# Keith's Kubes

### Task 192 ... Years 4 - 8

#### Summary

Students build in 3 dimensions to address the challenges in this task, but have to record in two dimensions. They are free to choose their own way of recording. The only criterion is that others must be able to interpret it. The task begins with an object made from 4 cubes which has a square plan view. It asks students to find all the possible ways of joining the 2 additional cubes. The basic objects that evolve are then used in two jigsaw-style spatial challenges designed to extend students' reasoning and spatial perception. There are two levels of this additional challenge to allow more opportunities for success.

This task was created by Keith Windsor, teacher and mathematics consultant, UK.

Keith's Kubes also appears on the Picture Puzzles Shape & Space A menu where the problem is presented using one screen, two learners, concrete materials and a challenge.

#### Materials

• Two sets of 6 linking cubes - 4 of one colour and 2 of another in each set

#### Content

• 2D representation of 3D objects
• combination theory / ordered arrangements
• concept of proof
• reasoning
• recording mathematics
• sorting, classifying, ordering
• spatial perception, 2D or 3D

#### Iceberg

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

Reasoning out Question 2 requires trying the additional cubes in all possible places that make one storey high 'buildings' and checking that there are no repeats. One way to do this is to fix the fifth cube (given the symmetry of the square there is only one way to do this) then find all the possible places to attach the sixth.

Unlike Task 103, Four Cube Houses, these objects are not houses sitting on a block of land. These ones can be rotated and flipped in the hand. This reduces the number of possibilities to just 8. The white cube is fixed and the yellow is taken for a clockwise walk. (Note: colour doesn't matter when considering the shape of the object.)

Question 3 encourages further thought about how to record the possibilities. Continuing the plan view approach above is a possibility, but extra coding needs to be added to indicate above and below. Isometric drawing could also be used, but there would be some objects in which one or more of the six cubes could not be seen. The students could also choose to use plan view with one or more elevation views. One way of using the plan view approach above shows that these are the 15 solutions. Read down each column to see how one is created from the other.

Then there are cases where where one cube is fixed on top.

 Variations on the two above result in repeats. We think we have found all the unique possibilities. But we are very happy to be corrected.

One answer to the first challenge question is:

There may be more. However, if not then we have 8 with all the cubes in the same plane and 15 others. Both these numbers are one less than a square number. Why? Does anyone have any explanation that could give meaning to that fact?

The second challenge, making the 6 x 6 square without repeating a shape, can be solved. We are saving a space here for the first student team to find a solution.

Jessica and Anki, Ashburton Primary School, tried Keith's Kubes in their first ever Task Centre lesson. They still have more to find out about the task, but the detail and presentation of the recording is good for a first try. (Click the image for a larger version.)

Read more about the work at Ashburton in Recording at Ashburton Primary School.

#### Extensions

1. Encourage an alternative 2D representation of at least a few of the solutions.
2. Choose 3 of the objects to be houses. Draw their isometric and elevation views. Design the interior of one of your chosen houses.
3. Explore making rectangles with the answers to Question 2. For example, 2 pieces is a total of 12 cubes, which could be arranged as a 3 x 4 rectangle. Are there two pieces which can be made into a 3 x 4 rectangle? What about 3 pieces = 18 cubes? 4 pieces? ...
4. Selecting from any of the objects, explore making cuboids (rectangular prisms) with all the objects.

#### 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.

All you need to turn this task into a whole class investigation is cubes which link in three dimensions and scissors for each person. Graph paper and isometric dot paper will be useful to aid recording. The purpose of the lesson is to develop skills in recording 3D objects in 2D in a context of developing organised reasoning and spatial perception. To begin the lesson each person needs four cubes of one colour and two of another. It is important to work in pairs so students can construct objects in what appears to be two different ways and then compare to discover if they are different.

Begin with the challenge of Question 2 without offering any assistance. Encourage recording on graph paper. When one pair thinks they have found all the answers, they compare with another pair. Try to allow the class to come to an agreement about the number of possibilities. This will take a little time as students convince each other about repeats. Record the agreed 8 on the board and discuss how they were found. Highlight how the search could have been done in an ordered way and complement those who took that approach. Also highlight the criterion for deciding that two objects are really the same.

The recording so far is for including in the student's journal, but the pieces are now going to be used to tackle the Challenge questions from the card. For this, each person needs a second piece of graph paper so that they can draw the objects using a scale of 2cm = 1 side length. Each person draws and cuts out four of the plan views of the objects. This is an opportunity to highlight that the plan view is only a representation of the object. It is also important to mention what it doesn't show, in this case the depth of the object.

It is possible to combine six of these objects to make a new object with a plan view that is a 6 x 6 square. One way to do it repeats just one object. Another way has no repeats. Your challenge is to find one or both ways.
Explore, discuss and record as appropriate. Remember we would be happy to record any solutions not already shown above.
Throughout this investigation we have been restricted to adding the extra cubes on the side of the square so that all the cubes are in the same plane. But a mathematician asks What happens if...? What is their likely question for this problem?
Investigate making objects with the extra cubes added to the top and bottom of the 'square'. There will be more discussion about same and different and in particular students need to realise that if two objects can be placed so they are mirror images, then they must be different because 'the world on the other side of the mirror is different'. This is the time to check that all students realise that their image in a mirror is a reverse handed version of themselves.

Continue the lesson by exploring alternative representations of the 3D objects in two dimensions. The final challenge of exploring cuboids that can be made from the objects (are there any?) can be on-going in a corner of the room. There are 23 different objects, so if each student makes one to finish the lesson, the pieces can be stored on a table to access in self-directed time.

At this stage, Keith's Kubes 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 Keith's Kubes task is an integral part of:

• MWA Space & Logic Years 7 & 8