Continue the Pattern for 6 Garages How Many Cubes Were Used

Iceberg

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

The opening question may seem a little straight forward, but its intent is to encourage students to see 'into' the scored cube - to deconstruct it. This visual separation of the whole into parts is important in developing the patterns in the main problem. If you have 2cm cubes (or similar) in the room it can be useful to ask students to show you how their calculation methods can be demonstrated. Two ways to calculate are:

• 3 layers, each with 9 unit cubes
• 9 towers each with 3 unit cubes

and there may be more. The background question here is the mathematician's question Can I check it another way? and as suggested by this extract from Maths300 ETuTE, Page 42, the exercise opens the door to further arithmetic in context. Counting cubes gives the following lines of the table:

Large Cube	3	2	1	0	Total
3	8	12	6	1	27
4	8	24	24	8	64
5	8	36	54	27	125
6	...	...	...	...	216

There may be a Size 10 cube available similar to the ones in the task, but it is unlikely that you will be able to find the other sizes. If you have 2cm wooden cubes, or linking cubes such as Multilink the students could make the Size 6 - or perhaps enough of it to be able to calculate. Alternatively, or as well, students can record on isometric paper and colour code the unit cubes. With one more line of data from the Size 6, patterns will start to occur to many students. But to tackle this problem as a number hunt is to miss some of its simple beauty - after all, wherever there is a number pattern there will be a corresponding visual pattern and vice versa. So perhaps looking at the geometry of the painted cube will help. For any size cube (Size n):

Drawn by Becky & Lydia, Settlebeck High School (see below)

• The unit cubes with 3 painted faces will always be at the corners. There are 8 corners, so that column will always be 8.
• The unit cubes with 2 faces painted will always be along the edges and will be two units less than the length of the edge because one edge cube is used for each corner. There are 12 edges on a cube so the second column will always be 12(n - 2).
• The unit cubes with 1 face painted will always be squares in the middle of each face. The side of the square will be (n - 2) because each side is stops when it gets to an edge. There are 6 faces on a cube so the third column will always be 6(n - 2)².
• The unit cubes with 0 faces painted will always be a cube in the middle of the bigger cube. The side of the cube will be (n - 2) because in each direction it is stopped when it gets to a face of the original cube. So the fourth column will always be (n - 2)³.

With this insight, the remainder of the table can be filled in more easily.

Extension

• The Size number can be paired with the corresponding number in any of the columns to make a set of ordered pairs. For example the ordered pairs for 2 painted faces are (3, 12), (4, 24), (5, 36) ...
  For each column make a set of ordered pairs and graph them. What can you learn from these graphs?
• Suppose you have 15,625 unit cubes. What size is the large cube? How many unit cubes have 3, 2, 1, 0 faces painted?

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 Painted Cubes task is an integral part of: MWA Pattern & Algebra Years 5 & 6 MWA Pattern & Algebra Years 9 & 10 The Painted Cubes lesson is an integral part of: MWA Pattern & Algebra Years 9 & 10

meadvengland.blogspot.com

Source: https://mathematicscentre.com/taskcentre/160paint.htm

Share this post