__Three Part Equations and Stuff__

I was in high school before I finally simultaneously had a math and a physics course which, though it was not taught as a specific idea, made it clear that beyond just counting and measuring, numbers and variable symbols can describe the real, physical world and events in it and enable us to figure out things we might not be able to directly measure. This is an idea that should be made available to the child from the beginning.

It is simpler than it may seem. Numbers describe reality. Count and measure. But The relationships between numbers also describe reality. Count, measure and model. Reality can be measured in many ways. What you can measure you can use, like flour and yeast.

So...

this = that

*the other thing*

**/**the slanted line there means divided by

Dr. Suess math

the equation also means

that = this X the other thing

the X there means, times, multiplied by

and so also

the other thing = that / this

Once you know arithmetic multiplication and division are no problem. Even before you know arithmetic fundamental quantities can be experienced and complex quantities named and talked about.

When things can be combined in groups of three with an "Equals" condition and an Operation between two of them, then any time you can physically measure any two of them you can figure out the third, describing an aspect of an object or energy that may not be immediately apparent or understandable to the senses. In the Child's Physics we present physical situations that make the operation concrete. The first three part equation, described in Sticks and Ticks, is...

Speed = distance

**/**time

There is a three part equation, called Ohm's Law, that describes Electricity...

Current = Voltage / Resistance

These three parts can be measured directly with a simple meter that only requires the child to set a knob, touch two points in a circuit with probes and watch the dial react, similar to using the battery tester. The primary point being that these aspects of Electricity exist, have names, can be measured and are directly related to each other. Batteries, being the power source, represent the idea of voltage, how hard you push. Elements in the circuit offer resistance, how hard it is to push. The circuit itself provides a flow of current, how much gets pushed. The concrete, physical experience of the elements in the three part equation can and should precede the equation itself and the arithmetic it expresses.

Three part equations can have elements that are built out of simpler equations.

Liquids are one of the four basic states of matter: solid, liquid, gas and plasma. Consider a three part equation for water in pipes, like a garden hose, a straw or a squirt gun. The impact of the water coming out,

*how it feels*when you get squirted, is called pressure. How hard it is being pushed out is called force, which is a mechanical analogy of voltage, and the cross section size of the pipe or tube or hose is called area which is like resistance. So we have...

Pressure = Force / Area

That word “Force” has another three part equation inside it. Equations get layered one in another like nested boxes or a file system's nested folders. So inside that three part equation is this...

Force = Mass x Acceleration

so

Pressure = (Mass X Acceleration) / Area

Force is how hard you push and Mass is how much stuff, weight, electricity, water...stuff....gets pushed. While Mass and weight are not identical, in practical matters on the surface of the Earth mass is measured is by weighing the object. The SI unit of mass is the kilogram. So in this sense and in the child's physics, mass is an irreducible primary physical measurement like space and time.

Acceleration , on the other hand, has another nested three part equation inside it...

Acceleration = change in Speed / Time = (speed 2 - speed 1)/time

but

Speed = Distance / Time

and

area = distance X distance

so we're back to the beginning, the primary measurements of space and time, Sticks and Ticks

so

Pressure = Force/Area

can be written

pressure = ( mass X (distance/time - distance/time)/time)/(distance X distance)

While gauges and sensors exist to give direct, calculated measurements of complex aspects like pressure and acceleration, the only raw measurements involved are weight, space and regular repeating movement. All three of which are well within the abilities of the primary age child to handle and count.

So to Sticks and Ticks we add Stuff.

We give the child numbered rods, clocks and scales.