Deconstructed Cubes
Introduction
In today's lesson you will get to use 6 square polydron pieces to build the nets of a cube. Each of the six squares corresponds to one face of the cube.
Task 1
The diagram on the right is one of the many nets for a cube (see here for an animation of why). There are other ways to arrange the six squares to make a net.
Can you find all of them?
Draw them on squared paper (PRINT ME) or on this from NCTM (make sure you take screenshots afterwards).
Use three colours to show opposite faces.
Not all arrangements can be folded into a cube. They are only called a net if they can be.
Hint: there are more than 5, but fewer than 15.
Once you think you have found all of them, use this from NCTM to check.
How can you do this systematically to make sure you have all the nets?
Hint: one approach is to look at an open box and see where the “lid” could be placed on the net.
Which nets have 4, 3, or 2 squares in a line?
Why not 5 or 6?
In theory there are 16 possibilities for nets with 4 cubes in row, so why are there only 6 in reality?
Task 2
What is the perimeter of each net?
Why isn't it 24 which is 6 squares x 4 sides per square?
How does the perimeter relate to the original cube?
Why must the perimeter always be an even number?
The lines inside the net form edges of the finished cube. The lines on the perimeter meet to form edges on the finished cube.
Put the same letter (from a to g) on the lines on the perimeter that meet to form the same edge.
How many edges are there on a finished cube?
Explain the connection between the perimeter of the net, the number of lines inside the net, and the number of edges on a cube.
How many vertices are there on a finished cube?
Task 3
If you arrange six squares at random with edges joined without overlaps, what is the probability that you will have a net of a cube?
Hint: How many different arrangements of the six squares are there. Click on hexominoes to find out.
In how many ways could you arrange the six squares so they won't make a cube when folded back together? F, for more advanced students: if the six squares are put together in a pattern chosen at random, what’s the probability it will be a net of a cube?
Further Practice
Some of the essential skills introduced in this lesson are "faces, edges and vertices" and "nets". The relevant skills can be found on DrFrostMaths, CorbettMaths, MyiMaths and Eedi. Watch any video and/or go through any online lesson as you see fit.
Extension
Transum
Some excellent cube net puzzles:
Net or Not- drag the nets into the corresponding panels to show whether they would fold to form a cube.
Coloured Cube 3D - colour in the remaining faces of the nets of the cubes to match the rotating three-dimensional picture.
Cube Face Meetings - visualise the cubes formed by the nets and paint the three faces meeting at a vertex.
Cubical Net Challenge- find all the ways of painting the faces of cubes using only two colours.
Dice Net Challenge - drag the numbers onto the net so that when it is folded to form a cube numbers on opposite faces add up to prime numbers.
Dice Nets - determine whether the given nets would fold to produce a dice.
Dice Reflections - a dice is reflected in two mirrors. What numbers can be seen?
Puzzle Cube Net - jumbled moving-block puzzle cube is shown as a net. Can you solve it?
For more goodies on 3D shapes on transum, click on the hyperlink.
Don Steward
Nets - a collection of activities by Don Steward. Some of which are not cubes!
nRich
Try these activities on nRich. In particular, try A Puzzling Cube (easy), Cut Nets (medium and introduces other 3D shapes) and Cubic Conundrum (hard)
Below are a few more:
Nets of a Cube - Problems.pdf