Monday, May 18, 2015

PARCC Practice Test Question 25 (Day 170)

I've discussed one of my favorite TV programs, The Simpsons, many times here on the blog. In particular, I told how according to the author Simon Singh, many of the show's writers are mathematicians, and they something sneak mathematical content onto the show. This isn't most episodes, of course, but a few episodes are highly mathematical.

Well, last night's episode of The Simpsons -- the Season 26 Finale -- was definitely one of the more mathematical episodes of the show. The episode was titled "Mathlete's Feat," and the "mathlete" in the title refers to -- who else -- Lisa Simpson. Because of its mathematical content, I want to discuss the episode in full.

The episode begins when Lisa and the rest of the math team from Springfield Elementary competes against their rivals at Waverly Hills Elementary. Springfield's team is wearing shirts that, when converted to ASCII, read "pi =/= a/b" (where =/= means "does not equal"). That is, the shirts read that pi is irrational (not of the form a/b for integers a and b). In case you're wondering, irrational numbers is an eighth grade standard under Common Core:

Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

The team from Waverly Hills is also wearing shirts containing the does not equal "=/=" sign, except that's the only symbol on their shirts. In this case, it means that the average income of a Waverly Hills resident (think Beverly Hills) does not equal that of the average Springfield resident. Outside the auditorium where the competition is about to take place, Homer reads a sign that contains some mathematical symbols, "sqrt(-1) 2^3 sigma pi," and laughs. It turns out that he was laughing at something else, not the sign, because it's a mathematical joke. Notice that sqrt(-1) = i, 2^3 = 8, and the sigma symbol denotes "sum." So the sign reads, "i 8 sum pi," or "I ate some pie." That joke existed well before this Simpsons episode -- for example, here's a link to an actual pie:

It's too bad that Homer didn't realize that it was a joke about eating pie, considering that he certainly enjoys eating pie.

Meanwhile, in the competition, Waverly Hills dominates the competition and wins 30-0. After being humiliated, Principal Skinner orders laptops for the entire school to help the students better, and the teachers destroy the textbooks. But then Nelson, the class bully, figures out how to watch cable TV on the laptops. The whole school follows suit, the school wi-fi is overloaded, and all the computers explode suddenly.

Lisa's teacher, Miss Hoover, has nothing to teach since there are no laptops and all of the texts have already been discarded. So Lisa just stares out the window and sees Groundskeeper Willie measure the schoolyard with a stick and knotted rope -- he says that this technique was used by his ancestors. the ancient Druids (as the groundskeeper is Scots). This gives Lisa the idea to convince Principal Skinner to convert Springfield Elementary into a Waldorf school.

There are a few issues raised in this episode so far that I want to discuss. First of all, many schools, including most notably in my area the LAUSD, have ordered laptops or tablets for every student. But the reason for the computers at LAUSD was not to win a math contest, like Springfield Elementary was trying to do, but for another high-stakes competition -- the Common Core SBAC exam. As it turns out, the LAUSD tablet rollout was extremely flawed -- and part of the problem was that students did use them to fool around and play games and videos, just like Nelson. And as we know, many people criticize the Common Core in general and the SBAC and PARCC exams in particular because of their reliance on computers that schools might not afford and distract the Nelsons in the class.

But I also want to discuss Waldorf education, because Lisa helps transform her school into one based on that philosophy. The idea of a Waldorf education goes back to the 19th-century Austrian philosopher Rudolf Steiner -- indeed, we see one of the teachers at Springfield read one of Steiner's books to learn how to implement Waldorf. As Lisa mentions in the episode, a Waldorf education emphasizes hands-on learning. Here is a link to a Waldorf-inspired charter school right here in Southern California, Ocean Charter School:

