This unit focused on developing the vocabulary of relations and ascribing meaning and context to relations.

The initial exercise on modeling the relation between height and width of paragraphs focused on the inverse relation and how inverse relations were associated with a conserved quantity spreading out or divided up evenly in one dimension.  

The next exercise focused on modeling the dependence of orbital period and average orbital speed on the radius of the orbit of the 8 planets and pluto.  The key conceptual understanding to be achieved was how the powerlaw relations for speed and period were determined by the constraints that their product would be 2Pi R1 and their quotient would vary as 1/R2 . The deep understanding to be reached was that the because of the way gravity spreads out into three dimensions the strength of the sun's gravitational field would vary as an inverse square and as a result the "turning force" on pluto would be so week that is would need to make a very very gentle turn. That is in addition to make a giant gradual turn, pluto also has to travel around that curve very very slowly (at least as compared with the speed of the earth in its orbit). So not only does Pluto have a much larger orbital distance to travel, it also does so at a slower speed, so two independent multiplicative factors are contributing to the length of Pluto's year.

The students then connected together the possible power law relations for average speed and travel time into an overarching pattern that includes most of the basic models of physics, given the constraint that  their product is proportional to the independent variable and their ratio is proportional to a whole power of the independent variable. 

This was a very interesting "big picture" in which the models for oscillatory motion, constant acceleration, turning at a constant speed, and planetary orbits, were connected to together into one paramaratize conceptual structure.

The best way to see how this worked is to look through the gallary of student whiteboards on the subject at:

Whiteboard Hall of Fame - Systems of Relations - Pendula, Cars and Planets

Other learning objectives to this unit were:

• Reenforce the concept that increasing mass increases the gravitational attration and inertia of an orbiting object proportionally,  compensating to have a null effect on the motion.
• Introduce the concept of turning force and acceleration associated with change of direction.
• Establish the connection btween inverse square laws and the 3 spatial dimensions.
• Reenforce the concept that a system has a center about which both parts move in an opposite dance as is apparent for binary stars.
• Introduce Newton's laws.
• Derive the relation mv2/r for the magnitude of the turning force.
• Observing that the scientific method sometimes proceeds by noticing holes in a pattern and then actively looking for cases to fill in missing places in a pattern, such as when chemistis gradually filled in the periodic table, or realizing as my students did that for circular motion at a constant speed, the turning force must fall off as 1/r according to the pattern on their whiteboards.