Thursday, January 29, 2015

Review for Chapter 11 Test (Day 102)

It's time to prepare for the next test. I'm labeling this as the "Chapter 11 Test," even though we didn't cover all of that chapter in the U of Chicago, and instead included sections from other chapters.

In particular, this test is based on the SPUR objectives for Chapter 11. As usual, I will discuss which items that I have decided to include and exclude, and the rationale for each:

Naturally, I had to exclude Objective G: equations for circles, which I take to be an Algebra II topic, not a Geometry topic. (If this had been an Integrated Math course, I would have delved more into graphing linear equations, as we covered this week.) Also, I left out Objective J, three-dimensional coordinates, as we haven't covered Chapters 9 or 10 on 3D geometry yet.

One major topic that I had to include is coordinate proof, as this appears in Common Core. I did squeeze in some coordinate proofs involving the Distance or Midpoint Formulas, but not slope. So therefore, the coordinate proofs included on this review worksheet all involve either distance or midpoint, not slope. The only proofs involving parallel lines had these lines either both vertical or both horizontal. Once again, a good coordinate proof would often set it up so that the parallel lines that matter are either horizontal or vertical.

What good are coordinate proofs, anyway? Well, a coordinate proof transforms a geometry problem into an algebra problem. Sometimes I can't see how to begin a synthetic geometry proof, so instead I just start labeling the points with coordinates and see what develops.

So coordinate geometry reduces an unknown problem (in geometry) to one whose answer is solved (in algebra, in this case). Mathematicians reduce problems to previously-solved ones all the time -- enough that some people make jokes about it:

I ended up including six straight problems -- Questions 8 through 13 from U of Chicago. Most of these questions are from Objective C -- the Midpoint Connector Theorem. The text covers this here in Chapter 11, but we actually covered it early, in our Similarity Unit, because we actually used the Midpoint Connector Theorem to start the proof of the basic properties of similarity. Still, this was recent enough to justify including it on the test.

Next are a few center of gravity problems. This is straightforward, since all we have to do is average the coordinates. Afterwards are a few midpoint problems, including two-step questions where one must calculate the distance or slope from one point to the midpoint of another segment.

Then there are a few more coordinate proofs where one has to set up the vertices -- notice that some hints are given in earlier questions. Finally, one must find the point where two lines intersect. This involves either substitution or elimination (from Algebra I), but these problems are both simple.

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