Friday, February 12, 2016

Chapter 11-12 Test (Day 102)

Today I'm posting the Chapter 11-12 Test. Last year, in addition to posting the answers for both Chapter 11 and 12, I wrote about several traditionalist topics. And so this will turn into yet another post to receive the "traditionalists" label. (Then again, I did say that on test days, there's often nothing for me to talk about besides the traditionalists!)

Once again, as much as I want to, I can't escape discussing politics, as even the name of this blog is political: "Right on the Left Coast: Views From a Conservative Teacher." But that's the problem -- the debate about Common Core is inherently political, and since the more liberal of the two major parties first proposed Common Core, naturally its opponents are more likely to be conservative.

The author is Darren Miller, and as the title of his blog implies, Miller is a West Coast teacher -- indeed, he is, like me, a high school math teacher from California. Here is the link to his blog entry from last year:

So Miller's district is, like many others in California, considering teaching Integrated Math. The district will try Integrated out this semester, and then make the final decision for this fall. Miller is clearly concerned about the transition to Integrated Math.

Let me remind you that Integrated Math is not a prerequisite of Common Core -- at least, it isn't here in California. Nonetheless, many California disticts are using Common Core as a reason to switch to Integrated Math. Therefore, my own opinions of Common Core and Integrated Math are separate -- I'm mixed on Common Core, but have nothing against Integrated Math.

Naturally, Miller is concerned that Integrated Math reflects a lowering of standards. Indeed, one of the commenters -- of course, that commenter is "Anonymous" -- writes that colleges won't want to accept students who took Integrated Math:

i feel bad for everyone who tries to apply to out-of-state schools that look for two clear algebra requirements and a clear geometry, with pre-cal, cal being an added bonus. how hard will it be to explain, "well, my classes were a mix"....
that might interrupt some students college dreams. not all states' colleges are friendly to california's changes.

Now here's the problem that I have with this commenter -- the United States is an outlier when it comes to traditionalist classes such as Algebra I, Geometry, and Algebra II. Most countries, including those considered to have superior math programs such as East Asia and Singapore, actually have Integrated Math programs. So introducing Integrated Math make us more like the nations that we want to emulate, not less!

Suppose an international student from East Asia or Singapore applies to a math program at an American college. Since secondary math is superior in those nations, one would think that an American college would leap out of its skin to admit the student. But based on what our anonymous commenter wrote, many colleges wouldn't admit the East Asian student, because the student's transcript doesn't contain "two clear algebra requirements and a clear geometry," classes that aren't taught in the student's homeland.

Once again, let me post the link to the Singapore secondary math standards, and I dare you to find the two clear algebra requirements and a clear geometry in the standards:

Now let's return to Miller's original post. We see that part of the reason for considering Integrated Math to be inferior is the ability of the students to reach AP Calculus:

Then we talked about all the courses we'll need to offer. It's more than Integrated 1, Integrated 2, Integrated 3, Pre-calculus, Stats, and Calculus. See, according to the Common Core standards and guidelines, students are not supposed to be accelerated in middle school. Get that? The smart kids will be kept back with everyone else, because, fairness! Our illustrious district will allow middle school students to accelerate one grade in middle school, meaning 8th graders will be allowed to take Integrated 1. (Those would be the smart kids; under California's old standards, those would be the on-track kids.) So if we want kids to be able to take AP Calculus AB and/or BC in high school, we need to accelerate them in high school.

And so we see the problem -- just as Common Core has become intertwined with Integrated Math, the latter is intertwined with the inability to reach AP Calculus. If Miller could see how other districts have already implemented Integrated Math -- such as my own district -- he would notice how freshmen in ninth grade are working out of eighth grade packets. He would be rightly concerned that these students have little shot at reaching calculus.

I don't necessarily agree with Miller's use of the word "on-track" to refer to California's old standards, in which the 8th grade standard was Algebra I. I don't like the idea that a student has to master the quadratic formula, which very few adults have mastered and only workers in STEM subjects actually need to know it to make a living, just to be considered "on-track" in eighth grade.

In the year since Miller posted this, I wonder, what classes is he teaching now? Well, last week he wrote about a test he gave to his Statistics students:

Perhaps this is how Miller decided to escape Integrated Math -- just teach Stats instead!

On the other hand, consider that in the last week-plus, I indirectly taught four different levels of freshman math courses -- two different special ed, regular, and honors. Back in October, I mentioned an Integrated Math text, Houghton Mifflin Harcourt, with 25 modules. We can check this list to see where the different levels of students are:

Moderate Special Ed. 2. Algebraic Models
Mild Special Ed: 3. Functions and Models
Regular Ed: 11. Solving Systems of Linear Equations
Honors Ed: 15. Exponential Equations and Models

Traditionalists like Miller would be glad to see that there are four different levels of math for the freshmen, rather than one level for them all. But they wouldn't like how it's not evident that even the Honors Integrated Math I leads to AP Calculus as a senior.

Since Math I freshmen work out of eighth grade MathLinks packets, the solution is obvious -- allow honors students to take Integrated Math II as freshmen. This would then allow them to reach AP Calculus as seniors.

Meanwhile, speaking of traditionalists, guess who wrote about Geometry as part of her Math Problem of the Week series? Of course, I'm talking about Dr. Katharine Beals:

Postulates and proofs: Let's take it to the courtroom!

In this unit, the process of solving proofs is practiced using the comparison and framework of a courtroom setting. Students will work in groups to solve a proof and then defend it in a class courtroom setting.
A lesson plan for grades 9–10 Mathematics

In this unit, the process of solving proofs is practiced using the comparison and framework of a courtroom setting. Students will work in groups to solve a proof and then defend it in “court.” This unit challenges and engages students, while building their confidence as they learn to support their arguments with sound, logical statements and reasons. Students will have both individual and group assessments during these lessons

This sounds interesting. We know that Michael Serra, when he finally gets to logical proof in his Discovering Geometry text (either Chapter 13 or 14), mentions Sherlock Holmes and the way he uses logic to solve his cases, so this is similar.

This is a group project, so we expect Beals and the other traditionalists not to like it. Let's take a look at the comments:

Unknown said...
One proof over 14 class periods? Yikes, that sounds like the winner of a competition to see who can waste the most time.

Auntie Ann said...
...and 14 class periods is nearly three weeks....and with a 180-day school calendar, that's 7.8% of the year: on one problem.

Dr. Barry Garelick, who often writes on Beals's blog, links to a more "efficiently" done version:

So it appears that as long as it's done individually and in one class period rather than over an entire unit, the traditionalists like Garelick have no problem with using court cases to introduce proofs.

Here are the answers to today's test:

1. Using the distance formula, two of the sides have the same length, namely sqrt(170). This is how we write the square root of 170 in ASCII. To the nearest hundredth, it is 13.04.

2. The slopes of the four sides are opposite reciprocals, 2 and -1/2. Yes, I included this question as it is specifically mentioned in the Common Core Standards!

3. Using the distance formula, all four sides have length sqrt(a^2 + b^2).

4. Using the distance formula, two of the medians have length sqrt(9a^2 + b^2).

5. 60.

6. From the Midpoint Connector Theorem, ZV | | YW. The result follows from the Corresponding Angles Parallel Consequence.

7. From the Midpoint Connector Theorem, BD | | EF. The result follows by definition of trapezoid.

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. (No, not 5250 ft. It's 7000 ft. from Euclid to Menelaus, not Euclid to Pythagoras.)

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!)

President's Day is on Monday, so the next post will be Tuesday, February 16th. On that day we'll begin Chapter 14, Trigonometry.

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