DragginMath: Flick Right to Inject

Most interactions with DragginMath use tree nodes already on the screen to rearrange, transform, and simplify expressions. For anything you drag, flick, or tap, DragginMath knows what is possible and what the results can be. All you have to do is move things in the right direction.

Sometimes, what is already on the screen just isn’t enough. Sometimes, you need other things to exist in an expression, where those things can’t be derived from what is already there. DragginMath doesn’t know what they are unless you tell it. Still, whatever you inject into an expression must follow the rules of algebra. You can make an expression look different, but it must still mean the same thing.

This kind of change actually requires two changes: one change that does something, and another change that either undoes the first change or otherwise balances it out. So these two changes end up causing no change at all. For example, you might add a new term in one location while subtracting the same term somewhere else. The net effect is nothing.

If the net effect is nothing, then why do this? Because these two parts still exist separately and can be used separately. What you do next with these separate parts may help you move forward. For example, to transform a↑2−b↑2 into (a+b)(a−b), inject two separate ab terms, adding one and subtracting the other. After expanding the Raise↑ operators and regrouping, this becomes (aa+ab)−(ab+bb). Factor a and b to make (a+b)a−(a+b)b, then factor again to make (a+b)(a−b). Without the invention and injection of ab-ab, there is no way to get to this result using only the existing symbols.

If this feels like a trick, that is understandable. This is just one example where math requires creativity, which is surprising to many people. For most, being able to imagine and do this kind of thing requires instruction and practice.

DragginMath calls this Balanced Injection. No, we didn’t invent the technique. But we did invent the name. We had to: we looked but didn’t find another name already in use. This is amazing, because people have been doing this for centuries. Dr. Sherman Stein of UC Davis thinks it should be called simply Wow! We appreciate his enthusiasm.

Think of Do the Same Thing to Both Sides of an Equation or the Properties of Equality that are at the root of basic algebra. These are special cases of Balanced Injection, which is a larger idea that works in any expression, not just equations.

To invoke Balanced Injection, flick right on a tree node to bring up the associated dialog. The text field there lets you enter an expression, which must be both correct and complete. This is the operand you intend to inject. It may be as simple as a single number or variable, or it may be quite complicated. Be sure to tap ↵ on the keyboard to finish entering it. You can Cancel at any time, but most buttons in this dialog are not enabled until you finish your expression with ↵.

You now have several options. They all allow you to inject elements into an expression in ways that are algebraically safe. The exact forms of these injections, and the labels on the buttons, are determined by where you flick right. For example:

Flick right on a in a+b=c, then enter x in the dialog. Tapping □+■−□ results in x+a−x+b=c.

Flick right on + in a+b=c, then enter x in the dialog. Tapping (■−□)+(□+■) results in (a−x)+(x+b)=c. 

Flick right on = in a+b=c, then enter x in the dialog. Tapping (■=■)+□ results in (a+b=c)+x, which distributes to a+b+x=c+x. This is the familiar Do the Same Thing to Both Sides or Additive Property of Equality.

These results all look different (and other options are available, too), but they all mean the same thing. They differ only because of what you might want to do next. If none of the options are exactly what you want, one of them is at least close, and you can rearrange any result to suit your needs because they all mean the same thing.

How can you understand the choices presented in these cryptic button labels? When you see ■, it represents something already in the expression, while □ represents the new thing you want to inject in a balanced way. Think of □ as the contents of the dialog’s text field.

Some labels don’t include □, for example – ■ = – ■. These buttons are always enabled. They inject only unary operators, so you don’t need to enter anything in the text field or tap ↵.

Some options cause operators or relations to invert, for example + to −, or ∗ to ÷, or < to >. Injections into the relations < ≤ ≥ > may invert or not depending on the sign of the injected operand, which may not even be known when the injection is performed. This is not a bug in DragginMath, but a complication inherent to algebra that you must learn and understand. It is your responsibility to always examine the results of this powerful tool to be sure it did what you wanted.

On iPhone’s small screen, some of the dialogs are cramped. Sorry. Play around to become familiar with the various options. Not every possibility is covered, just the ones we think you are most likely to use. If you think other options should be available, or if you have a better idea about how to present this technique, we will be happy to talk with you about it. This implementation of Balanced Injection involves difficult design tradeoffs, and your thoughts and feedback are important to any improvements we might make.

It all starts when you flick right.