1. Jim Tanton www.jamestanton.com The following calculations are not intended to replace the standard methods but rather to give clarity to the arithmetic that is involved.

2. There are many types of Dot Machines – Let’s Start with something simple like 1 ← 2 1 x 2 2 + 0 x 2 + 1 x 2 0 = 5 Start With 5 Dots

3. Let’s Consider a 1 ←10 Machine Start With 7431 Dots How many tens are in 7431? 743 and 1 left over How many tens are in 743?74 with 3 left over How many tens are there in 74? 7 with 4 left over 7 x 1000 + 4 x 100 + 3 x 10 + 1 = 7431

4. + = Adding Dot Machines The order of the carries is not an issue!

5. - = Subtracting Dot Machines The order of the borrows doesn’t matter!

6. Weird Multiplication

7. Division of Dot Machines

8. Dividing Polynomials with a 1← x Machine (x 3 + x 2 – 2x + 12)(x + 3) xx2x2 x3x3 Anti-Dots Dot and Anti-Dot Where does the dot go?The Result is x 2 – 2x + 4

9. What is this 101←2 Machine doing? What happens in a 2←1 Machine? Some Open Questions about Dot Machines What about a 2←3 Machine?