Archive for the ‘Math 104’ Category

Exam II. (Math 104.)

Polynomials. Quadratics in particular.
Bad news.

The campus-wide median score slipped
considerably and so did ours.

A lot of damage was done on the page
having problems of the forms
(a.) Complete the Square…
and
(b.) Use the Quadratic Formula…
(… to solve a given quadratic equation.).

I’ve written a little about QF in the past;
one of my most popular posts. And I like
getting attention as much as the next blogger.
Here are some remarks on “completing the square”.

So. Imagine me… imagine *us*…
scribbling on a blackboard and talking.

Let’s look at a quadratic equation
whose answer we *already know*.
Suppose that [ (x = 3) OR (x = -9)].
Then (“obviously”… check it!)
(x-3)(x+9) = 0.
“FOIL”
(or, finding this distasteful, expand
the product-of-binomials on the Left
Hand Side by some “other” algorithm):
x^2 + 6x – 27 = 0.

Now. Suppose it were some bloody
Exam Problem. “Complete the Square.”

Digress.

The trick is to find
a “perfect square trinomial”
having the same variable terms
as *our* trinomial (x^2 + 6x – 27).

We want
(x+?)^2 = x^2 + 6x + ___.
And the thing *here* is that
(x+?)^2 = x^2 + (2*?)*x + ?^2.

It follows that 2*? = 6, and so ? = 3.

Obviously, then, ?^2 = 9.
We’ve established that
(x+3)^2 = x^2 + 6x + 9
has the same variable terms
as our original polynomial.
We’re ready to go. (End digression.)

Start with x^2 + 6x – 27 = 0.
Add on both sides (to “isolate”
the variable terms):
x^2 + 6x = 27.

Here comes the magic…
the step that gives “Complete the Square”
its name. Add on both sides *again*:
x^2 + 6x + 9 = 27 + 9.

(the point here was to get the
“perfect square trinomial”
we computed in the digression;
the LHS has been transformed
into a easily-manipulated form
[abstraction to the rescue!].)

Now just use the “perfect square” property:
x^2 + 6x + 9 = 27 + 9
becomes
(x+3)^2 = 36. So
x+3 = plus-or-minus root-36;
x = -3 plus-or-minus 6;
finally
x = 3 or x = -9
(as we already knew from the
factored form we began with).

With the example in hand, the best
thing is to go off and *do* a bunch
of similar examples. And *then*
consider the “abstract” version
used in deriving the Quadratic Formula.

Never; the less.

Let A, B, and C be Complex Numbers.

(OK… let ’em be elements of your
favorite Field. Rationals, Reals,
and Complexes all work; so do many
other Fields not considered in
School Mathemathics (alas).
The point of using {\Bbb C}
the Complex Number field…
is that the “answers” will always “exist”.
Famously x^2 + 1 = 0 has no (so-called)
“real number” answers; this kind of thing
doesn’t happen in “algebraically closed”
fields like the Complex Numbers.)

Suppose further that A\not=0.
The Following Are Equivalent.
Ax^2 + Bx + C = 0
x^2 + (B/A)x = -C/A
x^2 + (B/A)x + (B/[2A])^2 = -C/A + B^2/[4A^2]
(x+ B/[2A])^2 = [-4CA]/[4A^2] + B^2/[4A^2]
x + B/[2A] = \pm \sqrt {[B^2 – 4AC]/[4A^2]
x = [-B \pm \sqrt {B^2 – 4AC}]/[2A].

In other words, when A is nonzero one has
Ax^2+Bx+C =0 \Leftrightarrow x =  {{-B \pm \sqrt{B^2 - 4AC}}\over{2A}}\,,
i.e., QF: we have derived
the Quadratic Formula.

(Remark: one need not be able to reproduce
this calculation to “prove” QF. Suppose you’ve
*memorized* it at some point but now you’re not
so sure. To check that *your* formula really is
the true “quadratic” formula, just choose either
the “plus” or the “minus” in the appropriate
place in the code [where i’ve abbreviated \pm
where i haven’t just written-it-out].
Then just “plug in” the whole mess on
Ax^2 + Bx + C and turn the crank until
out pops zero [if your formula and calculations
are correct]. It’s easier to *check*
that the formula works than to *derive* it.
It’s a good exercise, too, in my opinion…
but I don’t think I’ve ever assigned it
[or seen it assigned in a textbook].
)

Here’s a verbal recap.
We began by “dividing away the leading coefficient”.
This is where we “use” the condition A\not=0.