Our standards-based, Waldorf approach features a range of effective educational practices that contribute to our high student achievement and strong, stable learning community.
Elements of our effective learning practices:
  • A multi-sensory curriculum
  • Limitations on media and the use of technology
  • Integration of the arts into the full curriculum
  • Active learning incorporating movement
  • A curricula and method geared to the stages of child development
  • Science taught experientially
  • Thematic lessons scheduled into “Lesson Blocks”
  • Teachers looping with students
  • An educational culture of musical and theatrical performances, gardening and environmental stewardship, and seasonal festivals

There are a few practices that distinguish Waldorf from a more traditionalist education. Notice the item "Teachers looping with students" in the above list. Steiner divided childhood into three eras, or stages of development, each lasting seven years. The first era, from birth to age seven, focuses on a play-based education. The second era, ages 7-14, focuses on emotion and imagination. Teachers are supposed to focus on a student's learning style -- and the best way for teachers and students to have a deep, lasting relationship, according to Waldorf, is for them to have a single teacher loop with them during this entire second era.

Ages 7-14 correspond to grades 2-8 here in the United States. Theoretically, first grade should be grouped in the first era with preschool and kindergarten, with second grade as the first true grade of elementary school. Because this is awkward the way that grades are numbered in this country, Ocean Charter School instead has an extremely early cut-off date for kindergarten and first grade:

The minimum age of children applying to the school will be:
  • Transitional Kindergarten: age 5 on or before December 2nd of the year they enter school.
  • Kindergarten: age 5 on or before May 31st of the year they enter Kindergarten.
  • First Grade: age 6 before June 1 of the year they enter first grade.
  • All other grades: The First Grade age policy continues in this manner through all grades. This requirement helps ensure school readiness specific to the OCS, Waldorf informed, Common Core Curriculum and is consistent with non-discrimination statutes.
So we see that the Waldorf cut-off date is the last day of May -- a full three months earlier than the usual California cut-off date of September 1st. This means that students with summer birthdays are seven, not six, years during their entire first grade year. It also means that students with early June birthdays can't complete the 12th grade until they are 19 years old -- the last day of school is June 12th according to the link, so students born during the first 12 days old graduate at 19.

Notice the mention of a December 2nd cut-off date above. This refers to Transitional Kindergarten -- the new grade created when California moved its cut-off date three months earlier (so students born between the new and old cut-offs would attend TK instead of kinder). Waldorf has no control over the TK cut-off date, as it's determined by the state. What it does mean is that TK students can be born any time from June to November -- not September to November at a traditionalist school. So there are twice as many TK students at Waldorf than at a traditionalist school.

As I mentioned above, Waldorf teachers stay with their students from first up to eighth grade -- so Lisa would have been in Miss Hoover's class for years. Here is an explanation from Ocean Charter:

What is looping? In 1st grade, a class community of students is formed with a main lesson teacher. These students stay together as a class throughout their OCS experience and graduate together as 8th graders. Their teacher stays with them, “looping” for multiple grades with the class. Typically, a teacher stays with a class for 3-5 years, such as from 1st-5th grade. When appropriate for teacher and his/her class, a teacher may loop all the way from 1st to 8th grade with a class.

It's interesting that Ocean Charter would write that some teachers would stay with their classes only from first to fifth, rather than eighth grade. According to the following link, Ocean Charter is divided into two campuses -- and the grade ranges are apparently K-3 and 4-8. I've discussed schools where middle school starts at fifth grade -- here middle school starts at fourth grade:

According to link, at Waldorf schools, every day begins with a Main Lesson. We see that Main Lessons are on different subjects that last slightly less than a quaver:

What is a main lesson? Grades 1–8 commence each school morning with the “Main Lesson,” a two-hour period of intensive and focused thematic learning in which the core curriculum is presented when the students are most receptive and alert. The main lesson subjects (such as Greek history, Botany or American Geography) are taught in three- to four-week blocks. The teaching units are integrated and cross-curricular, and include activities to awaken and focus attention. This approach allows for freshness and enthusiasm, a concentrated in-depth experience, and gives the children time to integrate learning. Being taught in the morning Main Lesson enhances the knowledge acquisition of academic subjects, such as language arts, math, science and history. Daily math practice periods also occur in the morning when the students are most alert. The afternoons are often used for arts and crafts, outdoor activities, sport and practical work. Skill-building subjects that benefit from regular practice, such as music, movement, and world languages, are evenly scheduled throughout the middle of the day to the extent possible.

