Chapter 13 Test (Day 139)

Today on her Mathematics Calendar 2018, Theoni Pappas writes:

If the equation for this circle is

x^2 + 36 + y^2 + 2y = 12x

its center is located at (?, -1).

To solve this, we begin by subtracting 12x from both sides:

x^2 - 12x + 36 + y^2 + 2y = 0
(x - 6)^2 + y^2 + 2y = 0

We know that the equation of a circle begins with (x - h)^2, so h = 6, where h is the x-coordinate of the center of the circle. So we fill in the question mark -- the center is (6, -1), and of course today's date is the sixth.

We just did the minimum amount of work necessary to solve the problem, but we really should complete the square on y:

(x - 6)^2 + y^2 + 2y = 0
(x - 6)^2 + y^2 + 2y + 1 = 1
(x - 6)^2 + (y + 1)^2 = 1^2

Thus the center is (6, -1) and the radius is 1.

I wasn't sure whether to post this question at all. In the past, I would have counted this as an Algebra II question, yet we now know that it appears on the PARCC and SBAC as a Geometry question. I haven't really posted enough completing the square questions on the blog, and so I decided to post today's question.

For reasons that I explained in yesterday's post, today is the Chapter 13 Test.

This is what I wrote last year ago about today's test. As it turned out, I originally posted this test in 2015 on Friday the 13th, and I mentioned this fact throughout the test. Unfortunately, this year today isn't the 13th. But if it's any consolation, at least it's Friday the 13th next week -- too bad we can't delay the test until then:

Here's an answer key for the test:

1. a. 90 degrees. I could have made this one more difficult by choosing a heptagon, or even a triskaidecagon, but I just stuck with the easy square.

b. Here is the Logo program:
TO SQUARE
REPEAT 4 [FORWARD 13 RIGHT 90]
END

Notice that the side length is 13. I'll still find a way to sneak 13, if possible, into each problem.

2. a. If a person is not a Rhode Islander, then that person doesn't live in the U.S.
b. If a person doesn't live in the U.S., then that person isn't a Rhode Islander.
c. The inverse is false, while the contrapositive is true.

Notice that Rhode Island is the thirteenth state.

3. y = 10.

4. There is a line MN. (M is the thirteenth letter of the alphabet.)

5. Every name in this list is melodious.

6. The equation has no solution. (This question references 13, as 13x appears in the expansion.)

7. a. 13, 11, 9 (descending odds).
b. 13, 17, 19 (increasing primes -- of course, Euclid proved that this sequence is infinite).

8. a = 2, b = 1, c = 3.

9. kite.

10. I discussed this problem earlier this week. It is the same as the problem from the Glencoe text, except that I only drew half of the figure -- the part where a contradiction appears.

Assume that the figure is possible. Then ABC is isosceles, therefore angles A and C are each 40 degrees (as the third angle of the triangle is 100). Then ABO is isosceles (as it has two 40 degree angles), so AO = BO = 3. Then by the Triangle Inequality, 3 + 3 > 8, a contradiction.

11. Through any two points, there is exactly one line. (This is part of the Point-Line-Plane Postulate.)

12. a. KML measures 13 degrees.
b. K measures less than 167 degrees.
c. L measures less than 167 degrees. (This is the TEAI, Exterior Angle Inequality.)

13. a. Law of Ruling Out Possibilities.
b. You forgot to rule out another possibility -- that nothing bad will happen to you today. Hopefully, this will be true for you.

Since it's test day, this is a traditionalists post. Although Barry Garelick posted yesterday, it's his post from two days ago that has drawn more comments, including two from You Know Who.

https://traditionalmath.wordpress.com/2018/04/04/this-just-in-dept/

From “Education Dive” (as in “deep dive”, “deep understanding” and other ridiculous jargon which unfortunately permeates the edu-world), a summary of a shocking new study:

Less than 10% of math assignments in the middle grades require “high levels of cognitive demand,” and only about a third of tasks expect students to show their thinking when providing their answers, according to a new analysis of more than 1,800 assignments, released today by The Education Trust.

Garelick makes the standard complaints against this line of thinking. Basically, why should middle school assignments require "high levels of cognitive demand" instead of "low level" basic knowledge that is essential to finding the answers?

Here is SteveH's first comment:

SteveH:

“For this analysis, we reviewed over 1,800 middle-grades assignments from over 90 math courses from 12 middle schools in six districts across the country …”

They say nothing about textbooks, but I assume they are used. This could be a problem of teachers just assigning the basic skill sections of the textbook problem sets. I’m looking at my son’s old Glencoe Algebra I textbook and there many levels of questions, from basic skills to “Open Ended” questions just like they propose.

So SteveH writes about math texts here. He's endorsed the Glencoe texts before, and I've mentioned Glencoe texts for Algebra I and Geometry in previous posts.

