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

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.

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

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

Long division made exceptionally clear.

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

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

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

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

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

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.

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.