Monday, October 26, 2015

Lesson 7-3: Triangle Congruence Proofs (Day 42)

Lesson 7-3 of the U of Chicago text discusses triangle congruence proofs. Finally, this is what most Geometry students and teachers think of when they hear about "proofs."

There's actually not much that I have to say about this lesson. We already know much about how the proofs in this section go -- generally we are given two triangles with some of the corresponding sides and angles given as congruent. In the easiest examples, the given pairs are already enough to conclude that the triangles are congruent. The U of Chicago text points out that often we must work to get one of the needed pairs -- such as the Reflexive Property of Equality when the two triangles have a side in common, or perhaps the Vertical Angles Theorem to find a pair of congruent angles -- then we use SAS, SSS, or ASA to conclude that the two triangles are congruent. Finally, the students are usually asked to prove one more pair of parts to be congruent, which requires the use of CPCTC.

Now technically speaking, CPCTC was covered in Lesson 6-7, which we skipped over. The U of Chicago text uses the abbreviation CPCF, which stands for corresponding parts of congruent figures, for after all, the property applies to all polygons, not just triangles. But the abbreviation CPCTC is so well-known that I prefer to use CPCTC rather than CPCF. Even though we skipped Lesson 6-7, I believe that the students can figure out CPCTC quickly, so I incorporate it into today's lesson.

Last year I didn't create a worksheet, but instead just wrote down ten problems. This year, I will create a worksheet and include some of the problems from last year, but others I had to throw out. In particular, I had to drop the proof of the Converse of the Isosceles Triangle Theorem -- we certainly want to avoid the U of Chicago proof that uses AAS, since we haven't taught AAS yet. I want to cover it at the same time as the forward Isosceles Triangle Theorem -- I'm saving both for next week. I also had to throw out a few "review" questions that once again review lessons that we've skipped.

Notice that Question 5 mentions isosceles triangles, but it doesn't actually require either the Isosceles Triangle Theorem or its converse. We only need to use the definition of "isosceles" to get one pair of congruent sides -- the first S. The definition of "midpoint" gives us the second S, and the Reflexive Property of Congruence gives us the third S, so the triangles are congruent by SSS.

Indeed, this is a good time for me to bring up a point about two-column proofs. I can easily see a student being stuck on this question because none of the three S's are given -- they have to work to find all of them. One thing I like to point out is if the students see a long word like "isosceles" -- or even "midpoint" -- in a question, they will usually need the definition of that word. In particular, a big word in the "Given" usually leads to the students using the meaning half of the definition, and a big word in the "Prove" often needs the sufficient condition half. The meanings of the words "isosceles" and "midpoint" lead to two of the three S's, and then they'll probably see that the Reflexive Property gives the third S.

Of course, soon they'll learn about the Isosceles Triangle Theorem and its converse -- and once those appear, nearly every proof involving isosceles triangles will use one or the other. But before students learn about the theorems pertaining to a key term, it's the definition that appears in the proof. We've already seen a few proofs about parallel lines where it's the definition of "parallel" that matters -- but once we've proved the Parallel Tests and Consequences, we hardly ever use the definition of parallel.

Last thing -- in the U of Chicago text, the "Given" is never included as a separate step. I always include it as this is more common -- so our proofs will have one more step than those in the text.

OK, my very last post was about traditionalists vs. Common Core, so I shouldn't be posting another such post right away. But over the weekend, another elementary Common Core horror worksheet went viral, and I want to address it.

There was a third-grade assignment where students are supposed to multiply 5 times 3 using some repeated addition. A student wrote 5 + 5 + 5 = 15, but was marked wrong, because the answer intended by the teacher was 3 + 3 + 3 + 3 + 3 = 15. The next question was to multiply 4 times 6 using an array. The student drew six rows of four dots, but was marked wrong -- the answer intended by the teacher was four rows of six dots.

Just as with the infamous Common Core check, the mathematician Hemant Mehta came in to explain what is going on with these two problems. (Warning: Mehta's blog is mostly about religion and politics -- yet once again, it's the math post that draws the most comments.)

