Alien Counting

With the advent of New Maths in the 1960s, numbers in other bases came to prominence because it was thought that study of these would give students insights into the way our base 10 (written as base₁₀) place value system works. On the whole, it didn’t. Instead, it tended to confuse learners more than it enlightened
them, but that was probably because it was introduced at too early an age.
However, there are good reasons for studying numbers in other bases. Approached sensibly, they can lead to a better understanding of how our base₁₀ number system works. Another reason is that base₂ is part of our lives because it is fundamental to the way our electronic communications work. Electric current can be either on or off, so this property is used to denote the digits 0 and 1, which in turn allows quite complex communication. Our calculators and computers work on the base2 system, so the whole of communication technology depends on this base. A further reason that numbers in other bases are worth studying is that, once again, the notion of there always being one right answer in mathematics is undermined.
In Activity One, question 2, it should soon become clear, for example, that 7 + 6 can be 21 as well as 13 (although the 21, admittedly, should be written 216). What the correct sum is depends on the number system being used.
Questions 1 and 2 in this activity lead students to the understanding that the foundation of the place value system is that digits get progressively greater by powers of the base number as they move left and, conversely, get progressively smaller as they move right.
To illustrate: Take a number in base₁₀ (the base at the heart of our decimal system), such as 5. As the 5 moves left one column into the tens column, it becomes 10 times greater, and as it moves left one further column into the hundreds column, it becomes 10 times greater again (and hence 100 times greater than the original 5). Conversely, if the original 5 moves right one place to the tenths column, it becomes 10 times
smaller.
Teachers in earlier years have usually helped students to understand that in our usual base10 place value system, the first column is ones, the next to the left is tens, the next is hundreds, and so on. What your students now need to understand is that the first column is “lots of 1”, the second is “lots of 10”, the third is “lots of 100”, and so on. They also need to understand that these can be expressed in powers of 10. For
example, 100 is 10², 1 000 is 10³, and so on. The students could help construct a table of these understandings, using a number such as 6 452, perhaps as follows:

The useful thing about setting the table out in this form is that it mirrors the way we write numbers. (Note that this is the reverse of the way the table is set out in Activity One, question 4.) There, the table shows the potential for going on to higher place values. The setting out here focuses on the place values for a specific number. It shows clearly what the digits in 6 452 (in the bottom row of the table) stand for. The 6 signifies six lots of 1 000 or six lots of 10³ the 4 represents four lots of 100 or four lots of 10²; the 5 is five lots of 10 or five lots of 10¹; and the 2 is two lots of 1 or two lots of 10⁰.
You can approach the understanding of numbers in other bases in a similar way. For example, base₅ numbers would look like this:

The other useful thing about setting out the tables as shown above is that it can lead relatively easily to considering decimal fractions in terms of powers of 10. Thus tenths are denoted as 10^-1, hundredths as 10^-2 and so on. It then becomes possible to multiply (and divide) numbers involving decimal fractions by operating on the powers of the numbers, as in the activity on page 4 of the student book.
You could have an interesting discussion with your students about early counting strategies. They should have realised that the number of fingers each type of alien has is related to the number base that they use. Most humans use a base₁₀ system.

Answers to Activities

Activity One
1. Explanations need to include the fact that the Cartoks see the moons as one lot of 6 and 3 left over, as below.

2. Explanations need to include the fact that the Cartoks see the planets as two lots of 6 and 1 left over.  For example, tow of their sixes is like two of our tens.

The Cartoks' counting system is a base 6 number system.
3.

4.

5a. 27
b.32
c. 52. (1 x 36 + 2 x 6 + 4 x 1 = 52 or 1 x 62 + s x 6 +4 x 1 = 52)
d. 321. (1 x 216 + 2 x 36 + 5 x 6 + 3 x 1 = 321 or 1 x 63 + 2 x 62 + 5 x6 + 3 x 1= 321)

Activity Two
1. 118 moons in question 1 and 15₈ planets in question 2.  The Vanan counting system is a base 8 number system.
So 21 in Vanan counting would look like this:

2.

3. Repsonses will vary.