Indeed, my description above of Question #10 from today's test mentions "Glencoe." This actually refers to the Glencoe Geometry text. Three years ago, in a certain question, the Glencoe text gives a pair of triangles and students are asked to prove that they are congruent. But not only did Glencoe fail to provide enough measures to conclude that the triangles are congruent, the authors unwittingly gave enough measures to prove that they aren't congruent. And so I placed that question on today's test as an example of contradictions and indirect proof.

This just goes to show that all textbooks -- whether traditionalist or reformist -- contain errors. So far, I haven't proved myself correct or SteveH incorrect yet.

Oh, and before we leave our comparison of texts, notice what Garelick writes:

Look, I use a 1962 Dolciani algebra textbook to teach my algebra class. The word problems are plenty challenging for my students, though I’m fairly certain that the authors of said study would find such problems lacking in “real world relevancy” (as if my students care) and low cognitive demand.

As I've mentioned before, I believe that Dolciani is a coauthor of the Algebra I text I used back when I was a young student back in 1993. I doubt it was a 1962 edition, though -- I don't remember the age of the text, but it was within 15 years (probably less) of when I took the class. I'm one to talk about old texts when I keep quoting the Second Edition (1991) of the U of Chicago text, but traditionalists prefer to use books from at least half a century -- if not a full century -- ago.

SteveH:

“Assignments A and B are both opportunities for students to meet the standards. However, only providing students with problems in Assignment A limits their opportunity to engage in cognitively demanding tasks. …”

In my son’s Glencoe textbook, all question variations are right there, so what’s their real problem – that teachers give up if only basic skill questions are mastered? Are they suggesting Assignment B questions IN PLACE OF or before basic skill mastery questions? Go ahead and assign more homework. Set higher expectations, but what are they really saying here; that more is more or are they saying that swapping their Assignment B for Assignment A will get them something for nothing? Um, no.

My problem is not that teachers give up -- my problem is when students give up. And if the students see Question #1 or #2 and don't even know the first step to solving the problem, they're more likely to give up and not even look at the rest of the assignment.

And which sort of assignment are students likely to give up on -- Assignment A or B? Let's look at Garelick's example of an Assignment B problem:

Assignment B:
“The area of a rectangle is 24? What are the dimensions of the rectangle?”

Let's compare this to a corresponding Assignment A question:

Assignment A:
"The area of a rectangle is 24, and its width is 3. What is the length of the rectangle?"

Notice that assignment B, as an open-ended problem, has several answers -- the dimensions could be 24 * 1, or 12 * 2, or even 6 * 4. But Assignment A only permits one possibility -- the 8 * 3 figure. So with only one possible answer instead of four, students are more likely to hear "You're wrong!" from teachers who give Assignment A than those who give Assignment B.

Moreover, suppose a student asks the teacher for help on Assignment A. The first thing the teacher's likely to do is set up an equation like 3x = 24 (or maybe 24 = l(3), if we substitute directly into the area formula A = lw). And this might be from a student who received a low Algebra I grade (say C- or whatever the lowest allowable grade is to be passed into Geometry). The last thing the student wants to see is algebra, and here's the Geometry teacher "helping" the student by giving an equation.

So in the end, Assignment A is more likely to cause students to give up is Assignment B. This doesn't mean that we should never give Assignment A questions, but that I'm sympathetic to those who want to give Assignment B problems. (Npt to mention, I've said nothing about other complaints such as "Why must we write the area in square units?" that will come up regardless of whether we are giving Type A or Type B problems.)

We move on to SteveH's second comment:

SteveH:

At the end, they ask…

WHERE DO WE GO NEXT?
“This analysis of middle-grades math assignments show that schools and districts across the country are falling short when it comes to providing their students with high-quality math tasks that meet the demands of college- and career-ready standards.”

This is completely wrong. This is all about them. This is NOT about looking at what works and what doesn’t – or asking us parents of their best students what we have to do at home or with tutors. What do colleges and careers want? They want good grades in the AP Calculus track, good SAT I and II scores, and many colleges want to see your AMC test scores.

Hmm, that's a new one -- colleges want to see AMC test scores. I alluded to the AMC back in my December 5th post. The previous night, I had a nightmare that schools would force all students to take a high-level math test similar to the AMC, and even change the bell schedule to give the students enough time to take the test. If colleges really want to see AMC scores, then I wouldn't put it past some school or district to require the AMC test.

Back when I was in high school, I took the AHSME -- the predecessor of the AMC. I graduated in June 1999 and the first test under the AMC banner was February 2000, so if I had been one year younger, I would have taken the AMC as well. And I usually scored around 100, so I've taken the AIME as well. Just like the PSAT, the AMC is being pushed into lower and lower grades -- there is now AMC 12, AMC 10, and AMC 8.

