New Zealand Curriculum: Level 3 to early Level 4
Learning Progression Frameworks: Measurement sense, Signpost 5 to Signpost 7
Target students
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 characteristics of units:
 Relate to the attribute being measured (area is measured with a part of area, usually a square, volume is measured with a unit of volume, usually a cube)
 Identical
 Tiled with no gaps or overlaps to create a measure
 Subdividable (equally partitioned 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 most common metric units of length, metres, centimetres, and possibly millimetres, though they may not be able to convert measures (e.g., 45cm = 450mm). Students should have a partial or full grasp of their basic multiplication facts and the division equivalents.
The following diagnostic questions indicate students’ understanding of, and ability to find the volumes and surface area of cuboids (rectangular prisms). Allow access to pencil and paper and to a calculator if students need it (show diagnostic questions).
The questions should be presented orally and in a written form so that the student can refer to them. The questions have been posed without context but can be changed to other contexts that are engaging to your students.
 Here are three different cuboids. The shapes are also called rectangular prisms. The numbers show how many blocks are on each edge.
How many small cubes are in each of these cuboids?
Signs of fluency and understanding:
Using multiplication to count the number of small cubes, i.e., 2 x 3 x 4 = 24 for the blue cuboid. Recognises that the product of edges lengths gives the number of small cubes.
May notice that the blue and red cuboids have the same edge length so have the same volume. The blue and red cuboids are the same cuboid in different orientations.
What to notice if they don’t solve the problem fluently:
Counting, in ones, or by skip counting in twos, to find the total number of small in each layer. This indicates an inability to structure the cuboid in ‘layers or slices’ and/or no recognition that multiplication might be used to count the number of small cubes.
Supporting activity:
Finding volumes of cuboids (whole number edge lengths)
 Measure the edges of each cuboid shaped box in centimetres. Write the measurements down. Use the measurements to work out the volume of each box in cubic centimetres. Write the volumes using both numbers and units.
[Get two small grocery boxes of cuboid shape such as a spice box, a muesli bar packet, or a cracker biscuit packet. The edges should be whole numbers of centimetres. Have a calculator available.]
Signs of fluency and understanding:
Measure the edge lengths accurately to the nearest centimetre and write the measures. Apply multiplication to find and record the volumes, i.e., 14 x 4 x 12 = 672cm^{3}.
What to notice if they don’t solve the problem fluently:
Draw in or image marks to partition each edge into cubic units of 1cm^{3} then find a way to count all the small cubes. The strategy is unlikely to be successful so pause the student after a short time rather than let them continue. This indicates an inability to anticipate the structure of the cuboid in 3dimensional arrays of small cubes.
Apply additive strategies to find the number of cubic units, such as imagine a layer or ‘tower’ of cubic units and fit the layers or towers into the cuboid by addition. For example, a tower is 12 cubes so 12 + 12 + 12 + 12 = 48 makes a vertical layer. Adding 12 + 12 + 12 … (14 times) gives the volume. This might indicate that the student has yet to establish multiplication as a binary (two numbers at a time) operation or connect multiplication with the structure of 3dimensional arrays. This is also a cumbersome strategy so pause the student after a short period if they do this.
Supporting activity:
Finding the volumes of cuboids by measuring edge lengths.
 This cuboid has a volume of 3 x 8 x 4 = 96cm^{3}. The numbers show the length of each edge.
What is the surface area of the cuboid?
Have a calculator available. Ask students to record information and draw diagrams. You may need to explain that surface area is the sum of the areas of all the faces.).
Signs of fluency and understanding:
Calculate the area of each rectangular face fluently using multiplication, possibly using a calculator. Add the areas of the rectangles using whatever method they want (mental, written, calculator). For example, 3 x 8 = 24cm^{2}, 4 x 8 = 32cm^{2}, 3 x 4 = 12 cm^{2}, and 24 + 32 + 12 + 24 + 32 + 12 = 136cm^{2}.
Might notice identical parallel faces and multiply 2 x (24 + 32 + 12) = 136cm^{2}.
What to notice if they don’t solve the problem fluently:
Should use multiplication to find the area of each face. However, the student may not account for all faces. This suggests the student needs to work on the structure of cuboids, such as six rectangular faces, 3 pairs of identical faces parallel, rectangles have side lengths that are pairs formed from the three dimensions.
Supporting activity:
Finding surface areas
 Which cuboid has the greatest volume? Which shape has the greatest surface area?
Signs of fluency and understanding:
Calculate the volume and surface area of each cuboid using multiplication and addition, as appropriate. May use mental and paper calculation or a calculator combined with recording on paper. For example, the volume of the yellow cuboid is 2 x 3 x 10 = 60cm^{3} and the surface area is 2 x (6 + 20 + 30) = 112cm^{2}.
Recognises that surface area and volume are different attributes, so the measures are expressed using different units, area units for surface area (cm^{2}) and cubic units for volume (cm^{3}).
Answers:
Cuboid

Dimensions

Volume

Surface area

Blue

2cm x 5cm x 6cm

60cm^{3}

104cm^{2}

Red

4cm x 4cm x 4cm

64cm^{3}

96cm^{2}

Yellow

2cm x 3cm x 10cm

60cm^{3}

112cm^{2}

What to notice if they don’t solve the problem fluently:
May confuse the two attributes, volume, and surface area, when providing answers.
May have difficulty coordinating the information from the diagram to make appropriate calculations. This suggests that the student needs more experience representing 3dimensional models of cuboids with 2dimensional diagrams.
Supporting activity:
Working with volume and surface area together
 Use a ruler to measure the edge lengths first then find the volume and surface area of this box using a calculator.
[Choose a small grocery box that has edge lengths that are not all whole numbers, such as 20cm x 7.5cm x 6.3cm].
Signs of fluency and understanding:
Measure the edge lengths accurately to express those lengths as decimals, e.g., 18.5cm, 11.3cm, and 7.2cm.
Correctly enter an appropriate algorithm for find both volume and surface area. For volume work out 18.5 x 11.3 x 7.2 = 1567.785cm^{3}. For surface area work out the area of all three pairs of faces using multiplication then combine the measures to get a total.
Understand the size of the units by including the number and unit of measure, and explaining the size and nature of the unit, i.e., 1cm^{3} is a cube that is 1cm x 1cm x 1cm.
What to notice if they don’t solve the problem fluently:
May be unsure about how to accommodate decimal numbers of measurement units. This is likely to show in being unsure of how to express edge lengths, such as 18.5cm, 11.3cm and 7.2cm respectively. This indicates that the student needs experience with measuring lengths with greater accuracy using smaller units in the metric system.
May measure the side lengths accurately but either be unclear about how to find the volume and surface area by calculation and/or express the resulting measure using appropriate units. This indicates the student needs more experience with calculating volumes and surface areas with partial units.
Supporting activity:
Finding volumes and surface areas from decimal side lengths
Teaching activities