MathNotations

Some time ago, I posted a piece about math mnemonics. Buried near the bottom was my feeble attempt to make a table showing a well-known (?) fascinating pattern for sin and cos values for the common angles in Quadrant I. Over the years, some students have found this to be as useful as memorizing ordered pairs on the unit circle or deriving everything from 30-60-90 and 45-45-90 (which I still prefer personally). I've seen students make this table at the top of their trig unit exam - they figured it was worth the effort! I'm reprinting this today using an image created in LaTeX and the absolutely wonderful and easy to use Texify website . This has been a real boon for those using Blogger since LaTeX has not yet been supported. Many math bloggers have been using it for a while now and I'm sure they appreciate its power and simplicity as much as I do. Its author is Andrey Burkov and Ars Mathematica gives him proper credit here . Certainly if an old dog like me can learn new tricks like this, anyone can! By the way at the Texify site, there is an extremely well-written tutorial with many examples to follow. I suspect I will be using this a lot for my new posts and perhaps cleaning up my old. Let me know if the table below is as readable as it appears to me. and, of course, if you like the pattern, you can tell me that too!

The original post used the klutziest of notations and was barely readable. This should be a lot better! I omitted the row for the tan function which is just the quotient of rows 2 and 3:


\begin{matrix}&&0^\circ&&30^\circ&&45^\circ&&60^\circ&&90^\circ\\\sin&&\frac{\sqrt0}2&&\frac{\sqrt1}2&&\frac{\sqrt2}2&&\frac{\sqrt3}2&&\frac{\sqrt4}2\\\cos&&\frac{\sqrt4}2&&\frac{\sqrt3}2&&\frac{\sqrt2}2&&\frac{\sqrt1}2&&\frac{\sqrt0}2\end{matrix}

Posted by Dave Marain at 6:31 AM 4 comments

Labels: math mnemonics, Texify, trigonometry

Sorry for the silly title but when the temperature approaches 100, I start becoming delusional!
Algebra teachers know that the equations of horizontal and vertical lines (the coordinate axes in particular) are stumbling blocks for students and creative educators and desperate students often resort to clever mnemonics and other memory aids to recall these. I invite readers to share their favorite. The teachers in my department (former department that is -- be kind, I'm adjusting to retirement) became enamored of HOY-VUX. I'm not sure who originated it but this person deserves credit! The name is silly (reminiscent of horcruxes from Harry Potter), the students laugh at it, but when the test comes around, they write it at the top of their paper. Here's how it works:
HOY: Horizontal, slope 0, Y=...)
VUX: Vertical, Undefined slope, X=...)

Now I know that others out there have their favorite ways of teaching this so PLEASE SHARE!

Believe it or not, the above was not the intent of this post but it's probably more interesting than the technical stuff to follow! This discussion is intended for Algebra 2 students and beyond. A full treatment requires some vector analysis but I will avoid that for now. Unlike most of my offerings, I did not set this up as a worksheet but you'll get the idea. You may want to bookmark this and save it for when 3-dimensional graphing comes up in the curriculum.

Start with a horizontal number line: <------------------|---------------->x
Ask students to plot x = 3 on the line. No ambiguity here, right!?!
Thus, the 1-dimensional graph of x = 3 is a POINT! Easy, so far.

Now draw both coordinate axes. Plot the point at 3 on the x-axis, ask a student for both of its coordinates and ask if it satisfies the equation 1x + 0y = 3.
Confirm this: 1(3) + 0(0) = 3.
Students generally treat x = 3 as an exceptional case of the equation of a line, but having both variables may help them see it isn't that special (other than its slope of course!).

Ask students to verify that (3,1), (3,2), (3,-1), (3,-2) all satisfy the equation 1x + 0y = 3.
Have them plot these points.
Ask the class (I didn't feel like writing this in the form of a worksheet today) to verify that (3,k) satisfies this equation for any real value of k. Students need to understand this significance of the zero coefficient of y.
This should help them to recognize that the graph of x = 3 is a vertical line. Don't get me wrong. Understanding this does not necessarily lead to getting it right on a test! They still need survival gear (mnemonics) for that! HOY-VUX to the rescue!

BUT THERE'S MUCH MORE TO THIS!
Point out that in the equation 1x + 0y = 3, we see that the resulting line is PERPENDICULAR to the axis with the non-zero coefficient (x-axis here) and PARALLEL to the axis whose coefficient is zero (the y-axis in this case). We're not proving anything here or explaining why this is true, just making an observation that we will generalize later.


Ok, so where's the plane in all of this?
In 3-dimensional space, we examine the equation 1x + 0y + 0z = 3. We can still graph the point corresponding to 3 on the x-axis, the vertical line graphed above (y can be chosen arbitrarily) and now z can be any real number. Corresponding to each variable whose coefficient is zero, the graph will now be a plane PARALLEL to that axis and PERPENDICULAR to the axis whose coefficient is not zero. Thus, our graph is now a plane parallel to the y- and z-axes (therefore parallel to the yz-plane determined by these axes) and perpendicular to the x-axis. Of course, software like Mathematica, Derive, or even freeware available on the web will help students visualize this better. Cardboard or Styrofoam models are also highly effective here. The more the students construct these models and label the axes and planes, the better they will be able to make sense of all this.

So what is the graph of x = 3? All of the above!
Now have your students analyze the equation y = 3 following this model!

Posted by Dave Marain at 3:37 PM 4 comments

Labels: 3-dimensional graphing, algebra 2, investigations, linear equation, math mnemonics

[You may want to read the comments for this post. Some useful devices to help students recall important rules/facts from trig & calculus.]


Regardless of whether one approves of giving students mnemonics to help them recall various math facts or terms, students do use some of these and, in fact, don't we all! I know many math teachers detest PEMDAS because it can mislead students but the 'positives may outweigh the negatives'!
Here are a few of my favorites, some of which I've devised and some I've learned from creative teachers and students. I know some of you have your own pet phrases - pls share!!
With the May SATs only a few days away, perhaps one of these will stick in a student's head and help...

1. Zero is a WEIRDO (last 2 letters need to have a strikethrough)
Ok, here's how this works: Each letter helps students recall an important fact about the number ZERO which many students seem to forget almost daily! I'll start you off - try to guess the rest:
W: Whole (i.e., Zero belongs to the set of whole numbers)

2. Spell the word 'WHOLE'. The middle letter reminds us that ZERO is WHOLE and (E)VEN.

3. INTEGER (underline the N, E, and G) - to help students recall that integers can be NEGative.

4. PRIME (strikethrough the letter I, circle the last letter 'E.')
This may help students recall that 1 (the letter I) is not defined to be a prime number; further, there is only one (E)ven prime. Lame yes, but the lamer the better.

5. F)M Some students still listen to their favorite FM station.
This is to help them recall that a (F)ACTOR 'goes into' a (M)ULTIPLE. Thus, 4)12 suggests that 4 is a factor of 12, while 12 is a multiple of 4. Ok, stop groaning!

6. (From one of my outstanding Algebra teachers E.S.):
Permutations are Picky
Combinations don't Care (about order).

7. 'If you're Y's, you go to the top' or RISE rhymes with Y's (and things that RISE always end up on TOP).
These silly statements may help them recall that, in the formula for slope, the y's are in the numerator.

Now it won't be hard to top these, so go ahead...

Posted by Dave Marain at 7:01 AM 5 comments

Labels: math mnemonics, primes, SAT-type problems