Mehta explains that even though multiplication is commutative, drawing 4 times 6 as four rows of six rather than six rows of four is more consistent with the matrices that one learns in Algebra II:

What about the array problem? This one’s even more straightforward.
It actually matters which way you draw the picture. But it’s not something kids will understand until they start using matrices in algebra class.
Even if you’re unfamiliar with matrices, here’s what you need to know: There’s a difference between a 2 x 3 matrix and a 3 x 2 matrix.
So here Mehta argues that a third grade teacher who has her students distinguish between four rows of six and six rows of four will help prepare the students for higher math.

So now let's look at the first problem. Mehta tries to explain here why 5 * 3 should be written only as 3 + 3 + 3 + 3 + 3 and not as 5 + 5 + 5. But I disagree with Mehta here. For just as we appealed to higher math to explain how to draw the array, I can appeal to higher math to explain why, in fact, we should say that 5 + 5 + 5 is correct and 3 + 3 + 3 + 3 + 3 is wrong.

To find out why, the higher math we seek out is set theory -- the same theory that mathematicians all the way from Georg Cantor to Randall Holmes studied. In particular, we consider something called an "ordinal." Instead of 5 * 3, let's consider omega * 3, where omega -- the last letter of the Greek alphabet -- is a particular ordinal. Just as with matrix multiplication, ordinal multiplication is not commutative -- and in fact, omega * 3 is exactly omega + omega + omega. It is definitely not the same as 3 * omega. So using ordinal arithmetic, 5 * 3 = 5 + 5 + 5 is exactly right.

I've decided that it would detract from this Geometry blog to get into a deep discussion of what exactly ordinals are -- except that they have something to do with order (which helps to explain why the order matters). Here's a link that describes ordinal multiplication in more detail:

I can see why students would be frustrated when they lose points for multiply in the wrong order. As Geometry teachers, we can compare this to the student who complains about losing points for not putting square units for area or cubic units for volume. Of course, there are actual reasons that students must put square or cubic units. But juxtaposing these two situations in which students lose points even after doing all of the calculations correctly, we can see why from the students' perspective they are just losing points for silly reasons in both cases. This is indeed, exactly what happened today when I was subbing -- remember that today was my last day at a continuation school. A student -- who had been upset at something that had happened earlier in the day -- didn't want to listen to me when I told him to write down squared or cubed when calculating area or volume. (To believe that
students at a continuation school would listen to me just because I was "the sage on the stage"!)

Again, Mehta's post has drawn more than a thousand replies. In that comment thread, I noticed that one poster actually mentioned ordinal multiplication as an example! A poster named MNb wrote:

As a math teacher myself I think this case is very straightforward. The task explicitely was not about the answer, but about the method:
"Use the repeated addition strategy"
I do this all the time myself. If I give a quadratic equation and I ask them to use factorization then using the abc-formula is simply wrong. And I tell my pupils in advance.
[emphasis MNb's]

And it was a poster with the single letter b who responded with ordinals:

You are not teaching students a useful skill. You are teaching them an arbitrary rule that has no applicability outside passing a test that you wrote. No mathematician would say that pq can only mean p groups of q, never q groups of p. It can be either, or neither.
After thinking for a while, I've come up with two cases where a product has an asymmetrical grouping interpretation in real mathematics.
One is that matrices are more commonly divided into column vectors than row vectors, so if you think of a 5-component vector as a group of 5 things, then a 5×3 matrix is more likely to be 3 groups of 5 than 5 groups of 3.
The other is ordinal multiplication, where ω·3 means ω+ω+ω, while 3·ω means 3+3+3+... (ω times), which is less than ω+ω+ω.
Note that in both cases the order is the opposite of what you teach.
[end b's response]

I was actually looking for a simple traditionalist reply, but didn't find it quickly enough. Most traditionalists would say that the goal is for students to multiply quickly -- to be able to say that 3 times 5 is 15 in one second or so. All this about 3 groups of 5 vs. 5 groups of 3 should only be the means to that end. And I agree with traditionalism at that age (third grade).

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