But what is x? It is an unknown — just like in algebra.
The cells will contain dots with values 1, x, x2, x3 etc, but obviously exploding and unexploding are not possible. This corresponds to like terms when adding and subtractingin algebra—being unable to explode across cell boundaries is the same as being unable to combine unlike terms in an expression.
However, algebraic division works very well with these machines, with just one little change. When dividing with a 1 ← n machine, if it is not possible to complete a full pattern match, we resort to unexploding to change the dot distribution. With a 1 ← x machine, we instead add dot/anti-dot pairs to make the match possible. Since these pairs cancel each other out, the value is not being changed.
Click on the small dot/anti-dot button beneath a cell to add one of these pairs.
Choose a type of Exploding Dots machine using the buttons below.
The development of these exploding dots activities was inspired by James Tanton's resources available at G'Day Math.