Test Today, Plus Interim Grades

Test was on absolute value functions (equations, inequalities, graphs, piecewise functions, and step functions). I didn't think it was very tough, but I should not jinx myself.

For those of you who want to flex your algebra muscles again, here are some examples of problems I saw on the test.

Graph: f(x)=3[x-2]+2 (pretend those brackets are bars)

Or, "Explain in words (using complete sentences!) how you would graph the following piecewise function:"

f(x)=
{2x-1, if x is less than 1
{-x+3, if x is greater than or equal to 1

We also got back interim grades, and I got an A! an Actual A.... 94 percent, plus one percentage point extra credit for attending the tutoring sessions. Even in Fairfax, it's an A.

Twenty percent of the grade is homework. A lot of students struggle there. For me, if I don't do the homework, I forget everything I learned, since the class meets every other day.

By Michael Alison Chandler  |  November 7, 2008; 11:16 AM ET  | Category:  Class Time , Friday Quiz
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Comments



Something I seem to use almost every week in some form or another and which is associated with the absolute value is called the triangle inequality and it says that for any two numbers a and b:

[a+b] <= [a] + [b]

There is also another version:

[ [a] - [b] ] <= [a-b] or
[ [a] - [b] ] <= [a+b]

Also, on a somewhat unrelated note, an interesting way of thinking about the absolute value is like this:

[a] = Sqrt(a^2)

It's good to pay particular attention to these exercises on the absolute value because they contain concepts that are used heavily in Calculus and I feel that we end up losing a few of students each year who are uncomfortable with the absolute value concept.

Posted by: mathlete | November 7, 2008 1:55 PM | Report abuse

Careful with your square roots, mathlete. You wouldn't want to cause confusion between principal square roots, the "other" (negative) square root, and absolute value.

An interesting note on the triangle inequality: at my old job, where we were all a bunch of nerds, we used to like to ask people what the angle of a triangle with sides 1, 2, and 3 was. It was fun to watch them try to remember the laws of sines and cosines. Then, when you remind them about the triangle inequality, they get quite embarrassed. Or at least, I did. :-)

Anyway, to me, the more practical way of thinking about absolute value is as a distance from 0. That will be useful when a class on vector analysis introduces the concept of the norm (which often uses the same notation).

Also, speaking of symbols, just a quick note: It's generally tough to type good mathematical notation, especially when we're limited to one font. But in the case of absolute value, we have the "pipe" character. It's the shifted character on the backslash key for English keyboards, and looks like this: |

[x] could be confused with brackets very easily, but |x| should be clear.

Posted by: tomsing | November 7, 2008 4:49 PM | Report abuse

tomsing:

Thank you. Both good points and potential sources of confusion. On the first point, Sqrt(x) is typical thought of as being positive by convention, at least in my circles. On the second point, I was just trying to keep Michael's notation.

You are quite right. I agree it's more useful to think of absolute value as measuring distance.

Posted by: mathlete | November 8, 2008 2:57 PM | Report abuse

Michael-
As a history teacher who also enjoys the math world (but not enough to teach it day in and day out), I wanted to comment and say that this blog is the highlight of my daily washpost fix. As you share more complex equations with us throughout the year, you ought to see if a program called MathType will translate to the blogging script. It's what many math teachers use to make their fractions and exponents clear to the reader when they write out their own problems.

Posted by: HistoryTchr | November 9, 2008 5:16 PM | Report abuse

okay. but i'd sure be more impressed
with a student who could *actually*
draw the graph (but couldn't explain it)
than a student who could (somehow)
"explain how" to draw such graphs
but consistently drew 'em wrong.

getting students to write coherently
is *very hard* ... and likely to become
a distraction from the (all-important)
mathematics itself. of coure this is not
to say that writing should have *no* place
in math classes. but it's very easy
to overdo it ...

p.s. |x| = \sqrt(x^2)
is in *no way* careless.
the "square root" symbol
(which i've here denoted "\sqrt")
unambiguously *denotes*
the "principal square root".
that's why, for example, there's
a "plus/minus" in the quadratic formula,
for example.


Posted by: vlorbik | November 10, 2008 4:34 PM | Report abuse

vlorbik, there are benefits to having kids write in a math class. One of the standards of the National Council of Teachers of Mathematics is to promote clear communications about math....I've been in a classroom and overheard students tell one another "Oh, you just move the thingy next to the other thingy"....how can anyone understand and remember what to do with that explanation?

Having students write explanations allows them to demonstrate they do know how to do a problem even if they consistently make calculation errors. It is difficult at first, but with practice, it becomes easier for students and they gain confidence in their math abilities as their ability to discuss and explain problems improves.

Posted by: lolee6241 | November 11, 2008 6:14 PM | Report abuse

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