Thursday, June 2, 2016

PARCC Practice Test Question 31 (Day 174)

Today I subbed in the classroom. Since the day I was hired as a middle school math teacher, this is actually the first time I subbed in a middle school math class! In May, I spent a whole week in a high school math class, and I also spent another week in a middle school science class (which had nothing to do with math, so I didn't mention it on the blog), but only today am I actually subbing for a class that I am getting ready to teach full-time.

This teacher has three sections of Common Core Math 8 and two sections of (Honors?) Common Core Math 7. Both classes take quizzes today -- the seventh graders' quiz is on the Chapter 8 of their text, on Measure Figures. On their review worksheet, they find the circumference and area of a circle as well as the volume of a box and pyramid. But on the quiz there are no pyramid problems. The class clearly ran out of time before reaching the lesson on the volume of a pyramid, to say nothing of Chapters 9 and 10 of their text (on Probability and Statistics). The students appear to understand most of the quiz, but there was one tricky question on the area of a semicircle. (Thank you, Dido!)

I usually want this blog to focus on the class that is studying geometric concepts -- so in today's post, the focus is on the Math 7 class. But I can't help but point out a few things about Math 8. These students take the quiz on the ninth and final chapter of their book, Scatter Plots and Data Analysis. I observe how much these students struggle on this quiz -- especially the question where they must identify and write an equation for a trend line. (Considering how much the eighth graders struggled with stats, maybe it's a good thing that the seventh graders didn't make it to their own stats chapter!)

There is also an incident in fifth period, when a student announced that her 15th birthday is coming up this summer -- even though she is just now completing the eighth grade. She explains that she flunked a grade because she didn't know her times tables (meaning that she most likely had to repeat the third grade). But then she says that she still doesn't know her times tables -- and she announces it proudly, almost as if she's saying, "Ha! I'm graduating eighth grade and I didn't have to learn your stupid multiplication tables!"

That line, of course, is something that will annoy the traditionalists. In fact, I've decided to give today's post this week's traditionalists label. Traditionalists complain when students not only don't know how to multiply, but are allowed to pass despite not knowing how to multiply.

Speaking of traditionalists, Dr. Katharine Beals is at it again:

This time Beals doesn't say a word, not even her usual sarcastic "Extra Credit Question." Instead, she links to a 4th grade question from her home state of Pennsylvania. There is a question about three angles whose sum is 180, with two of the angles given as 78 and 75. Students must explain why Liban's answer of 177 is incorrect. The response "He did add 75 degrees instead of subtract it" is given as a correct response, while "He didn't know how to subtract and add" is marked incorrect -- and Beals is left wondering why the first is right and the second wrong.

Well, I'm not sure whether the girl in my class knows how to subtract and add, but she admits that she doesn't know how to multiply. This is the closest I've ever come to actually using the word "dren" in the classroom -- I won't use it as a sub (since no one knows what it means), but I did tell her that it is she, not the system, that has the problem when she can't multiply. Her reply was that at least she has a smart phone that can multiply 12 * 8 for her.

How would Beals respond to someone like this? Here's a link to the only post on the Beals blog that has the "multiplication tables" label:

What with the proliferating meme that calculators can substitute for most real-world human calculations, and with restaurant bills that increasingly calculate the various tip possibilities for you (15%, 20%, 25%), it's harder and harder to find examples of cases in which, say, there's no good substitute for knowing your multiplication tables.

But on a recent airplane flight, one such case suddenly jumped out at me. Before the flight took off, the flight attendants were asked to verify that all the passengers were on board. There were six seats per row on what seemed to be a maxed-out flight, and one of the older flight attendants knew exactly what to do. Walking down the aisle, she rapidly counted by 6's: 6, 12, 18, 24, etc...

Think about it... There's no faster way to count rows of airplane passengers than to apply your memorized multiplication tables.

How much longer would it have taken for the flight to take off, I couldn't help wondering, if one of the younger flight attendants had taken charge?

Notice how Beals mentions the generation gap -- "one of the younger flight attendants." This implication is that it's our generation -- the Millennials or the "Dumbest Generation" of Mark Bauerlein (which, as the title of his book implies, refers to anyone born within 30 years of the publishing of the book, 1978-2008) -- that has the problem here. This girl, like many other members of our generation, believes that technology obviates the need to learn anything higher than second grade math. Instead, all she does is make us look bad to those born before 1978 -- technology is a crutch without which we can't do easy arithmetic.

