Lesson 12-10: The Side-Splitting Theorem (Day 130)

Lesson 12-10 of the U of Chicago text is called "The Side-Splitting Theorem." This is what I wrote last year about today's lesson:

The U of Chicago version of the theorem is:

Side-Splitting Theorem:
If a line is parallel to a side of a triangle and intersects the other two sides in distinct points, it "splits" these sides into proportional segments.

And here's Dr. Wu's version of the theorem:

Theorem 24. Let triangle OPQ be given, and let P' be a point on the ray OP not equal toO. Suppose a line parallel to PQ and passing through P' intersects OQ at Q'. ThenOP'/OP = OQ'/OQ = P'Q'/PQ.

Notice that while the U of Chicago theorem only states that the two sides are split proportionally, Wu's version states that all three corresponding sides of both sides are proportional.

Moreover, the two proofs are very different. The U of Chicago proof appears to be a straightforward application of the Corresponding Angles Parallel Consequence and AA Similarity. But on this blog, we have yet to give the AA Similarity Theorem. So how can Wu prove his theorem?

We've seen several examples during the first semester -- a theorem may be proved in a traditionalist text using SSS, SAS, or ASA Congruence, but these three in Common Core Geometry are theorems whose proofs go back to reflections, rotations, and translations. Instead, here we skip the middle man and prove the original high-level theorem directly from the transformations. We saw this both with the Isosceles Triangle Theorem (proved from reflections in the U of Chicago) and the Parallelogram Consequences (proved from rotations in Wu).

So we shouldn't be surprised that Wu proves his version of the Side-Splitting Theorem using transformations as well. Naturally, Wu uses dilations. In fact, the names that Wu gives the points gives the game away -- O will be the center of the dilation, and P and Q are the preimage points, while P' and Q' are the images.

Here is Wu's proof: He considers the case where point P' lies on OP -- that is, the ratio OP'/OP, which he labels r, is less than one. This is mainly because this case is the easiest to draw, but the proof works even if r is greater than unity. Let's write what follows as a two-column proof:

Given: P' on OP, Q' on OQ, PQ | | P'Q', r = OP'/OP
Prove: OP'/OP = OQ'/OQ = P'Q'/PQ

Statements                                           Reasons
1. P' on OP, Q' on OQ, PQ | | P'Q'      1. Given
2. OP' = r * OP                                    2. Multiplication Property of Equality
3. Exists Q0 such that OQ0 = r * OQ  3. Point-Line/Ruler Postulate
4. For D dilation with scale factor r,    4. Definition of dilation
    D(Q) = Q0, D(P) = P'
5. P'Q0 | | PQ, P'Q0 = r * PQ               5. Fundamental Theorem of Similarity
6. Lines P'Q0 and P'Q' are identical      6. Uniqueness of Parallels Theorem (Playfair)
7. Points Q0 and Q' are identical          7. Line Intersection Theorem
8. OQ' = r * OQ, OP' = r * OP,            8. Substitution (Q' for Q0)
    P'Q' = r * PQ
9. OP'/OP = OQ'/OQ = P'Q'/PQ = r    9. Division Property of Equality

Now the U of Chicago text also provides a converse to its Side-Splitting Theorem:

Side-Splitting Converse:
If a line intersects rays OP and OQ in distinct points X and Y so that OX/XP = OY/YQ, then XY | | PQ.

The Side-Splitting Converse isn't used that often, but it can be used to prove yet another possible construction for parallel lines:

To draw a line through P parallel to line l:
1. Let X, Y be any two points on line l.
2. Draw line XP.
3. Use compass to locate O on line XP such that OX = XP.
4. Draw line OY.
5. Use compass to locate Q on line OY such that OY = YQ.
6. Draw line PQ, the line through P parallel to line l.

This works because OX = XP and OY = YQ, so OX/XP = OY/YQ = 1.

The U of Chicago uses SAS Similarity to prove the Side-Splitting Converse, but Wu doesn't prove any sort of converse to his Theorem 24 at all. Notice that many of our previous theorems for which we used transformations to skip the middle-man, yet the proofs of their converses revert to the traditionalist proof -- once again, the Parallelogram Consequences.

Another difference between U of Chicago and Wu is that the former focuses on the two segments into which the side of the larger triangle is split, while Wu looks at the entire sides of the larger and smaller triangles. This is often tricky for students solving similarity problems!

Now it's time to give the Chapter 12 Test. Again, it's awkward to combine Lesson 12-10 with the test, but that's the way it goes. I can argue that by doing so, we're actually following the modern Third Edition of the text. Yesterday's lesson on SAS~ and AA~ is the final lesson of Chapter 12 in the newer version (Lesson 12-7). In the Third Edition of the text, the Side-Splitting Theorem is the first lesson of the next chapter, Lesson 13-1.

