubes to confirm this. Therefore, 60 cubes have a volume of 60 x 8 = 480 cm3. Compare that volume to a litre (1000 cm3).
- You are working with volumes of just under half one litre.
3.
Ask students to make different-sized cuboids that contain exactly 60 connecting cubes, so have a volume of 480cm3.
- Let’s see how many cuboids we can make. How can you record the different cuboids you find?
Organise students in groupings that will encourage peer scaffolding and extension, as well as productive learning conversations. Together, you might make a shared list of methods for recording the cuboids.
Roam as students work, and look for students to:
- anticipate dimensions that will work using calculations and diagrams
- apply multiplicative thinking to find possible dimensions. For example, if the length is set at 20 cm then each vertical layer must have 480 ÷ 20 = 24 cm3.
4.
Gather the group and discuss the cuboids that have been made.
- Are these cuboids distinct?
That means they are different in their dimensions – they are not just turned around or flipped copies.
Create a group collection of cuboids (physical models). Express the dimensions in centimetres.
Examples might be:
- Have we found all the possibilities?
- How do you know?
The prime factors of 60 are 2 x 2 x 3 x 5. Considering all the possible trios formed from these factors gives all the cuboids that can be made.
In the examples above, the possibility of having 2 cm as a dimension has not been considered. Using 2 cm as a dimension gives six other possible cuboids:
- 2 cm x 2 cm x 120 cm
- 2 cm x 4 cm x 60 cm
- 2 cm x 6 cm x 40 cm
- 2 cm x 8 cm x 30 cm
- 2 cm x 10 cm x 24 cm
- 2 cm x 12 cm x 20 cm.
5.
If you have sufficient cubes, make all the 10 possible unique models (as above).
- All these cuboids have the same volume. Do they have the same surface area?
Let students discuss the question and come to a consensus about surface areas.
Allocate the cuboids and ask the students to use their rulers and calculators to find the surface areas. Organise the data in a table, as shown below:
Dimensions (cm) |
Calculation for surface area |
Surface area (cm2) |
|---|---|---|
4x12x10 |
2x (4x12+4x10+12x10) |
416 cm2 |
4x4x30 |
2x (4x4+4x30+4x30) |
512 cm2 |
4x20x6 |
2x (4x20+4x6+20x6) |
448 cm2 |
6x8x10 |
2x (6x8+6x10+8x10) |
376 cm2 |
2x2x120 |
2x (2x2+2x120+2x120) |
968 cm2 |
2x4x60 |
2x (2x4+2x60+4x60) |
736 cm2 |
2x6x40 |
2x (2x6+2x40+6x40) |
664 cm2 |
2x8x30 |
2x (2x8+2x30+8x30) |
632 cm2 |
2x10x24 |
2x (2x10+2x24+10x24) |
616 cm2 |
2x12x20 |
2x (2x12+2x20+12x20) |
608 cm2 |
6.
- What do you notice?
Students should comment that cuboids with equal volume need not have equal surface area. Some may notice that long, thin cuboids have greater surface areas than those with dimensions that are similar lengths such as 6 cm x 8 cm x 10 cm.
1.
Extend interpretation of volume and surface area with problems where surface area or volume is given but a dimension (edge length) is not.
Two examples might be:
- The surface area is 312 cm2. What is the dimension labelled h?
- The volume of the cuboid is 420 cm3. The surface area is 349 cm2.
- What is the dimension (edge length) marked l?
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