During the fourth period conference, I subbed in a special ed math class. The students were in sixth grade, but of course the math they were working on was below sixth grade level (but at least some of it was multiplying fractions, a fifth grade standard). Right outside the classroom, the band members were out practicing, and they kept doing exercises which repeated the following notes (where the letter "b" is ASCII for the "flat" in music):

Bb-A-Bb, Bb-Ab-Bb, Bb-G-Bb, Bb-Gb-Bb, Bb-F-Bb, Bb-E-Bb, Bb-Eb-Bb.

(For those of you who don't know music, but recognize patterns and wonder why I skipped Fb, actually Fb is the same as E in music.)

This disturbed some of the students enough so that they couldn't concentrate on math (or at least they gave that as an excuse). There might have been a similar note pattern that went F-Gb-F, F-G-F, and so on, but it was the Bb-A-Bb pattern that was played the most.

Believe it or not, Beals actually has something to say about such musical drills:

In his A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form, Paul Lockhart opens with an allegory about a musician, who awakens from a nightmare in which the “curious black dots and lines” that “must constitute the ‘language of music’” become the center piece of what has become a universally mandated music curriculum. He proceeds to describe just how tedious this curriculum is for all concerned:
It is imperative that students become fluent in this language if they are to attain any degree of musical competence; indeed, it would be ludicrous to expect a child to sing a song or play an instrument without having a thorough grounding in music notation and theory.
And then, of course, he famously proceeds to connect this musical nightmare to the way K12 mathematics is supposedly actually taught: all meaningless, mindless drill.

As Alfred North Whitehead writes back in 1911, however, mindlessness is often a virtue:
It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle — they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.
Furthermore, while no sane piano teacher would ever make "curious black dots and lines" the centerpiece of music instruction, some of the best ones give top priority to mind-and-soul-numbingly tedious muscle exercises. I was reminded of this reading an accomplished pianist's recent New Yorker memoir about his "Life in Piano Lessons." Here's an excerpt (the student/narrator is Jeremy Denk, and the teacher is William Leland):
Learning to play the piano is learning to reason with your muscles. One of the recurring story lines of my first years with Leland was learning how to cross my thumb smoothly under the rest of my hand in scales and arpeggios. He devised a symmetrical, synchronous, soul-destroying exercise for this, in which the right and left thumbs reached under the other fingers, crab-like, for ever more distant notes. Exercises like this are crucial and yet seem intended to quell any natural enthusiasm for music, or possibly even for life. As you deal with thumb-crossings, or fingerings for the F-shart-minor scale, or chromatic scales in double thirds, it is hard to accept that these will eventually allow you to probe eternity in the final movement of Beethoven's last sonata. Imagine that you are scrubbing the group in your bathroom and are told that removing every last particular of mildew will somehow enable you to deliver the Gettysburg Address.
Of course, a certain amount of grit and gruel also underlies good writing. The Gettysburg Address doesn't just happen, either.

On the piano, drills consist of practicing scales such as F# minor, but on band instruments, apparently drills consist of those Bb-A-Bb patterns. The point Beals is trying to make is that both the fourth period band's Bb-A-Bb and the fifth period girl's 12 * 8 drills are neither easy nor fun. Yet the band musicians are willing to slog through Bb-A-Bb in order to get to more interesting music, while the girl isn't willing to slog through 12 * 8 in order to get to more interesting math.

Meanwhile, Question 31 of the PARCC Practice Exam contains a proof:

31. Given: In triangle ABC shown, BA = BC
Prove: Angle A = Angle C

Statements                    Reasons
1)                                  1) Given
2)                                  2)
3)                                  3) Definition of midpoint
4)                                  4)
5)                                  5)
6)                                  6)

Part A
Select from the drop-down menus to correctly complete Step 2 of the proof.
Statement: Choose...
let D be the midpoint of segment AB
let D be the midpoint of segment AC
let D be the midpoint of segment BC
Reason: Choose...
every line has exactly one midpoint
every segment has exactly one midpoint
every triangle has exactly one midpoint

Part B
Select from the drop-down menus to correctly complete Step 4 of the proof.
Statement: Choose...
angle ADB is congruent to angle CDB
triangle ADB is congruent to triangle CDB
line segment BD is congruent to line segment BD
Reason: Choose...
reflexive property of congruence
definition of perpendicular bisector
Side Angle Side congruence postulate