(Of course, it also makes sense to say that we “used”
this condition already in calling Ax^2 + Bx + C = 0
a quadratic equation in the first place…
if A were 0, one would have the *linear* equation
Bx+C=0. (We can handle these *without* memorizing
the “Linear Formula” [the equation is equivalent…
when B\not=0… to x = -C/B]. Instead one learns
“steps” [subtract C from both sides; divide by B].
For that matter, while I’m thinking about it,
neither does one expect students to learn that
“y = Mx + B” equivalences-to “x = (y-B)/M”
[when, dammit, M\not=0… I can’t help myself…];
rather, again, one simply carries out certain “steps”
to *dervive* this result whenever it’s needed.
Formulas-versus-procedures is a major battleground
in the Math Ed Wars so I think about this stuff.
A great deal sometimes.
)

Having “divided through by A” we get
(the so-called “monic” polynomial…
having lead coefficient 1
is a big enough deal to deserve
its own name…) x^2 + (B/A)x + C/A = 0;
subtracting the constant on sides gives
x^2 + (B/A)x = -C/A.

The number (B/A) is in the position of
the “6” in our first example.
Just as it turned out in the example,
where we add 9 to both sides because
9 is (6/2)^2, in *every* instance
of “complete the square” we add
the square of half of the constant-coefficient
(of the monic form) to both sides of the equation.
x^2 + (B/A)x + (B/[2A])^2 = -C/A + B^2/[4A^2]

Factor on the left; take square roots
on both sides (slightly tricky;
don’t forget the \pm!); clean up.

Coffee!

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the skin of our teeth

tonight we looked at a *great deal*
of material-from-the-syllabus.

too much by almost any standard,
i think. and obviously, missing a day…
half a week, really… doesn’t help.

and yet. it’s built into the course.
never mind snow days: if you’re getting
most of your ideas about What Algebra Is
*or* How To Do It, *from this presentation*,
then you’re almost sure to be left behind
(& very quickly) in the nature of the case.
this is College Math: “way too late
and much too fast”.

(meanwhile. *read*, dammit! and *talk* to each other.
for hecksake! do you think you’ll be young *forever*?)

okay. everybody gets their expectations
pushed-and-pulled around (and otherwise distorted)
and somehow we work out some way of getting along;
some way of talking about what it all might mean.
if there’s scribbling-on-the-board involved…
if there’s *symbolism*… then that’s what i call
“Doing The Math”. okay okay. don’t hate me for it.

quadratic equations, *again*? well, yes.
maybe you’ve missed something.

it’s much better if *you* take a piece
of chalk: *what* were you saying, again?

why have the deepest-thinking sages
returned to suchlike issues again and again
throughout all recorded time? (no… really:
why?)

because, yeah, duh: “theology”, so called.
“ontology”, forsooth.

it’s mostly transparent, though, if we agree
that “appeals to authority” are even more
contemptable than “outright lies”.
funding, funding, funding. everybody
talks about the weather. fuck the
god damn weather.

philosophy. feh.

quadratic equations.

*every* quadratic equation…

Ax^2 + Bx + C = 0…

can be solved (in the appropriate “domain”)
by the famous “Quadratic Formula”
(cf: QF Lore [a popular piece
from my blogging heyday]). To wit.

let D = B^2 – 4AC.

(D… or \sqrt{D}… i forget…
is the *discriminant* of our function…
we *were* talking about a function, right?
let’s see. let f(x) = Ax^2 + Bx + C, where
A, B, and C denote “numbers” [i.e., elements
of the Domain of Discourse] and “x” is
an “indeterminant”.)

we are not data.

