# Thinking with boxes

Well, we did meet in the Coffee Box. But beyond that, there was a history of exploring reasoning with boxes.

- Thinking about circuits - the AVOW box
- Thinking about momentum, where the sides of the box are velocity and mass
- Thinking about energy, where many trade-offs can be reasoned about by assigning say, force, to one side of the box and distance to another.

### What not to do

Since pretty much any three-term multiplicative relationship can be construed as a box, and there are plenty of them (*F* = *m* × *a*, *P* = *F* × *A*, *x* = *v* × *t*, and so on), this could get exhausting.

So, three constraints to give a direction.

- The product of the terms should be a physical quantity and be orthogonal, so could be varied independently
- Comparing the areas of boxes of different dimensions should yield insight appropriate to the effort needed to encode the relationship as a box.
- The box should represent a quantity which can be compared at different times or places (different
may-be

's).

### What we are trying-out boxes on

**Right Lines: ** Conservation laws

Conservation laws are a special way of thinking about the

may-be

's.
Of the six conservation laws in physics, we chose to explore mass and energy separately (thinking of current pre-19 practice in schools) and momentum because it is a vector. Angular momentum did not make the cut because it is not widely exploited, and we did not thinking reasoning with boxes had much to say for the conservation of charge, lepton number or baryon number.
So, here follows an exploration of reasoning with boxes about:

- mass (introducing reasoning with boxes)
- momentum (boxes for vector quantities)
- Energy and power (boxes with sides representing different quantities, but where the areas represent the same quantity). Met as the trade-off in Supporting Physics Teaching.

We're after exploring boxes as tools for thinking with

The boxes contribute to the quantity to be compared across may-be

's, allowing insight into the way that the value of the quantities can be altered. But comparing the boxes directly across may-be

's is not, in our experience, as a result of considering a number of examples, fruitful.