This is what I wrote last year about today's lesson:
When I was young, a local PBS station aired a show called Homework Hotline. After school, middle and high school students would call in their homework questions in math and English, and some would be chosen to have their questions answered on the air by special teachers. Even when I was in elementary school, I often followed the geometry proofs that were called in, and more often than not, there were triangle congruence two-column proofs where the Reason for a step was often CPCTC. So this was where I saw the abbreviation CPCTC for the first time. (By the time I reached high school, a few calculus problems were called in to the show. Nowadays, with the advent of the Internet, the show has become obsolete.)
Here's a link to an old LA Times article about Homework Hotline:
http://articles.latimes.com/1992-02-09/news/tv-3184_1_homework-hotline
When I reached geometry, our text usually either wrote out "corresponding parts in congruent triangles are congruent," or abbreviated as "corr. parts of cong. tri. are cong.," probably with a symbol for congruent and possibly for triangle as well. But our teacher used the abbreviation CPCTC. Now most texts use the abbreviation CPCTC -- except the U of Chicago, that is. It's the only text where I see the abbreviation CPCF instead.
Dr. Franklin Mason, meanwhile, has changed his online text several times. In his latest version, Dr. M uses the abbreviation CPCTE, "corresponding parts of congruent triangles are equal."
Well, I'm going to use CPCTC in my worksheets, despite their being based on a text that uses the abbreviation CPCF instead, because CPCTC is so well known.
Once again, it all goes back to what is most easily understood by the students. Using CPCTC would confuse students if they often had to prove congruence of figures other than triangles. But as we all know, in practice the vast majority of figures to be proved congruent are triangles. In this case, using CPCF is far more confusing. Why should students had to learn the abbreviation CPCF -- especially if they have already seen CPCTC before (possibly by transferring from another class that uses a text with CPCTC, or possibly even in the eighth grade math course) -- for the sole purpose of proving the congruence of non-triangles, which they'd rarely do anyway?
So it's settled. On my worksheet, I only use CPCTC.
Notice that for many texts, CPCTC is a definition -- it's the meaning half of the old definition of congruent polygons (those having all segments and angles congruent). But for us, it's truly a theorem, as it follows from the fact that isometries preserve distance and angle measure.
Another issue that comes up is the definition of the word "corresponding." Notice that by using isometries, it's now plain what "corresponding" parts are. Corresponding parts are the preimage and image of some isometry. Unfortunately, we use the word "corresponding angles" to mean two different things in geometry. When two lines are cut by a transversal and, "corresponding angles" are congruent, the lines are parallel, but when two triangles are congruent, "corresponding angles" (and sides) are congruent as well. The phrase "corresponding angles" has two different meanings here! Of course, one could unify the two definitions by noting that the corresponding angles at a transversal are the preimage and image under some isometry. I tried this earlier, remember? It turns out that the necessary isometry is a translation. This is one of the reasons that I proved the Corresponding Angles Test using translations -- it now becomes obvious what "corresponding angles" really are. I mentioned yesterday, however, that in many ways using translations to prove Corresponding Angles is a bit awkward since it took so much work to avoid circularity. (This is why some authors, like Dr. Hung-Hsi Wu, uses rotations to prove Alternate Interior Angles instead.)
In past years, I wrote about how many teachers often given multi-day activities, but I've never done so on the blog. And so today I post such an activity:
Monday: Lesson 6-6 (Day 66)
Tuesday: Lesson 6-7 and Begin Multi-Day Activity (Day 67)
Wednesday: Finish Multi-Day Activity (Day 68)
Thursday: Review for Chapter 6 Test
Friday: Chapter 6 Test
This activity is based on the Exploration Question in today's text:
a. Find three characteristics that make Figure I not congruent to Figure II.
b. Make up a puzzle like the one in part a, or find such a puzzle in a newspaper or magazine.
Students can perform part a today and part b ("Make up a puzzle") tomorrow. Of course, if they wish, they can do the "find such a puzzle" part tonight. With this plan, we return to having only one day for review -- a formal test review to be given on Thursday.
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