Exploding Dots

Overview

Here is a story that isn’t true:

When I was a young child I invented a machine (not true) that was nothing more than a series of boxes that could hold dots. And these dots would, upon certain actions, explode. And with this machine, in this non-true story, I realized I could explain true things! In one fell swoop I explained all the mathematics of arithmetic I learnt in grade school (true), all of the polynomial algebra I was to learn in high-school (true), elements of calculus and number theory I was to learn in university (true), and begin to explore unanswered research questions intriguing mathematicians to this day (also true)!

Let me share this story with you. See how simple and elegant ideas connect to elegant and profound ideas in mathematics as a whole. See how these ideas will COMPLETELY REVOLUTIONIZE your thinking of school arithmetic and algebra and beyond! This mathematical story will knock your socks off!

 

FLASH NEWS! This story is the featured topic for Global Math Week 2017 of the The Global Math Project. All teacher guides to delivering this material with, and without, technology in the classroom appear in the opening Welcome section of each experience. Also, sprinkled throughout the appropriate lessons are additional videos and features created by our world-wide community of users. We give particular thanks to Goldfish and Robin and their friends for their spectacular “kids explain math for kids” videos, doing a better job than me in explaining the mathematics!  (It is worth fishing through all the content here just to find their videos!)  

Wanna see most all of Exploding Dots in one hit? Here is a filmed one-hour lecture of me delivering the story.

Wanna try a wordless puzzle for fun? See if you can figure out what this is all about? Have a look at this animation: 

 

 

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Lessons

1.0 EXPERIENCE ONE: The Machinesarrow
1.1 Welcomearrow
1.2 The \(1 \leftarrow 2\) Machinearrow
1.3 Other Machinesarrow
1.4 The \(1 \leftarrow 10\) Machinearrow
1.5 Wild Explorationsarrow
1.6 Solutionsarrow
2.0 EXPERIENCE TWO: Insightarrow
2.1 Welcomearrow
2.2 Explaining the \(1 \leftarrow 2\) Machinearrow
2.3 Explaining More Machinesarrow
2.4 We Speak \(1 \leftarrow 10\) Machinearrow
2.5 Wild Explorationsarrow
2.6 Solutionsarrow
3.0 EXPERIENCE THREE: Addition and Multiplicationarrow
3.1 Welcomearrow
3.2 Additionarrow
3.3 (Optional) The Traditional Algorithmarrow
3.4 Multiplicationarrow
3.5 (Optional) Multiplication by 10arrow
3.6 (Optional) Long Multiplicationarrow
3.7 Wild Explorationsarrow
3.8 Solutionsarrow
4.0 EXPERIENCE FOUR: Subtractionarrow
4.1 Welcomearrow
4.2 Piles and Holes; Dots and Antidotsarrow
4.3 Subtractionarrow
4.4 (Optional) The Traditional Algorithmarrow
4.5 Wild Explorationsarrow
4.6 Solutionsarrow
5.0 EXPERIENCE FIVE: Divisionarrow
5.1 Welcomearrow
5.2 Divisionarrow
5.3 (Optional) Division by 10arrow
5.4 Remaindersarrow
5.5 (Optional) Deep Explanationarrow
5.6 (Optional) The Traditional Algorithmarrow
5.7 Wild Explorationsarrow
5.8 Solutionsarrow
6.0 EXPERIENCE SIX: All Bases, All at Once: Polynomialsarrow
6.1 Welcomearrow
6.2 Division in Any Basearrow
6.3 A Problem!arrow
6.4 Resolutionarrow
6.5 (Optional) Remaindersarrow
6.6 (Optional) The Remainder Theoremarrow
6.7 (Optional) Multiplying, Adding, and Subtracting Polynomialsarrow
6.8 Wild Explorationsarrow
6.9 Solutionsarrow
7.0 EXPERIENCE SEVEN: Infinite Sumsarrow
7.1 Welcomearrow
7.2 Infinite Sumsarrow
7.3 (Optional) Should we believe infinite sums?arrow
8.0 EXPERIENCE EIGHT: Decimalsarrow
8.1 Welcomearrow
8.2 Decimalsarrow
8.3 Adding and Subtracting Decimalsarrow
8.4 Multiplying and Dividing Decimalsarrow
8.5 Converting Fractions into Decimalsarrow
8.6 Irrational Numbersarrow
8.7 Decimals in Other Basesarrow
8.8 Wild Explorationsarrow
8.9 Solutionsarrow
9.0 EXPERIENCE NINE: Weird and Wild Machinesarrow
9.1 Welcomearrow
9.2 Base One-and-a-half?arrow
9.3 Does Order Matter?arrow
9.4 Base Two? Base Three?arrow
9.5 Going Really Wildarrow
100.0 Where did these ideas come from?arrow
100.1 To answer …arrow
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