*obviously* there isn’t-and-can-never-be
any “quadratic formula” for life itself:
“if you act *this* way, life will
work itself out in *that* way!”…
and it’s halfpast time we stopped
thinking that “math’, all by itself,
could ever fool anybody into thinking
that it’ll even ever’ve been a good
idea to *try*…

still. dammit.

suppose A \not= 0.
(otherwise, our equation
isn’t quadratic at all
[rightly so-called] but
merely linear; refer to
some already-well-understood
theory, duh). then 1/A
is a number. multiply
both sides of Ax^2 + Bx + C = 0
to obtain (the “monic” equation…
leading coefficient equal to 1…)
x^2 + (B/A)x + (C/A) = 0.

one “easily” applies the technique
called “completing the square”
at this point (add-and-subtract
the square of “half the middle coefficient”;
“regroup” the pieces and rewrite
the appropriate bit of code
as a “perfect square”).

clean it up and show
(at some appropriate level
of rigor) that the equation
Ax^2 + Bx + C = 0
(by the godlike authority
of faith-in-perfect-clarity
[work it out!]) is equivalent
(when A \not= 0; when the domain
of discourse allows the relevant
operations) to
x \in { (-B +\sqrt{D})/(2A), (-B -\sqrt{D})/(2A) }.

nobody wants to talk about the set theory here.
wait. that’s false. *i* want to talk about
the set theory here. can i get a witness?

but i’m here of course. in a short while i’ll go see
who *else* made it through the ice and snow;
naturally i’ll try to give ’em something extra.

but “time listening to owen rambling about math”
doesn’t seem to be *widely* understood as
“something extra”. so we’ll see.

there’s a new edition of MEdZ #1.
an 8-page digest. much easier
to read. order now.

Blogging 104. Exam 1.

104 links.

The first exam was last week. I’ve posted grades into the appropriate intranet doo-hickey; students can look up their own grades, administrators can tell I’m actually doing the part of the job that matters most (to the machine), and I have some insurance against the almost-unthinkable fate of losing my gradebook. I haven’t been posting Quiz or Homework grades since these are subject to tweaks. The “daytime” classes have Lecture and Recitation meetings, with Q’s & HW’s in the recitations; in my “evening” class, I’m both Lecturer and Recitation Coach (both “sage on the stage” and “guide on the side”… holy moley, I’m beside myself). Anyhow, this situation, as you can imagine, improves the communication between Lecturer and Recitation Coach immeasurably and it’s, well, better for morale if I hold back on posting the “recitation” grades.

Alas, there’s been a marked tendency on the part of a sizable fraction of the class to treat Tuesdays (when I administer Q’s and pick up HW’s) as the “recitation” section and Thursdays as the “lecture”… and then skip the lectures. Attendance is a lot better on Tuesdays, in other words. For this and other reasons, I didn’t do much “post mortem” work (“going over the Exam”) when last I met the steady-attenders on Thursday. Here’s a rundown now.

Very pleasant for me overall. We use the same Exams as the Day Class versions of 104, so until the exact day, I didn’t know exactly what to expect. It’s a real good test: do-able in an hour by appropriately-prepared students. The temptation to use tricky questions has been suppressed to my relief. This is somewhat in defiance of a claim in the syllabus to the effect that any HW problem is as likely to appear on an Exam as any other. So be it. A custom more honored in the breach than the observance, sez I. I got a pretty typical distribution… too small (n = 15) to be a good fit to “the curve”, but with the much-to-be-expected “lots in the middle and a few at each end” property just the same. One perfect score; two failing scores. Mean \mu = 75.4; Median Q_2 = 78.5.

Solutions to “two linear equations in two variables”; check. The class-as-a-whole has, anyway, learned the basic moves for both the “addition method” and the “substitution method”, and proved it on the first page. One variable in one solution had a value of 0; this causes more confusion than one would like; the “no solution” and the “infinitely many solutions” cases are typically more confusing still… so I don’t say we’ve mastered the methods. But everybody’s at least prepared to have a conversation about how this stuff works (if there were world enough and time).

On the more abstract interpret-the-graphs version of the “systems of two linear equations” situation, there is, predictably, more confusion. The same page revealed more of the zero-versus-nothing bug… a common difficulty for learners of math (ever since the introduction of “zero”… and yet, confusing as this seems to be for the laity, “zero” is one of the best ideas of all time…). A classic instance of the classic general complaint of teachers about students, “they don’t want to think”. But we’ve got, anyway, kind of a grip… somewhat tenuous I have to admit… on translating from data presented graphically to equations and inequalities. (Going the other way one has the Graphing Calculator so this is the hard way.)

