The Exploding Dots story comes in 12 short-and-swift explorations

What to do:

Start with Exploration 1 and move on through! Go as far as you and your students desire.

At each stage…

1.) Share and discuss the videos with your students

2.) Share the written materials and handouts with your students (Click here for non-English materials.)

3.) And perhaps guide your students to our interactive EXPLODING DOTS web app too

Or

You can just conduct your own classroom experience using nothing but a chalkboard following what you see in the videos.

Exploding Dots What To Do

Here we will introduce and interact with a mysterious machine that sets the scene for the entire EXPLODING DOTS story and the transformational mindset to mathematics

Topic: Discovering place-value

Grades: All

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Lesson 1.1

Getting started with the story.

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Lesson 1.2

Encountering our first machine.

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Lesson 1.3

Encountering more machines.

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Lesson 1.4

Okay. Let’s now go wild.

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Here we figure out what these machines are actually doing and discover place-value. We now have a powerful visual understanding of this subtle concept that will profoundly help grade-school, middle-school, and high-school students alike.

Topic: Deeply understanding place-value. Understanding why we humans are drawn to base-ten

Grades: All

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Lesson 2.1

Explain the machines.

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Lesson 2.2

Explaining 1 < 2 machine.

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Here we make profound sense of the long addition and multiplication algorithms.

Topic: Long Addition. Multiplication.

Grades: All, but of particular interest to K-6.

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Lesson 3.1

Welcome!

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Lesson 3.2

Addition

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Lesson 3.3

The Traditional Algorithm

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Lesson 3.4

Multiplication

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See subtraction as the addition of the opposite. This insight, coupled with the Exploding Dots machinery, provides natural and deep insight into long subtraction.

Topic: Negative numbers. Long Subtraction.

Grades: All, but of particular interest to K-6.

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Lesson 4.1

Welcome!

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Lesson 4.2

Piles and Holes; Dots and Antidots

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Lesson 4.3

Subtraction

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Lesson 4.4

The Traditional Algorithm

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Long division made exceptionally clear.

Topic: Long Division. Long Division with remainders.

Grades: All, but of particular interest to K-6.

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Lesson 5.1

Welcome!

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Lesson 5.2

Division

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Lesson 5.3

Remainders

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Lesson 5.4

The Traditional Algorithm

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Here we discover that all the work of K-6 arithmetic is essentially high-school polynomial algebra.

Topic: Polynomials. Division of Polynomials. Synthetic division in disguise.

Grades: All, but of particular interest to 7-12

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Lesson 6.1

Welcome!

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Lesson 6.2

Division in Any Base.

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Lesson 6.3

A Problem!

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Lesson 6.4

Resolution

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Here we play with “infinitely long polynomials,” discover the geometric series formula, and more.

Topic: The geometric series formula.

Grades: All, but of particular interest to 7-12

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Lesson 7.1

Welcome!

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Lesson 7.2

Infinite Sums

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Another infinite process has introduced more boxes into the machines. We discover decimal numbers.

Topic: Decimals. Converting fractions into decimals. Constructing irrational numbers.

Grades: All, but of particular interest to 7-12

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Lesson 8.1

Welcome!

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Lesson 8.2

Decimals

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Lesson 8.3

Adding and Subtracting Decimals

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Lesson 8.4

Multiplying and Dividing Decimals

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Lesson 8.5

Converting Fractions into Decimals

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Lesson 8.6

Irrational Numbers

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Now we are just playing. We discover fractional bases, negative bases, and more. Each usually comes with unsolved research questions.

Topic: Base-one-and-a-half and other fractional bases. Negative and other bases.

Grades: All who want to have wild intellectual fun

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Lesson 9.1

Welcome

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Lesson 9.2

Base One-and-a-half?

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Is 0.9999… equal to one or does it not equal one? We explore this question and an even wilder one than that!

Topic: Convergence of infinite processes (limits). p-adic numbers.

Grades: All who want to have wild intellectual fun

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Lesson 10.1

A Troubling Number for our Usual Mathematics

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Lesson 10.2

A Troubling Number for our Usual Mathematics Rejects

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Lesson 10.3

Some Unusual Mathematics for Unusual Numbers

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Lesson 10.4

A Serious Flaw of Our Ten-adic Numbers

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Lesson 10.5

Who Really Cares about Ten-adic and other "adic" Number Systems?

Making Exploding Dots two-dimensional allows you to do algebra with polynomials of two variables: x and y. Scottish mathematician of the 1600s essentially did this with his multiplication checkerboard.

Topic: Napier’s historical approach to multiplication and division. Polynomial algebra with two-variables.

Grades: All who want to have wild intellectual fun

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Here is a growing collection of puzzles – some classic, some not-so-classic, some new – that can be explained with Exploding Dots. Puzzles of all levels for all ages can be found here.

Topic: Puzzles galore!

Grades: Absolutely everyone.

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