# NumberPad - Playing with Numbers

- 5 mins# Toys in NumberPad

Numbers can represent almost anything — saliva production per day, the cost of a pool on the moon, or the volume of earth’s viruses. Yet, on their own they are kinda boring.

\[487\]NumberPad is for playing with numbers. (Check out this previous post to see the basics of NumberPad.) How can we show numbers in a less boring way? Here is one small experiment of tying those numbers to “toys”.

This football is tied to `x`

and `y`

. When those numbers change, the
football moves to match.

We can make `x`

and `y`

both depend on `time`

, then make the football move both
directions at once. I set up the equations \(x = velocity_x * time\) and
\(y = time^{power}\). With `time`

selected, I can change the value and have
it update both `x`

and `y`

.

If we wanted to tweak `power`

and see how it changes the arc we could:

- Select
`power`

. - Change it from \(y = time^2\) to \(y = time^{2.5}\)
- Select
`time`

- Scrub back and forth to see the new effect

The turnaround makes it hard to see the effects of playing with a number. A
better way is to see the entire trajectory of the football all at once, so we
can instantly see the impact of a change. Here is a visualization of the
football trajectory as I change `power`

.

What you are seeing are “ghost” versions of the football at `time + 1`

,
`time + 2`

, `time + 3`

, etc as well as `time - 1`

, `time - 2`

, `time - 3`

, etc.

The strength of NumberPad is that you can change any value without solving to
isolate the variable. In this example, that means we can grab the football
and have it change `velocity_x`

and `power`

automatically. As I drag the
football and change `x`

and `y`

, NumberPad calculates
\(velocity_x = \frac{x}{time}\) and \(power = log_{time}y\).

NumberPad tries to make equations concrete by always showing one solution to the
equation because most equations are easier with one solid example. Sometimes
a single example is not enough. The most important aspect of an equation might
be how it *changes*. Showing the trajectory of the football is higher up the
ladder of abstraction. From this
vantage point we can see how our equation is about motion. We can also see how
the motion itself changes with different values.

We can also use this abstraction to jump to different concrete solutions. Tap
on one of the ghost footballs to jump to a specific time. Here I tap to jump
to the concrete solutions where `time`

is `2`

, `5`

, `6`

, `7`

, `9`

, or `11`

.

Putting this all together, we can play with the equation for projectile motion
with gravity. We don’t have to change `x`

, but the equation for `y`

is now
\(time * velocity_y + time^2 * gravity\). It looks more complicated in
NumberPad because it shows step of the calculation.

Once the equation is set up, I can select `time`

and scrub it to see the
football fly through the air.

It is fun to play around with. I can grab the football have gravity change until we have the perfect arc.

Or, we change only the \(velocity_x\) and see that the ghost football stays in the air for the same amount of time. We can tell because it is always the same ghost football, which represents the football at \(time = 8.7\) which is touching the ground.

Toying with numbers is a fun way to discover relationships and get a feel for equations. Tying the numbers to a physical representation lets us visualize them in a concrete way, and the “ghost” effect lets us see the relationships at a higher level of abstraction.

I’m still at the stage where I’m prototyping these concepts in NumberPad. If you’ve got ideas for practical applications I’d love to hear them @bridgermax.