Part C
Select from the drop-down menus to correctly complete Step 5 of the proof.
Statement: Choose...
triangle ABD is similar to triangle CBD
triangle ABD is congruent to triangle CBD
angle ABD is congruent to angle CBD
Reason: Choose...
Angle Angle similarity postulate
Side Side Side congruence postulate
Side Angle Side congruence postulate

Part D
What is correct reason for the statement in step 6?
A. the transitive property of congruence
B. base angles of isosceles triangles are congruent
C. corresponding parts of congruent triangles are congruent
D. vertical angles are congruent

This is clearly a proof of the Isosceles Triangle Theorem. Back in Euclid's day, this was known as the Pons Asinorum, or "bridge of donkeys." It is stated in Lesson 5-1 of the U of Chicago text:

Isosceles Triangle Theorem
If a triangle has two equal sides, then the angles opposite them are equal.

There are many ways to prove this theorem. The U of Chicago text proves this theorem directly from reflections and symmetry without appealing to any congruence theorem. I've also mentioned the Pappus proof, where we prove that triangle ABC is congruent to CBA via SAS.

Most modern texts use an auxiliary line in this proof. This line turns out to be the same as the symmetry line of the triangle, except that it's labeled BD, where D is a point on AC. We could let BD be the bisector of angle B, then we can prove that ABD and CBD are congruent via SAS. Or we could let BD be the altitude to side AC, then we can use HL to complete the proof. Finally, there's one more possibility -- and it's the one that appears on the PARCC. Here is the completed proof:

Statements                                                 Reasons
1) BA = BC                                                1) Given
2) Let D be the midpoint of segment AC. 2) Every segment has exactly one midpoint.
3) AD = CD                                               3) Definition of midpoint
4) BD = BD                                               4) Reflexive property of congruence
5) Triangle ABD is congruent to CBD.     5) Side Side Side congruence postulate
6) Angle A = Angle C                               6) Corresponding parts of congruent triangles are congruent.

So in this proof, we must draw in the median BD and use SSS to complete the proof. Here is a common wrong answer -- students might choose B for the last part instead of C, since after all, "base angles of an isosceles triangle are congruent" is exactly what we're trying to prove!

Back in November, I wrote extensively about the Isosceles Triangle Theorem proof and the PARCC, and I might as well repeat it now:

So now we ask, which of these proofs should we teach? We might say that the PARCC encourages the median-SSS proof. Indeed, we see that the PARCC places a special emphasis on medians -- even though in theory any of the concurrency theorems can appear on the PARCC, only the concurrency of the medians -- the centroid -- actually appears on any released practice questions.

But who is the PARCC to dictate which ITT proof to use? Of course, some proof of the ITT has to appear on the PARCC, and so students should be ready to give whichever proof any test expects them to give. It's most important that students simply know any proof of the ITT -- again, our focus should be on which the students understand most easily. So if a standardized test like the PARCC has to ask for any proof, it should be the one that is best for the students.

I tried to find out via a Google search which of the proofs median-SSS and angle bisector-SAS is more common, or is easier for students to understand, But it's inclusive -- I see both proofs at numerous websites. On the other hand, altitude-HL doesn't really appear much at all.

Notice that just as there are three ways to prove the Isosceles Triangle Theorem using an auxiliary line, there are two ways to prove its converse. The U of Chicago text uses angle bisector-AAS, while it's also possible to use altitude-AAS -- both methods use AAS because the congruent sides that we get for free (using the reflexive property) are always opposite the given congruent angles. But we can't use the medians because this leads to SSA, and it's impossible to convert this SSA to a valid congruence such as HL or SsA.

PARCC Practice EOY Question 31
U of Chicago Correspondence: Lesson 5-1, The Isosceles Triangle Theorem

Key Theorem: Isosceles Triangle Theorem

If a triangle has two equal sides, then the angles opposite them are equal.

Common Core Standard:
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

Commentary: There really isn't much I can come up with for Exercises on this worksheet -- what, prove the ITT using the angle bisector-SAS method? So instead, I posted the review worksheet for the quiz that the Math 7 students are taking today. The seventh graders don't need to know the volume of a pyramid, but our Geometry students do, so of course I retain these questions on the worksheet.

No comments:

Post a Comment