WEEK 13

 

6.2 Perimeter, area and volume

6..2.1 Understand the concept of volume

Here we look at the volume (or capacity) of 3D shapes, and briefly consider surface area. Rather than an in-depth exploration of volume, the emphasis is on visualising and problem solving.

Monday: Here we have the net of an open box. The box is fairly easy to visualise. A neat way to solve the task is to think of the box as being filled with six 1 cm thick vertical slices, each consisting of 12 unit cubes: if we increase the 6 cm width of the box by one cm, it will hold an extra slice of 12 unit cubes.

It is interesting to see whether students think in this visual way, which leads to a modified version, as shown below, of the original net. A variant of this would be to see the unit cubes as being arranged in horizontal layers each containing 24 unit cubes: raising the height of the box by 0.5 cm would accommodate 12 more cubes (if we're allowed to cut them in half).

The given net produces a 3 cm by 4 cm by 6 cm box. By seeing 3×4×6 as 12×6, some students might realise that changing the 6 cm length to 7 cm will result in the desired capacity, without necessarily visualising the boxes in detail. Other students might calculate 3×4×6 = 72 and thus look for dimensions that would give 84 without relating their new box and net to the original box and net. So they might end up with a 2 cm by 6 cm by 7 cm box and, for example, a net like this:

Tuesday: Students might find it quite challenging in this task to visualise the original block: if we un-glue the L-shape we need to turn one of the pieces over to re-form the block.

The original block was 15 cm long (10+5 = 7+8 = 15).  The plan view of the block is shown below. Cutting the block along the dotted line would produce two pieces with the same volume as cutting along the brown line. So the volume of the pieces is 2×3×8.5 cm³ and 2×3×6.5 cm³ or 51 cm³ and 39 cm³.

 

Wednesday: Here we consider surface area rather than volume.

Consider the front and side elevations of the larger of the two pieces (see below). The cut has exposed various vertical faces with a total surface area of 7×5 cm² + 5×5 cm² = 60 cm². The cut will have exposed the same total surface area on the smaller piece, making a grand total of 120 cm².

 

Thursday: This is a fairly straightforward task, once students have visualised the situation. Though there would seem to be several ways of cutting the loaf, if we are dealing with a standard-shaped loaf and it is cut in the usual way, then the cut will be parallel to a 10 cm by 15 cm face. So the slice will be roughly in the shape of a 1 cm by 10 cm by 15 cm cuboid, which has a volume of 150 cm³.

Friday: The wood size 'four by two' traditionally referred to inches, although planed wood is usually smaller than the given dimensions. This applies here, as 47 mm is about 1.85 inches and 100 mm is about 3.9 inches.

Students will need to decide whether to work in mm, cm or m, and the associated volumes. None of the options is particularly nice!
If for example students choose cm, we get this:
The given piece of wood has a volume of 4.7 × 10 × 300 cm³ = 14100 cm³. A cubic metre is the same as 1 million cm³ (which might surprise some students). So a cubic metre of the wood, priced at the same rate as our piece costing £12.42, would cost £12.42 × 1000000 ÷ 14100 = £880.85.