Now heres something completely different EXPLODING DOTS John

Now here’s something completely different … EXPLODING DOTS! John Rodger, GMP Ambassador

Introducing Dots and Boxes

A long, time ago … my friend, James, invented this rather strange machine … The Two – One Machine 1 … 2

Numbers versus Codes … The number 1 is represented by 1 dot in the right-most box. … 1 0 The number 2 is represented by 1 dot in the box adjacent to the right-most box, and 0 dots in the right-most box.

Number 1 2 Machine Code 1 1 2 10

Number 1 2 Machine Code 1 1 2 10 3 4 11 ?

Number 1 2 Machine Code 1 1 2 3 10 11 4 5 6 7 100 101 110 111 https: //www. youtube. com/watch? v=q. SQyct. FGMxc

Time to Explore 1. What is the 1 2 machine code for the number 13? 2. Which number has a code of 11001 in a 1 2 machine? 3. What number does a code of 102101 represent in a 1 2 machine? Why? (assuming we have done all possible explosions)

Then one day … James had a flash of insight … “I could have just as much fun coding numbers using a …” Three – One Machine 1 … 3

Machine Code for … 14 : 112 … … … 1 3

More Questions to Ponder … 1. What is the 1 3 machine code for the number 20? 2. Which number has a code of 1022 in a 1 3 machine? 20 : 202 35 : 1022

It’s time to go wild! 1 10 Let’s go all the way up to a Ten – One machine and put 273 dots into the machine. What do you think the machine code 1 10 will be for the number 273 ?

1 273: machine code? 10 … … … 2 7 3

Number : 1 273 : 273 10 machine code? Whoa! What are these machines really doing?

Machines and Place Value 1 2 … 1 … 2 1 … 4

Connecting Numbers and Machine Codes 1 number: machine code 2 1 1 0 1 8 4 2 1 … 1 1 0 1 x x 8 4 2 1 = = 8 4 0 1 13 13 : 1101

Place Value and Powers 1 3 3 1 … 81 27 9 1 10 … 10000 100 10 1

Speaking Our Language 1 10 … 2 7 3 In our language, 273 means: Two HUNDRED(S) Seventy (seven TENS) Three (ONES)

New Ways of Looking at Arithmetic Addition and Multiplication

Addition Let’s continue to work with 1 our 10 machine as we consider the problem: 371 + 425 796

What about a … not so nice … addition problem? 168 + 395 4 15 13 Four hundred and fifteenty thirteen

168 + 395 4 15 13 How do we simplify our answer … for society’s sake?

168 + 395 563 5 5 4 15 6 13 3

Traditional Approach 1 Dots-and-Boxes Approach 1 168 + 395 563 Both approaches are mathematically correct … it’s really a matter of personal choice!

Multiplication What’s the product of: 1529 x 3 ? 3 15 6 27 We can see that the answer is: Three thousands, Fifteen (hundreds), Sixty (tens), Twenty- seven (ones)

1 5 2 9 3 15 6 27 34 15 5 68 6 27 27 7 So, 1529 x 3 = 4587 x 3

More Computing Conumdrums? 1. Compute 2714 x 6 2. What is 32148 x 10 ? How easily could you show the result 321 480 using dot-and-boxes? 3. How would you approach two-digit multiplication … say 43 x 26 ?

Subtraction Introducing Anti-dots

Let’s look at how we can deal with subtraction using our dots-and-boxes machine … beginning with the machine: 1 10 = dot = antidot + = 0

What is Subtraction … really? When we consider the problem, 564 – 213, in the traditional way … we focus on “taking away” 564 - 213 3 51

564 - 213 351

Subtraction: where un-explosions are necessary 512 - 347 12 -3 6 7 -5 5

Subtraction: How does the Dots-and-Boxes approach compare with the traditional algorithm? q The traditional approach begins on the “right” … q It has you first try to “take 2 away from 7” which you can’t do … q It has you “borrow” a unit from the ten’s column, and writing a 1 in front of the 2 in the one’s column … 512 -347 165

Subtraction – continued: 6328 - 4469 Is it easier to un-explode from left to right, or from right to left? Do you think you could become just as fast & efficient with the dots-and-boxes approach to subtraction as you are with the traditional approach? Why?

Subtraction – what about negative answers? 165 - 497 -3 -3 -2 332

Division A different perspective: finding groups of dots

Starting Slowly … with a division problem whose answer might be obvious, just by inspection 3906 3 = 1000 3906 3000 + 300 900 + 2 6 1 3 0 2

Division – by a single digit, where equal groups cannot be found in each box

Long Division – what does it look like?

Long Division – continued:

Try this one:

Long Division - what do we do with remainders? Eric Answer: 121 remainder : 1100 Hugo Answer: 211 remainder: 20

All Bases, All At Once Connecting arithmetic with algebra

What if we did a dots-and-boxes division question … but I wouldn’t tell you what base we were working in? How would you represent the problem? 1 1 … X 3

Now, it’s time for - Advanced Algebra! Compute: What would this look like on an 1 machine? X

Representing the problem in an 1 machine. X

The division problem is asking us to find copies of in the picture . Answer: Stare at this answer for a minute … does it look familiar?

In a machine: 1 10 SAME PICTURE In an machine: 1 X

So Much More to Explore! Adding & Subtracting Polynomials Multiplying Polynomials Remainder Theorem Number Theory Infinite Sums Dots-and-Boxes for Decimals & Fractions Understanding Irrational Numbers Decimals in other bases Weird & Wild Machines (e. g. 2 3)

Exploding Dots and The GLOBAL MATH PROJECT Who, What, Where, When, Why … and how you can get your students involved.

How You Can Get Involved Website: http: //theglobalmathproject. org

What we need to do for mathematics is … place it within the reach of everyone, and especially those who have long given up on connecting with the beauty, elegance and simplicity of the subject.

Thanks for listening!