Music is a regular part of a Waldorf curriculum. Students in grades 1-3 learn how to play the pentatonic flute. A few weeks ago I linked to a flutist who played, not 12-tone scale, but a 19-tone scale -- well, a pentatonic scale is the exact opposite, as there are only five notes per scale. The C major scale goes C, D, E, F, G, A, B, C, but the C major pentatonic scale omits F and B, so it goes C, D, E, G, A, and then back to C. The black keys of the piano also form a pentatonic scale, called F# (or Gb) major pentatonic. Recall that notes that sound good together (called "consonant" notes) form simple ratios such as 3:2 or 5:4, while notes that sound bad together (called "dissonant" notes) do not form such ratios. The most dissonant intervals are the semitone (such as E to F or B to C) and the tritone (F to B) -- and all of these involve notes that are skipped in the pentatonic scale. Thus the Waldorf flute sounds more soothing as it avoids these hard dissonances.

Students in Waldorf schools also participate in dance -- in particular, there is a special type of Waldorf dance called eurythmy.

Recall that in the episode, Springfield Elementary changed after losing its technology. As it turns out, the perfect type of education for a school without technology is Waldorf:

Lower Grades:  OCS involves parents in their child’s learning goals and in creating a supportive home environment, including support of a healthy home diet, monitoring the influence of media and video games and ensuring appropriate use of technology. Like their teaching colleagues around the country, teachers at Ocean Charter School have been concerned about excess media exposure and the effect on students’ learning outcomes.Our teachers have found that when media exposure is limited to weekends only, students are more focused, calm and engaged in their academic learning. When students watch television and play video games during the school week, the teachers at Ocean Charter School have noted a qualitative difference in students’ behavior, including: exclusive rather than cooperative play, a lack of socialization experiences, a lack of imagination in their work, and an enhanced need for immediate gratification. Therefore, to engender an environment in which learning best occurs, Ocean Charter School has adopted a policy of limiting media exposure for students.

In some ways, this is antithetical to the idea of Common Core, with its computerized testing. As a charter, Ocean is still required to teach Common Core and give the SBAC exam. As much as it would like to have students avoid computers until sixth grade, the school can't escape having third graders take the SBAC on computers.

The relationship between Waldorf and Common Core is quite complex. Here is a link to Journey School, another Waldorf charter school here in Southern California. It discusses how well the standards between Waldorf and Common Core are aligned:

Each Common Core Standard from grades K-8 is listed. The letter "Y" indicates that yes, the standards are aligned. Otherwise the grade when it's covered in Waldorf -- and invariably, the Waldorf grade is level later than the Common Core grade level. For example, we see that among the kindergarten math standards, only the counting standards stay in kindergarten. The others are pushed back anywhere from first to fourth grades. One kinder ELA standard involves computers and so is delayed all the way to seventh grade. Meanwhile, Waldorf catches up to Common Core by eighth grade, as all of the eighth grade standards are listed with the letter "Y."

In some ways, as much as traditionalists oppose Common Core, Waldorf is even further away from the traditionalist ideal. Waldorf delays some of the math standards later, including some seventh grade Core standards into eighth grade. Traditionalists want to do the opposite -- they want to push eighth grade Core standards down into seventh grade so that students can have a full Algebra I course in eighth grade and a full Calculus course in 12th grade. Moreover, the idea of having the same teacher all the way from 1st to 8th grades is the opposite of the traditionalist ideal. Traditionalists often don't like how some elementary teachers don't like math and subconsciously encourage their students to hate the subject as well. They sometimes advocate that even the higher elementary grades should have their math taught by math majors only. Waldorf does the opposite -- it takes the same math-phobic elementary teacher and has them teach eighth graders, who ought to be be learning fully rigorous Algebra I from a math teacher, not the same math-phobic teacher they've had since first grade.