Test Answers:

10. b.

11. Yes, by AA Similarity. (The angles of a triangle add up to 180 degrees.)

12. Yes, by SAS Similarity. (The two sides of length 4 don't correspond to each other.)

13. Hint: Use Corresponding Angles Consequence and AA Similarity.

14. Hint: Use Reflexive Angles Property and AA Similarity.

15. 3000 ft., if you choose to include this question. It's based on today's Lesson 12-10.

16. 9 in. (No, not 4 in. 6 in. is the shorter dimension, not the longer.)

17. 2.6 m, to the nearest tenth. (No, not 1.5 m. 2 m is the height, not the length.)

18. 10 m. (No, not 40 m. 20 m is the height, not the length.)

19. $3.60. (No, not $2.50. $3 is for five pounds, not six.)

20. 32 in. (No, not 24.5 in. 28 in. is the width, not the diagonal. I had to change this question because HD TV's didn't exist when the U of Chicago text was written. My own TV is a 32 in. model!)

This is a traditionalist post. Let's look at Barry Garelick's latest post:

https://traditionalmath.wordpress.com/2018/03/13/what-you-learn-in-ed-school-dept/

For those who are wondering what future math teachers learn in ed school, here is a concise summary:

Traditional mathematical teaching has never worked and has failed thousands of students.

The standard ed school catechism is that traditional math teaching is based on rote memorization with no understanding, and no connection between concepts.  Another is that the conceptual underpinning of math procedures are not explained. According to ed school teaching, procedures are presented as a “bag of tricks” (such as “keep, change, flip” for dividing fractions). The evidence presented is simply that many adults do not remember how to solve certain problems. This stands as proof that the traditional methods are not effective–if they were, they would “stay with us.”

Of course, Garelick disagrees with this "catechism." So far, this post has drawn five comments. Wow, SteveH must be losing his edge -- he wrote only one comment to this post. Before we look at SteveH's comment, let's look at the previous commenter. This commenter tells a Common Core horror story, and there's some confusion as to what he's describing. So he clarifies himself in a subsequent comment:

I think I have jumped over the actual test item.
The aim was to achieve the solution to an arithmetical 2 digit multiplication by the “area strategy”, and the student was expected to follow the strategy exactly, with a sequence of steps.
Marks: 1 to succeed totally, 0 for inadequate explanation,

Here is the relevant Common Core Standard:

CCSS.MATH.CONTENT.4.NBT.B.5
Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

We've seen enough of these stories to know what's happening here. The fourth graders are taught some "area strategy" in order to multiply a pair of two-digit numbers. Let's say they are asked to multiply 98 by 39. I suspect the "area strategy" means that they're supposed to draw a rectangle whose dimensions are 98 and 39. The length is to be divided into 90 and 8, while the width is to divided into 30 and 9. These produces four rectangles whose area they find separately, and finally they add up the areas:

(98)(39) = (90 + 8)(30 + 9)
               = (90)(30) + (90)(9) + (8)(30) + (8)(9)
               = 2700 + 810 + 240 + 72
               = 3822

The author tells us the rubric here. He strongly implies that if students use any method other than the area strategy -- like the standard algorithm, of course -- then they receive a score of 0, even if the final answer is correct. And naturally, the standard algorithm isn't taught until fifth grade:

CCSS.MATH.CONTENT.5.NBT.B.5
Fluently multiply multi-digit whole numbers using the standard algorithm.

We already know what the traditionalists would prefer -- just teach the standard algorithm in to fourth grade and skip the "area strategy" altogether. I'm sympathetic to this idea, but I still believe that some students learn multiplication more easily using the lattice method (the "rectangular arrays" mentioned in the standard?) than using the standard algorithm.

Then again, one solution that should suit both traditionalists and reformers alike is simply to ask for the product of 98 and 39, and award one point if the answer is correct, regardless of what algorithm is used to find the solution. Then fourth grade teachers might teach the lattice method and fifth grade teachers the standard algorithm, and students can choose which method they prefer without penalty (including fourth graders who know the standard algorithm).

OK, let's see what SteveH has to say in the comment thread:

SteveH:
That’s just plain bad teaching and you offer no description or indication of anything better. The traditional AP Calculus (or IB) track in high school is the only success path for proper STEM careers. Other “integrated” high school math options have lost the battle and are not what colleges are looking for.

Here we go again -- SteveH claims that colleges aren't looking for Integrated Math students. We know this is false, because colleges admit thousands of international students, all of whom (except for the Vietnamese) took Integrated Math in high school.