Speaking of inequalities: interval notation was the source of the worst difficulties here. Our examiner very tastefully avoided absolute value inequalities altogether. I don’t say that this topic is more trouble than its worth; I do say I’ve seldom seen it done right. Our text has rather a high-concept presentation that I liked… but I sure didn’t feel like we’d taken enough time to’ve improved the overall skill level in this area by much if at all. I note with pleasure that we skip the section on graphing-systems-of-inequalities. This topic was very badly handled at Crosstown Community College (in a mostly-very-different course, also called 104): in particular, there was a flat-out mistake, for years, on the key to the Final Exam (concerning the “shading” of a certain “boundary point”); nobody seems to have been scandalized by this but me: this is one of those areas (like “set-builder notation”) where the instructors quite often aren’t qualified to present the standard treatment of the material. The industry is aware of the situation and appears to like it that way.

The slope of y=3x+7 is 3. Not “3x” (dammit). If the variable were part of the slope, we couldn’t get the slope by “plugging (four) numbers into (all four variables of) the well-known formula for the slope”. (Now, could we? Think, doggone it! Think!) But… of course… the question of “how variables work” is one of the trickiest of all: this takes practice. Of course the classic area for confusion-as-to-the-nature-of-variables is “word problems” and we’re still seeing quite a bit of it here. It pleases me much more than it should that we had a “mixture” problem since I laid a lot of stress on those; my class would have done even worse on some of the other problem “types” (precisely, on my model, because of the nature-of-variables issues; problems-by-type is essentially a way of “routing around” students’ astonishingly-stubborn refusal to discuss what variables mean and how).

Intercepts are points, not numbers (when we’re being careful). It’s common enough when talking to say, for example, that “3” is “the intercept” for “5x + 3”, when we mean “(0,3) is the y-intercept for [the graph of] the equation y = 5x+3″. The fact that we require more precise language in certain contexts than in others should create no confusion.

But “should” has nothing to do with much of anything, and it turns out that this “be more formal in work to be handed in than when you’re banging away talking and calculating” thing has been a major problem for a sizable fraction of any class I’ve taught at this level of the game. They don’t want to do it and think I’m just being mean. “Well, I meant…” [such-and-such], they’ll tell me, refusing to believe that it’s my duty to grade what they wrote and not what they meant. This misses rather a big part of the whole point of bothering with “mathematical precision” at all: code can be perfect.. and “perfect” is a lot better than “almost perfect”. It should be helpful to think of computer interfaces here: one wrong mousetweak can botch the whole environment. But somehow it never is. Helpful, that is. “Should”. Feh.

Hey. Madeline just woke up. See you later.

remarks on recent work

there’s info on assignments here;
i’m hoping everyone knows this already.

i only graded three homework problems.
three points possible apiece, plus one
for appearing-to-have-done-most-of-the-rest:
ten points possible. the 5-day-a-week classes
have a scale of “grade *two* problems at
*two* points apiece, plus one for completeness”.
so, while any slight mistake (using my scale) is
already 10% of the grade on the whole paper,
this is a heck of a lot better than 20%.

when i noticed something without even trying…
an inappropriate long-digit decimal approximation,
for example… on an *ungraded* problem, i went
ahead an remarked on it. but there’ll be quite a
bit of interesting work going *unremarked* here
of course. ideally, every student would talk over
the homework with at least one other student…

a few papers had “no slope” for “zero slope”;
don’t. i let these go by… but, unfortunately,
some writers use “no slope” for *undefined* slope.
so it’s best not to use this language at all.

using graph *paper* for graphs seems to correlate
(positively) with “good grades on HW1”.

intercepts are points, not numbers…
the x-intercept might be (3,0) for example
(not 3). not an enormous big deal…
but it pays to try to be as precise
as we know how.

there was a system-of-equations having
*no solution* on the quiz. i gave full
credit for “parallel lines” in one case…
but we’re looking for “no solution” here.

at least one student panicked pretty badly
on this problem… and *erased* what looks
to’ve been pretty good progress toward the
answer. when you get an equation that
*can’t be solved*… remember that this
doesn’t necessarily mean you’ve done
anything wrong! (and *whatever* you
do… don’t “blank out” on a problem!).

when there *is* a solution for a system,
we’ll prefer *ordered pair* solutions
(for “abstract” problems like the ones at
hand… for “word” problems, it is of course
more appropriate to give “word” answers
[typically including units]).

fractions are *more algebraic* than decimals
and much to be preferred. the calculator is
pretty good at making the conversions, too.
so-called “mixed numbers” like 3{1\over8} are
much harder to work with than (so-called)
“improper fractions” like 25/8. students
of algebra should make the effort to get
used to this situation. again, the calculator
can be very helpful.

