and weird multiplication Jim Tanton wwwjamestantoncom The following calculations are not intended to replace the standard methods but rather to give clarity to the arithmetic that is involved ID: 270490
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Slide1
Exploding Dots Machinesand weird multiplication
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.Slide2
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 DotsSlide3
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 = 7431Slide4
+
=
Adding Dot Machines
The order of the carries is not an issue!Slide5
-
=
Subtracting Dot Machines
The order of the borrows doesn’t matter!Slide6
Weird MultiplicationSlide7
Division of Dot MachinesSlide8
Dividing Polynomials with a
1← x Machine(x
3
+ x
2
– 2x + 12)
(x + 3)
x
x
2
x
3
Anti-Dots
Dot and Anti-Dot
Where does the dot go?
The Result is x
2
– 2x + 4Slide9
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?