On the other hand, some opponents of Common Core feel that the Core is too traditionalist and doesn't allow for more progressive alternatives, such as Waldorf. We've already seen how Common Core doesn't allow for the Waldorf ideal of avoiding computers until middle school. Here's a link to an article which argues that Common Core interferes with Waldorf and other alternative philosophies:

So Common Core is opposed on two different fronts here. But there is one thing that traditionalists and Waldorf agree with against Common Core -- avoiding technology in the early grades, including calculators in early math classes.

Let's get back to the Simpsons episode. Groundskeeper Willy is in charge of the math team, and as captain he chooses -- Bart Simpson? Well, he saw Bart launch an egg at a perfect 45-degree angle in order to hit the superintendent's car, and so he believes that Bart can be a valuable asset to the team.

And so the rematch between Springfield and Waverly Hills begins. At the top Simpsons message board, called the No Homer's Club, there is a link to screen grabs of some of the problems featured on the show:

Scroll down to the post by the username pkkao. This blog entry is already long enough, so I don't want to discuss the problems one by one, so you can just read pkkao's post instead. He solves all of the problems and notes that one problem doesn't seem appropriate for the students:

S = {x | -3 <= x <= 4}. What is S, expressed in interval notation.

Here is pkkao's response:

Odd second grade question seeing as interval notation isn't usually seen until calculus. The R just means all real numbers. Ans = [-3,4]

Here pkkao mentions "second grade" because Lisa is an eight-year-old girl in the 2nd grade. Notice that in all school episodes, Lisa is 8 and in the 2nd grade. We would normally expect a second grader to be 7 at the start of the school year and turn 8 during the year, not be 8 the whole year. Then again, this actually fits the Waldorf school ages. Likewise Bart is a fourth-grader and is 10 years old during the entire school year.

Then again, we expect students on a math team to be well above grade level. According to pkkao's public profile, he is 15 years old and therefore presumably not yet in a Calculus class, yet he was able to figure out the Calculus interval notation.

As you can see, most posters at No Homer's Club did not like this episode very much -- most of them rated it only two or three out of five stars. We must recall that The Simpsons has just completed its 26th season, and so it's inevitable that for anyone who's been watching since the beginning, the newest episodes aren't as good as the earliest episodes. Just as Waldorf divides children into seven-year eras, so we may do the same with The Simpsons. The first seven (sometimes 8 or 9) seasons of the show are considered the "classic era." Complaints about last night's episode include the plot jumping back and forth between the math contest and Waldorf (had this been the classic era, the episode likely would have focused one or the other) and silly time-fillers near the end (after the contest, Lisa was drunk on Mountain "Doo" for no particular reason, and then the whole family started playing hillbilly music for no particular reason). Another difference between the classic era and now is that many of the classic writers have left the show -- and the latest news is that one of the star voice actors, Harry Shearer (voice of Homer's boss and the family's neighbor, among others), plans on leaving the show, which is scheduled to go on for at least two more seasons without him.

Before I leave the topic of The Simpsons, let me mention a few more episodes that are also relevant to the Common Core Debate. In Season 20, there was an episode "How the Test Was Won," which has the students take a national high-stakes test, due to the "No Child Left Alone" law. Bart is sent on a field trip with Principal Skinner in order to avoid dragging down the test scores -- because on the practice test, he had filled in the bubbles to spell out vulgar words. They end up learning about the Law of Angular Momentum (yes, rotational symmetry!) in order to escape a garbage barge. Lisa, meanwhile, is psyched out by the whole test that she ends up leaving it blank.

Of course, this episode is inaccurate. National high-stakes tests didn't exist at the time the episode air, as they didn't arise until this year's Common Core Tests. They are taken on the computer, so Bart can't bubble in random words (but this was a concern before Common Core). And second-graders like Lisa don't have to take the test -- probably because youngsters may end up being psyched out on the test, just like Lisa. The final message of both episodes is the same, and it's one that eschews both traditionalist education and high-stakes testing.

