# Land of ET

### Task 145 ... Years 4 - 10

#### Summary

An imaginative story shell drives this task, but far from being trivial, it leads to the very core of the number system. In fact, beyond that to structures that link apparently different parts of mathematics. And yet, because of the colour, visual, concrete aspect of the presentation it can be tackled successfully from middle primary. The idea is simple - if a little unreal. A kingdom which has an alphabet of only two letters. That makes for some very long words. The king and queen are finding this all a bit tough, so they introduce rules to shorten words. Perhaps by accident, perhaps through regal wisdom, their rules result in surprisingly few words.

#### Materials

• 30 counters in two colours - about 15 of each

#### Content

• properties of the number
• mathematical structures
• laws of arithmetic
• logic and reasoning
• transformations of equilateral triangles

#### Iceberg

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

The first question on the card is about encouraging exploration of these new language rules. The simplifications are:
 BYYBYYBBY BBBBY BY BYYBYYBBBYBY BYYBYYYBY BBYBY BYY B BBYBY BYY B

Students are then encouraged to explore words of their own and finally challenged to find out just how many unique words remain the language. Most will begin randomly making strings of letters and trying to shorten them. Perhaps they will have to be encouraged to reread the rules that say 'removed (or included)'. There are some words that can only be simplified by using inclusion at some point. For example:
 BBY BB + BYB = BBBYB YB YBB BYB + BB = BYBBB BY
It may be useful to encourage students to break into parts the problem of simplifying the apparently infinite collection of words.

• 1 letter words?
... B and Y (and at some point students will come across the Nothing word).
• 2 letter words?
... BB, BY, YB
• 3 letter words?
... let them explore; they will eventually find that all three letter words simplify
• 4 letter words?
...
In the end, counting the Nothing word (N), there are only six words left in the Land of ET!

### Extensions

What happens if we add two of the six words together?
What are the possibilities?

 2nd 1st N B BB Y YB BY N B BB Y YB BY

Wow, another surprise! The answer is always one of the six words. Well perhaps not such a surprise since we were pretty sure they were the only words in the language. Are there any patterns in the table?

The structure of the ET language has now become closed. That's different from our number system which is quite open; in fact, infinite.
Could we create a closed number system parallel to the ET words?
Perhaps if we try thinking of a circular timer that only has 6 equal spaced markings from 0 - 5.

 2nd 1st 0 1 2 3 4 5 0 1 2 3 4 5

With the imagined clock timer, if it is already at 5 and you add one more section you get to 0.
So in this closed arithmetic system 5 + 1 = 0. Mmm, that's another surprise ... and the mathematician is starting to sense that there are structures underlying apparently different sections of mathematics. So if 'behaviour' in one section is explored it must have implications for behaviour in the other.

And then one day the mathematician might be playing around with an equilateral triangle and realise that no matter how it is:

• rotated in multiples of 60o around the centre, or
• flipped around a perpendicular from a vertex

Hey, that's a closed system too. I wonder how many different transformations there are in this system? ... and what happens when they are combined?

#### 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 turn this into a whole class lesson, the objects you use need to be yellow and blue. Not many schools will have sufficient counters or cubes, or tiles of these two colours, however schools with Poly Plug have just the right resource. One yellow/blue board between two will do the job. Use the lesson to highlight:

• that mathematicians love to find structures underlying mathematics.
• similarities and differences between the ET system and our number system.
Is there an argument for introducing the investigation through physical involvement? That is, each student has a square of either yellow or blue paper to hold in front of their chest and words are created by students lining up.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 65, Land of ET, which also includes an investigation guide for the transformations of the equilateral triangle.

Visit Land of ET on Poly Plug & Tasks.

#### 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 Land of ET task is an integral part of:

• MWA Pattern & Algebra Years 3 & 4
• MWA Space & Logic Years 9 & 10
The Land of ET lesson is an integral part of:
• MWA Space & Logic Years 5 & 6
• MWA Space & Logic Years 9 & 10