Pack The Box

Task 20 ... Years 2 - 10

Summary

First make the box. it will be a cube. Instructions are provided and it need only be done once and held with tape. Then the challenge is to fit six identical blocks inside the dimensions of the box. There is enough space... They should fit ... but it is so ... infuriating!

Note: This is an Ikea-style task. It is shipped flat-packed. Teachers are expected to make the box first and tape it together. Otherwise students have to struggle to hold the box together and solve the problem at the same time, as in the photos below.

Materials

Five rectangles to create the box
Instructions for making the box
Six cuboid blocks

Content

3D spatial perception
measurement - length, volume
logical reasoning
identifying position in 3D space - ordered triples (vectors)

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

Students at East Devonport Primary.
Their teacher has organised
Mathematician Teams.

The hint on the card leads to realising that the box would be filled with 27 cubes, but the six blocks only supply 24 of those. The other three must be 'empty cubes'. How can this empty space be best distributed through the box? Consider these possibilities.
1 'empty cube' placed in the bottom layer:

There will be 8 cube spaces left.
One block will use 2 or 4 of these depending how it is placed.
Therefore combinations of blocks could fill the 8 cube spaces.
Then look at consequent option in the next layer.

2 'empty cubes' placed in the bottom layer:

There will be 7 cube spaces left.
One block will use 2 or 4 of these depending how it is placed.
Therefore no combination of blocks could fill the 7 remaining cube spaces.
Therefore 2 'empty cubes' in the bottom layer can't work.

3 'empty cubes' placed in the bottom layer:

There will be 6 cube spaces left.
These 6 can only be filled with 3 sets of 2 or a combination of a 4 and a 2.
Trying these shows that each leads to problems in the other layers.

So there must only be one 'empty cube' in the bottom layer and following through the possibilities from there shows that there needs to be one 'empty cube' at each level. But where at each level?

How do you describe to someone else where the empty spaces are?
In how many ways can you describe their position?

Here is one way:

3	3	2
3	2	3
2	3	3

Here is another:

Starting from the bottom left front corner of the box, the points (0,0,0) ... (1,1,1) ... (2,2,2) ... (3,3,3) lie on a line which passes through the diagonally opposite corners of each empty cube space.

These ordered triples measure distances in three directions (dimensions), ie: breadth, depth and height. In general such measurements could be represented by the triple (b,d,h) or (x,y,z). But the numbers in each triple above obey the same rule. In each case:

Double the first one, then take away the second and the third and the answer is zero.

In symbols this is: 2x - y - z = 0 which, given the starting point for measurement as above, is the algebraic rule for this particular straight line in three dimensions.

There are strong links between this task and Task 156, Photo Angles, which also involves identifying the position of points in 3D space.

Whole Class Investigation
Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.
To make this task the focus of a whole class investigation, you will need boxes and blocks for, at least, each pair of students. The task provides instructions for making these. In co-operation with the woodwork/craft teacher, why not make this a project in which the students each make their own puzzle, solve it, find a way to record an explanation of the solution, and then gift wrap the puzzle and give it to someone special at Christmas?
Note: We have seen a 'life size' version of this puzzle which scaled up the box to about hip height for an upper primary child, and scaled up the blocks in the same way. The blocks were made as hollow closed cuboids with hand holes cut in opposite end faces. The whole thing was pretty heavy since it was made of wood, but perhaps it offers thoughts for a scaled up version in cardboard.
At this stage, Pack The Box does not have a matching lesson on Maths300.

Is it in Maths With Attitude?
Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.
The Pack The Box task is an integral part of:

MWA Chance & Measurement Years 3 & 4

MWA Chance & Measurement Years 7 & 8

Follow this link to Task Centre Home page.