All Circles Are Similar

This lesson does two things at once, and it is the first time the course really touches transformations. It is quiet groundwork for the work that comes in Unit 10. The surprise at its center is genuine and worth preserving: every circle is similar to every other circle. Not most circles, not circles of related sizes, all of them. I rewrote the launch this year. Instead of handing students the transformation and asking them to carry it out, I ask them to try to build two circles that are not similar. The failure to do it is the engine of the whole lesson.

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The resolution is almost too simple, slide one center onto the other, then dilate by the ratio of the radii. The question worth lingering on is not how but why it always works. A circle has exactly one degree of freedom: once you fix the radius, the shape is locked, and there is nothing else about it that could make it a different shape. I push this by asking groups what other shapes are always similar. When students started writing shapes like octagons (they were only picturing stop signs, forgetting that was not the only form of the shape that met the definition of octagon) I asked if they remembered the “guess my rule game” from the first day of class. One student immediately said “we aren’t checking out assumptions are we?” With that I sent them to look for counter examples to things in their list.

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The triangle work is where past learning resurfaced in lovely ways. Most groups translated and then dilated correctly. The standout was a group that located a point of dilation and a scale factor and moved their triangle that way, instead of by left–right and up–down the reached back to an earlier lesson entirely unprompted. Two slips are worth watching for: groups that dilate without saying from where, and groups that align different pairs of corresponding points. Showing the class that both correct methods land in exactly the same place is worth doing out loud.

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In consolidation I used Desmos to show the translation as a vector, and — yet again this year — the vector turned out to be the hypotenuse of a hidden right triangle, a thread that keeps reappearing across the course. This is the natural place to introduce the ⟨a, b⟩ component notation for vectors.

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The π thread runs underneath all of it. Because every circle is similar to every other, the ratio of circumference to diameter is the same for all of them, that constant is π. I ask “what is pi?” every single day of this unit, trying to retire “3.14,” which captures nothing meaningful, in favor of “the ratio of a circumference to its diameter.” Once that is the working definition, the circumference formula stops being a thing to memorize: just solve π = C/d for C