Geometry CO.6-8 and 8th grade connections as well as "prove" standards

[tsebring]

In 8th grade, a foundation of isometry is laid with 8.G.3 and 8.G.4. The Geometry standards that relate are CO.6-8. I am trying to discern the intent of CO.6 and its relation to 8.G.3 and 8.G.4. For example the wording in 8.G.3 specifically mentions coordinates whereas the wording of CO.6 does not use “coordinates”. Would predicting “the effect of a given rigid motion on a given figure” include a student giving the coordinates of a point on a figure that has undergone a reflection, rotation or translation? As CO.7 refers to congruence I am assuming the predicted effect is not that the figures are congruent. As CO.5 includes naming the transformation I also did not that as as “the effect”.

Lastly, the Geometry standards have many “prove theorems”. I have two questions about these standards. The application of several of these theorems is not always seen in the standards, is there a reason for that? Or would that be “implied”? And lastly, I have seen some books that named “corresponding angles are congruent” as a postulate. What resource would you suggest using as an authority for what is a theorem and what is a postulate?

[Dr. M]

I’ll respond to the question about parallels.

There’s no one right answer about what to call a postulate and what to call a theorem. Euclid offered a proof of the SAS triangle congruence principle in his Elements, but Hilbert, in his Foundations of Geometry, made that principle is postulate. Neither is right. Neither is wrong. They simply made different decisions.

But there are certain principles that should guide the choice of a set of postulates. One is that it should not include superfluous postulates, i.e. postulates that can be proven on the basis of other postulates already in place. I do think that a bit of that is fine in a high school geometry classroom, but it should be kept to a minimum. That’s why I so dislike, for instance, when a text makes both SAS and SSS triangle congruence postulates. SAS can be used in a relatively straightforward proof of SSS!

This principle implies that we should not make the statement below (or any other equivalent to it) a postulate:

When a transversal cuts a pair of lines so that alternate interior angles are congruent, then those lines are parallel.

Why not? It’s provable! See Book I, Proposition 27 of the Elements.

A second principle that should guide the choice of a postulate set is that the postulates chosen should be both simple and obviously true. (I mean this to hold only for the high school classroom. These requirements are dropped at higher levels.) That’s why, when I teach parallels, I choose the Playfair Postulate. (Through a point not on a line, there’s at most one line parallel to that given line.) It’s clear and (to students’ minds) obviously true. Together with the proposition above, it can be used to prove that if a pair of parallel lines are cut by a transversal, then alternate interior angles are congruent. (Here’s a quick sketch of the proof. Assume that point P does not lie on line m. Construct line n parallel to m through P. Construct a transversal to m and n through P. If alternate interior angles are not congruent, then we can construct a second line r through P for which they are. But then this line r will be parallel to m, and so we then have two lines through P parallel to m. This contradicts the Playfair Postulate. Hence alternate interior angles are in fact congruent.)

[Dr. M]

Let me add that I find Harold Jacobs text Geometry: Seeing, Doing, Understanding just about right for a high school classroom. The one real deficiency from the point of view of the CCSS is that it makes SAS and a postulate. That’s a problem, though, that’s relatively easily rectified. (He also gives Euclid’s proof of the AA triangle similarity principle. By my lights, that’s fine. But one might like to do it instead with some sort of scale transformation postulate.)

[Bill McCallum]

First, I would say the isometry standards in Grade 8 are 8.G.1–3. 8.G.4 is about similarity transformations.

On the question about G-CO.6, I think the words “geometric descriptions of rigid motions” are important. In Grade 8 students might have a fairly intuitive notion of rigid motions; in high school they work with definitions of rigid motions in geometric terms. For example, in Grade 8 you might say that a reflection about a line takes every point to the point on the other side at the same distance from the line. In high school you might say that reflection about the line $\ell$ takes a point P to itself, if P is on $\ell$, and otherwise takes to a different point P’ such that the line through P and P’ is perpendicular to l, intersecting it at O, and PO is congruent to OP’. Predicting effects using this description is partly simply a matter of grasping which rigid motion is being described.

I also agree that predicting effects could include naming the coordinates. It could also include other observations: for example, reflecting about the side of a triangle produces a triangle that shares a side with the original.

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