Blogging 104. Week One.

(Homework and other administrative stuff is here.)

Blogging 104. Week One.

There were an unusual number of walkouts on the first night… and a much smaller class the second night. We’ll see how things shake out when the points-for-a-grade start going in the log next week.

For all I know, One-Oh-Anything students at Big State might habitually “shop around” on the first night of these Night-for-Day classes (most of the sections meet a Lecturer in a big lecture hall and a TA in smaller “breakout” sessions… I do, in effect, the lecture and the recitation [with less time to do ’em in; the big classes have extra sessions for exams (all the sections get the same exam), whereas I’ve gotta give up class time…].).

And, of course, for all I know, I’m just the world’s worst lecturer-slash-recitation-coach and they’re just running away fast for their own good. You’ll forgive me if I find this option hard to believe. Things’re going pretty well by my lights. Several students responded early on to questions I tossed out to the general room. In some cases (when I needed to refer again to the result), I asked their name (and then immediately used their name in referring to the result). In principle, this is part of my getting-to-know-you process but in practice I tend to forget the names and it’s really about letting ’em know they can and should speak up when they have something to say (and that it’s good idea to know who the other students are and what they can do).

Then the much-smaller Day Two class was all over me with the questions. Actually, it was almost entirely the same three people during the “lecture” bit… but I spoke with several more on the “problem solving” bit.

And the problem solving itself? Well. Too soon to tell. No scarier than one should expect, I suppose. One is mostly doing recap for most of the students here or they wouldn’t stand a chance at this pace. This week we “did” four textbook Sections, “covering” slope-of-a-line, forms of (two-variable) linear equations (general, slope-intercept, and point-slope), and systems of (two) such linear equations, via (a.) algebra (“substitution method” only; the “elimination method” is in Section 5 [but naturally I looked ahead and did one this way]) and (b.) the Graphing Calculator.

The second half (five weeks) of Math 102 at Crosstown Community College where I’ve presented the same material countless times to students mostly even more doomed than, let’s say, the bottom quartile of the 104 students here. I may be misunderestimating something or somebody somehow of course but make no mistake. There is a great deal of doom in these courses in their nature. Math departments pay the rent by “weeding out” students in “required” courses… in programs where Algebra plays no other part. It feels like telling tales out of school putting it thus bluntly. But actually, outside of school it’s pretty well understood. It’s just taboo in school because teachers are incredibly touchy about their grading practices (and administrators are worse).

So. My team on the first day… those who stayed to the end and handed in a Problem Of The Day… did “Find an equation for the line through (-1, 3) and (2, -6)” [or somesuch pair of points]; half of ’em got it perfect. Of course, one should routinely check such work (and not hand in until you know it’s right), but this is still a pretty good sign. What’s more, there were no blind-fumbling-with-formuli papers at all (handed in). All but two or three appeared to have a pretty good handle on the nature of the procedure.

And last night? The shrunken class made it easy and comfortable to talk over most of the papers with their authors and it felt pretty right. I plotted (-7, 11) against some co-ordinate axes and sketched the horizontal and vertical lines through (-7, 11); also the line through (-7, 11) and the origin; find the equations. This used to give 102-103 students fits back at Crosstown. Also a straight-up “solve the system”.

And, like I say, so far so good. On the actual quiz Tuesday? The “special case” stuff… systems with no solution or infinitely many solutions, for example… will probably throw more than a handful for a loop (despite my having… emphatically… “reviewed” it just before the quiz [as I intend to do]). Beyond that, I’m unwilling to predict. Don’t want to jinx it.

Oh, and I’ll draw a map of the room just before they take the quiz and study the names. If I time it right, I might be able to take my “quiz number one” and recite ’em all off (with the “map” hidden from my eyes). Yay, small classes.

Math 104 Course Page. Section 19291. Calendar (pdf).
Carmen (log in for grades).
MSLC (“learning center”) 104 page (MTWR 10:30–4:30).
Homework (pdf).

Math 104 blog archive.