# Station I**Wild Exploration**

Here are some “big question” ideas you might want to explore, or just think about.

Have fun!

Station I

Here are some “big question” ideas you might want to explore, or just think about.

Have fun!

Here is an addition problem in a $1\leftarrow 5$ machine. (That is, it is a problem in base five.) This is not a $1\leftarrow 10$ machine addition.

What number has code $20413$ in a $1\leftarrow 5$ machine?

What number has code $13244$ in a $1\leftarrow 5$ machine?

What is the $1\leftarrow 5$ machine answer?

Let’s now work with a $1\leftarrow 3$ machine.

Find $1202\times 3$ as a base three problem.

Also, what are $111\times 3$ and $2002\times 3$?

Can you explain what you notice?

(In base three we should really be writing $10$ rather than $3$ in each of these problems.)

Let’s now work with a $1\leftarrow 4$ machine.

What is $133\times 4$?

What is $2011\times 4$?

What is $22\times 4$?

Can you explain what you notice?

(In base four we should really be writing $10$ rather than $4$ in each of these problems.)

In general, if we are in a $1 \leftarrow b$ machine, can you explain why multiplying a number in base $b$ by $b$ returns the original number with a zero tacked on to its right?