Notice that the AMC 12 contains no Calculus, while the AMC 10 contains no Algebra II. Thus it's unnecessary to be on the "AP Calculus track" in order to be successful on the AMC.

SteveH:
The traditional AP Calculus track works very well. The problem is how to best support ALL students on that track starting from Kindergarten. Educators and CCSS systemically fail this task, and no amount of understanding or rigor blather can hide that fact.

Here SteveH implies that he wants ALL (emphasis his) students to be on the AP Calculus track. I strongly disagree that ALL students should take AP Calculus. I can only imagine how low the graduation rates would be if ALL students were forced on the AP Calculus track. Regardless of how traditionalist the curriculum is starting from kindergarten, there will be some students who have neither the skill set nor the desire to be on the AP Calc track.

In a previous post, Garelick acknowledged that not all students need to take Algebra II. But for SteveH, it's AP Calc or bust.

SteveH:
It’s right there in black and white – CCSS NO STEM low slope math in K-8 and magic fairy dust Pre-AP math in ninth grade where students have to double-up in math. Then they have the audacity to talk about “social justice.” This report offers absolutely no critical thinking. They just claim to know what is needed for college and career readiness. Hello! At least look at what colleges want to see in applications for those applying to STEM programs like engineering. Then work backwards to the fairyland of K-8.

So what math should those who are not applying to STEM or engineering take? A place where ALL students want to be on the STEM track -- that's where the real "fairyland" is.

The rest of the posts in the comment thread are about another issue -- homework. For example, one commenter writes:

Homework shouldn’t exist at the lower grades, IMO. At the primary level, the only learning that should be occurring, is in the classroom, being taught by the knowledgeable teacher.

Another commenter disagrees:

The issue I have with the “no homework” policy our school has implemented is that parents have little idea what their kiddos are learning unless they take the initiative. For my son, it would be a no-learning zone if we didn’t also teach at home right now. He loves his special time with us too. Art of Problem Solving will launch their online Beast Academy very soon and he is so excited.

I've mentioned the Art of Problem Solving website on the blog -- this is where Southern California math professor Kent Merryfield sometimes posts, and there are forums devoted to AMC and other math contests.

Of course, both sides agree that there should be homework in middle and high school. As Garelick himself writes in the comments:

Barry Garelick:
Agree for lower grades K-5, say. But for middle and high school, homework plays an important role in building confidence and gaining practice as math grows more complex. Anti-homework advocates view homework at all grades as “busy work” which is definitely not the case.

When I was a young elementary student, our teachers assigned only weekly spelling homework -- so Monday nights we wrote each word five times each, on Tuesday nights we wrote each word in a sentence, and so on in preparation for the Friday spelling test. This was the only homework I ever had in Grades K-3. Beginning in fourth grade, we'd have projects for homework, such as the famous California missions project. The first time I ever had math homework was seventh grade -- recall that sixth grade was still considered elementary school in our district.

Yesterday's post from Garelick is basically a follow-up to Wednesday's. In the new post, Garelick posts another example of Assignments A and B:

https://traditionalmath.wordpress.com/2018/04/05/more-just-in-dept/

A: Factor completely, and state for each stem what type of factoring you are using.   

x4 + x3 – 6x2

B: Create expressions that can be factored according to the following criteria. Explain the process you used to create your expression.

A quadratic trinomial with a leading coefficient of 1 that can first be factored using greatest common factoring. The greatest common factor should be 2x.

Garelick writes:

"I find the wording of Assignment B a bit confusing but that’s besides the point."

OK, I agree that the wording is confusing. In fact, a quadratic trinomial with a leading coefficient of 1 can't possibly have a GCF of 2x. Most likely, the intention is for the trinomial to be quadratic monic after 2x has been factored out -- that is, the leading term is 2x^3. This isn't a quadratic trinomial with a leading coefficient of 1 -- it's a cubic trinomial with a leading coefficient of 2. But once again, the idea is the same -- students are less likely to hear "You're wrong!" on Assignment B than A.

Later on, Garelick gives some more examples from Dolciani 1962 -- and in the comments, he explains how he obtained the old texts.

Garelick concludes his post:

My students are not concerned with the “relevance” of the problem or whether it meets “real world” criteria. They want to solve such problems and draw on solid, explicit instruction and mastery of procedures in order to do so.

Meanwhile, the rest of us hear the students say "When will we use this in real life?" all the time. I wonder how Garelick would recommend that we answer that question.

(A quick Google search: when will we use this in real life -- all ten results on the first page are related to math. Only on the second page do we see other subjects -- Chemistry and Computer Science, with even the Comp Sci one going back to proofs in math. And as I typed the phrase in to Google, the suggested search terms are all related to math. Thus a few seconds on Google is enough to demonstrate that Garelick's students aren't typical.)