Thursday, February 11, 2016

Review for Chapter 11-12 Test (Day 101)

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 lessons from Chapter 12.

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 save for the final unit of the year on circles along with the rest of Chapter 15. (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 four straight problems -- Questions 8 through 11 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.

Now we move on to Chapter 12. Notice that I created this year's test by combining the first page of last year's Chapter 11 test with the second page of last year's Chapter 12 test. By doing so, I ended up leaving out most of the dilation questions -- much to the delight of the traditionalists -- and instead included most of the classical similarity problems. (It also means that the question numbers skip from 7 to 10 -- oops! Just give the students two extra questions or something.)

As usual, most of these problems will be from the "Chapter Review: Question on SPUR Objectives" section of the U of Chicago text. I had to skip around in order to avoid the problems that relate to area or volume, since these involved the skipped Chapters 8-10. In particular, I included ten straight problems from the text, all in the thirties, since this stretch of questions avoids area and volume.

Here is the test review:

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