For example, MIT famously sent out its admission letters this week. Of course, data for this year's class isn't available yet, but here's a link to data for the current freshman class:

http://mitadmissions.org/apply/process/profile

According to the link, 11% of the incoming class is international. Let's give SteveH the benefit of the doubt and assume that 1% of the class is from Vietnam (likely an overestimate). So this leaves 10% of the class having taken Integrated Math. And it goes without saying that many if not most MIT students are STEM majors. So even though Integrated Math isn't "what colleges are looking for," there surely are a lot of MIT admits who took Integrated Math. (Again, MIT admits typically have taken Calculus, but I'm talking about the classes they took in the years before Calculus.)

SteveH:
My son will be graduating from college in May and one of his degrees is in abstract math. He went through a proper AP Calculus track in high school and he now sits next to the best math students from around the world.

The best students from around the world have taken Integrated Math in their home countries.

SteveH:
They offer no understanding advantage (everything they do is proof-based) and they all went through “traditional” math programs. You can’t just throw out a pejorative complaint without backup. If you offer a process or pedagogy that works better than the traditional math curriculum track, I might be your biggest supporter. I won’t hold my breath.

If we're using results (international math scores, admissions to MIT) to measure, then Integrated Math clearly "works better than the traditional math curriculum track."

Returning to the original Garelick post, I notice that he writes about tracking. As usual, it's impossible to write about tracking without writing about race (but I'm protected by the "traditionalists" label):

Traditional math failed to adequately address the realities of educating a large, diverse, and rapidly changing population during decades of technological innovation and social upheaval.

This argument relies on the tracking argument, when many minority students (principally African Americans) were placed into lower level math classes in high school through courses such as business math. It goes something like this: “Most students did not go on in math beyond algebra, if that, and there were more than enough jobs that didn’t even require a high school diploma.  Few went to college.  Now most students must take advanced math, so opting out is not an option for them like it was for so many in the past.”

First, in light of the tracking of students which prevailed in the past, the traditional method could be said to have failed thousands of students because those students who were sorted into general and vocational tracks weren’t given the chance to take the higher level math classes in the first place — the instructional method had nothing to do with it.  Also, I don’t know that most students must take advanced math in order to enter the job market. And I don’t think that everyone needs to take Algebra 2 in order to be viable in the job market.

OK, notice that Garelick concedes that Algebra II isn't necessary for everyone -- as opposed to SteveH, for whom it seems to be AP Calculus or bust.

Indeed, Garelick proceeds to write about traditionalism not in Algebra, but in Pre-Algebra:

Secondly, while students only had to take two years of math to graduate, and algebra was not a requirement as it is now, many of today’s students entering high school are very weak with fractions, math facts and general problem solving techniques. Many are counting on their fingers to add and rely on calculators for the simplest of multiplication or division problems.  In the days of tracking and weaker graduation requirements, more students entering high school than now had mastery of math facts and procedures including fractions, decimals and percents.

I think about the three eighth graders I see in the morning on this, my third day of subbing for the digital film teacher. Since I already introduced race in this post, I might as well point out that the three students are white, black, and Asian. I help the black student with her Common Core 8 homework on the Pythagorean Theorem. She's confused when she reaches the questions that ask her to find a leg rather than the hypotenuse. Yet she does very well in performing all the arithmetic by hand, only needing a calculator to find the square roots. This includes the one question with decimals. This is a very traditional worksheet -- the Pythagorean Theorem is a topic that's impossible to teach any other way than traditionally, using the formula.

The white and Asian students are in Algebra I. Their assignment is on finding angle measures. One worksheet is more like a puzzle -- there's a single diagram with dozens of angles to find, using linear pairs, vertical angles, and parallel lines cut by a transversal. Both of them add and subtract the angle measures without a calculator. When I check their homework, I notice that their answers are slightly different -- some of them are off by three degrees. Unfortunately, I'm stumped as well -- the puzzle format makes it difficult to tell where the errors are made, and in fact, I wonder whether one of the angles given by the teacher is off by three degrees.

When the two of them come in later at lunch to work on their video assignment, I ask them what their math teacher said about the homework. They inform me that she showed them a worksheet with the correct answers, so apparently they made a mistake somewhere that I can't find. These puzzle problems are tricky -- an early student mistake ruins the entire worksheet. But I know that some traditionalists are fond of this type of worksheet -- this is the deeper thinking that they want to see, as opposed to "explain what method you used to get your answers."

All three students have strong arithmetic skills. I can't be sure about their ability to handle fractions, since there are no fractions on either worksheet.

OK, that's all I have to say about traditionalists. Here are today's worksheets -- one for Lesson 12-10 and the other for the Chapter 12 Test.