Since Simon Singh's book The Simpsons and Their Mathematical Secrets is still on my mind, let me discuss another chapter of his book. Chapter 14 of Singh's book is about the other show that Matt Groening created, Futurama. Even though the couch gag from last night's Simpsons episode was about the TV show Rick and Morty, there were scenes from Futurama in the background. The same math majors wrote for both Groening shows, and so Singh describes some of the mathematical jokes that appear on Futurama, which is set in the year 3000. These include a BASIC program written at the home of the robot Bender:

30 GOTO 10

an alien language, based on a cipher created by the Italian mathematician Girolamo Cardano (who was also one of the mathematicians mixed up with the cubic formula), and, to get this back to geometry, the successor of Madison Square Garden, called Madison Cube Garden.

Oops! I got myself distracted again! I spent the entire post discussing The Simpsons and Waldorf schools that I've said nothing about the PARCC. (Fortunately, the geometry student I tutor is on a school trip this week, so I won't have to Well, today's question should be easy -- at least for those students who have seen straightedge-and-compass constructions. Question 25 of the PARCC practice exam is all about constructing angle bisectors.

The figure shows the result of a geometric construction.

(Of course, the figure shows the construction of an angle bisector.)

Part A

The first step of the construction is to draw an arc centered at A that passes through point B and point C. What is established by the first step?

(A) AB is congruent to BC
(B) AB is congruent to AC
(C) AD is congruent to AC
(D) BD is congruent to CD

Part B

The construction creates congruent triangles. Triangle ABD is congruent to ACD (not shown). Which statement provides evidence that ray AD is the angle bisector of Angle BAC?

(A) Angle ACD is congruent to Angle ABD
(B) Angle BAC is congruent to Angle BDC
(C) Angle BAD is congruent to Angle CAD
(D) Angle BAD is congruent to Angle ABD

To answer Part A, we first notice that since the arc centered at A passes through B and C, we must clearly have AB = AC. So this gives us (B) for Part A. For Part B, we only need to think about the definition of angle bisector -- recall that whenever we see a long word like "bisector," half the battle is just giving the definition of that long word. Thus Ray AD must divide BAC into two parts -- and those two parts are BAD and CAD. So this gives us (C) for Part B.

The U of Chicago considers the construction of an angle bisector to be an afterthought -- it spends much more time on perpendicular bisectors. The construction of an angle bisector is hidden in two exercises in Section 4-7, on reflection-symmetric figures (because an angle is reflection-symmetric, and the line of symmetry contains the Angle Bisector). Indeed, the text directs the students to draw the perpendicular bisector of BC, which is also the angle bisector of Angle A. The proof that this gives the angle bisector is not given in the text. The traditionalist proof is hinted at in Parts A and B of this problem -- the second step of the construction establishes BD = CD, and of course AD = AD, so we have the two triangles congruent by SSS. Then the two angles mentioned in Part B are then congruent by CPCTC.

I've posted this before, but let me post it again -- the Square One TV episode where doctors at General Mathpital must perform a "bisectomy" on an angle (skip to about 11 1/2 minutes in):

Today's question is fair provided that constructions are taught in class. If constructions are not taught, then this question should be dropped.

So who won the rematch between Springfield and Waverly Hills? It was Springfield -- the final question of the competition was to create triangles using a minimum of line segments. Bart was the one who solved this one -- by drawing the triangles out of the "M" on Homer's bald head!

PARCC Practice EOY Exam Question 25
U of Chicago Correspondence: Section 4-7, Reflection-Symmetric Figures
Key Theorem: Angle Symmetry Theorem

The line containing the bisector of an angle is a symmetry line of the angle.

Common Core Standard:
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Commentary: Questions 16 and 17 in this section involve angle bisectors. In the Skills section of SPUR, Questions 9 and 10 ask students to draw symmetry lines of angles -- which really means bisect them.

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