Using these kinds of relationships you can make models that represent situations rather than processes.

You are working in this quadrant of the modeling space:

Let's start with a simple situation: comparing the impulse with the change in momentum. Here's a simple model which has the two quantities, and allows you to vary their values by dragging. But there are no rules in the imagined world: there are no rules connecting the two quantities:

There are deep reasons for thinking that these two will always have identical values, that is both the numerical values and the units will be equal.

WordFraction{number for impulse}{unit for impulse}}{WordFraction{number for change in momentum = unit for change in momentum}

That is, the two are constrained. Vary the impulse and you necessarily alter the change the momentum **and** vary the change in momentum and you necessarily alter the impulse. It is a genuine assertion of equality, not an assertion of assignment.

change in momentum = impulse

Is identical in meaning to

impulse = change in momentum

(There is evidence that this is not how children read and use the equals symbol, and that this distorts their understanding of the snapshot-like nature of situations: so the distinction is important in teaching physics).

Fortunately we can model this. Here is a constrained pair: they go up and down together. Drag either to vary the value and the other alters.

You can present this as either an empirical discovery about the world, or an a priori truth: but whichever way you look at it, it's a rather significant and reliable relationship. So worth treating with respect—it was a hard won discovery.

Let's start with an old chestnut.

Fm = a

But respecting the insights into the abuse of the equals sign, perhaps exacerbated by the use of triangles

, let's represent these three quantities, at first in an model where there are no rules.

You might not like the arrangement of the representations here—the force is not exerted on the mass—but it suited me to arrange it to match the relationship above. If you object strongly enough , adapt the code(EFTR).

Now you could add some rules into the imagined world, representing the discovery that *a*,*F* and *m* are bound together in a permanent relationship. Here the mass is kept constant, so that you can explore the effect of one independent variable on one dependent variable, again by dragging.

If you' d like to change the quantity that is locked, adapt the code.

Again, whether an empirical discovery about the world, or an a priori truth: another rather significant and reliable relationship. It's not just a calculating device: it's a relationship between important quantities in the physics world.

Constraining is all about exploring what could or could not be the case in a situation. Some possibilities are ruled out, because of the rules that we've learnt to be reliable guides to action (the 'laws 'of physics). So you could make a bit of a fuss about this, exploring first the possible values when the quantities are unconstrained, and then the possibilities when the quantities are constrained.

So we got back to linear graphs, but by an insightful route.

Constraints apply to many different relationships, not just to two- and three-term relationships: some you'd expect to come across appear in the categorisation above.

Two different circuits can be designed so that values of pd and current are such that they switch the same power.

Reasoning about transformers builds on this, where the conservation of energy underpins the relationship.

ProductABCD{SymbolSub{I}{1}}{SymbolSub{V}{1}}{SymbolSub{I}{2}}{SymbolSub{V}{2}}

So create the quantities and then constrain them, to give a model:

Here you can drag both currents to set the values, but neither pd. That allows you to have one independent and one dependent variable. To change these, clone the code and alter the model.

**Back to SPT**

This idea of constraint relationships has been in SPT for a while, and, for those of you who prefer a more visual, but less flexible and more limited modeling environment, there was Quantities with Constraints(QWC), a Flash-based modelling tool, now deprecated.

**Even further back**

If you prefer something more flexible, but still graphical, again there was Variables and Relationships(VnR), although its metaphor does not do such a good job on constraining.