Wednesday, April 4, 2018

Lesson 13-7: Exterior Angles (Day 137)

This is my milestone 700th post. Last year, I tried to make my 500th post on Pi Day and failed. This year, I'm even further away from the big math holiday as my 700th post is three weeks late.

Anyway, I decided to change the blog background for the special 700th post. My original plan was to change the background for Pi Day last year -- then I ended up not even posting that day. This year, I posted on Pi Day, but still kept the old background. I don't wish to go a third year with this same background, so I finally changed it -- to a bookshelf. I recall seeing this same bookshelf on another blog and decided I liked it. Perhaps I'll fall into the habit of changing it at every milestone post.

Anyway, this milestone does fall on a special date -- the 50th anniversary of the MLK assassination. I've obviously mentioned the civil rights leader's birthday on the blog each year (due to the school holiday), but this is the first time I've mentioned his death day.

(Google, meanwhile, typically celebrates birthdays, not death days. This is why the Google Doodle for today is of Maya Angelou, not Martin Luther King, Jr.)

Lesson 13-7 of the U of Chicago text is called "Exterior Angles." In the modern Third Edition of the text, the content of this lesson has been split. Exterior angles themselves now appear in Lesson 5-7 (which, just like the same numbered lesson in the Second Edition, is on corollaries to Triangle Sum), while Unequal Sides and Unequal Angles now appear in Lesson 6-2 (as the scalene analogs of the Isosceles Triangle Theorem).

This is what I wrote last year about today's lesson:

Unequal Sides Theorem (Triangle Side-Angle Inequality, TSAI):
If two sides of a triangle are not congruent, then the angles opposite them are not congruent, and the larger angle is opposite the longer side.

Given: Triangle ABC with BA > BC
Prove: angle C > angle A

Proof:
Statements                                     Reasons
1. Triangle ABC with BA > BC     1. Given
2. Identify point C' on ray BA       2. On a ray, there is exactly one point at a given distance from
with BC' = BC                               an endpoint.
3. angle 1 = angle 2                       3. Isosceles Triangle Theorem
4. angle 2 > angle A                       4. Exterior Angle Inequality (with triangle CC'A)
5. angle 1 > angle A                       5. Substitution (step 3 into step 4)
6. angle 1 + angle 3 = angle BCA  6. Angle Addition Postulate
7. angle BCA > angle 1                  7. Equation to Inequality Property
8. angle BCA > angle A                  8. Transitive Property of Inequality (steps 5 and 7)

The next theorem is proved only informally in the U of Chicago. The informal discussion leads to an indirect proof.

Unequal Angles Theorem (Triangle Angle-Side Inequality, TASI):
If two angles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the larger angle.

Indirect Proof:
The contrapositive of the Isosceles Triangle Theorem is: If two angles in a triangle are not congruent, then sides opposite them are not congruent. But which side is opposite the larger angle? Because of the Unequal Sides Theorem, the larger side cannot be opposite the smaller angle. All possibilities but one have been ruled out. The larger side must be opposite the larger angle. QED

In the new Third Edition of the text, not only is Unequal Sides proved right after the Isosceles Triangle Theorem, but both converses (Unequal Angles and Converse Isosceles Triangle) are proved all in the same lesson.

I've also seen -- and mentioned numerous times on the blog -- a theorem order that's completely different from either the Second or Third Editions of the U of Chicago text. The Triangle Exterior Angle Inequality is proved first -- not as a trivial corollary of the Exterior Angle Equality, but as a separate theorem a la Euclid. Then TEAI is used to prove Unequal Sides and Angles just as is done in this lesson. Finally, Unequal Angles is used to prove the Triangle Inequality of Chapter 1.

As I've written before, using TEAI to prove the other theorems could be interesting, but this year we're following only the U of Chicago text. (Yes, my blog is all about going by the book -- now you can see why I chose to have books in the background!)

Since today is the MLK assassination anniversary, it's a good day to continue the discussion I started last Palm Sunday about tracking and race. But I don't wish to dig too deeply -- even though my new district is still on spring break, this is officially labeled as a school year post in my old district, and so I'm to keep race and politics to a minimum in school year posts.

I do want to make two more comments:

• Traditionalists argue that students learn more effectively under tracking, but opponents counter that tracking leads to racial segregation. Traditionalists point out that students placed on the low track feel more comfortable and are happier when they actually understand the material rather than forced to move faster. But parents often object when their own children are placed on a lower track. So perhaps a partial solution to the tracking problem is to have a low track, but parents must consent for their children to be placed there. Will any parent ever say, "Place my own child on the low track, for he/she will feel more comfortable and will be happier to understand the material..."? If this happens, then the racial argument vanishes -- I'd have no problem with everyone on the low track being of the same race if I knew that it was their own parents placing them there.
• Tracking isn't currently in the news -- but racial disparities in school discipline are. The current Secretary of Education, Betsy DeVos, is joining the big debate. Instead of pursuing this further in my post, let me just link to the relevant New York Times article:

According to the article, it's not just a racial matter -- disability and orientation make a difference in suspension rates as well. In the end, though, it's all about civil rights -- and that's exactly what Martin Luther King stood for half a century ago.