MathNotations

Don't miss registering for MathNotation's First Math Contest. Registration is as simple as emailing me (dmarain "at" "gmail dot com") to request a form and the Rules. The contest is team-based (up to 6 students), is designed for both middle and high school students and should take 45 minutes or less (extra time is provided for students to enter their answers/solutions on the official answer form in Word). Look here for further info.

I would also like to thank the following blogs and/or webmasters for their graciousness in spreading the word about our first math contest:
Let's Play Math!
MathNexus
Wild About Math
Vlorbik
jd2718
Note: Take a look at jd2718 to see the latest Carnival of Mathematics. Another excellent job by Jonathan!
Homeschool Math Blog

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While we're waiting for the Inauguration on 1-20-09 (12,009 = 3 x 4003 of course), today is Dr. King's birthday, 1-19-09 and 11,909 is prime as it should be! How appropriate it is that we should be honoring today the man who paved the way for our new President...
The title of this post reminds me of an old Johnny Carson routine: Which one doesn't belong with the others! In fact, we can probably make connections among all of these if you're willing to play with words...
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In case you thought that the Math Contest would lead to a hiatus in publishing investigations and instructional strategy articles, fear not! Today we will once again examine the raison d'etre of this blog:
TEACHING BOTH PROCEDURALLY AND FOR MEANING


Part I
Consider the equation
\sqrt{x+2}=\sqrt{x}+2

To reinforce multiple representations (Rule of Four) we can ask students to:

Explain or show why this equation has no real solutions
(a) Graphically
(b) Numerically (TABLE)
(c) Algebraically


At this point I am including some ScreenShots from the TI-84. The bold graph is Y1:











Part II - The Extension!
Consider the equation
\sqrt{x+k}=\sqrt{x}+2


(a) For what value(s) of k will the above equation have one real solution? In this case, also determine an expression for that solution in terms of k. Show method clearly.

(b) For what value(s) of k will the above equation have no real solutions. Show method clearly.

(c) Demonstrate your results in (a) and (b) by choosing specific values of k for each case. Use both a graph and a TABLE to support your argument. [Use of the graphing calculator makes sense here.]

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Pedagogy
Which do you think is more helpful to students -- the graph or the TABLE? From my experience I find that both are important for comprehension and concept. They not only complement each other but each contributes something by itself. The graph not only suggests (not prove!) that the two graphs in part I do not intersect but it leads to a natural questions like: Why is the graph of y = √x + 2 above the the graph of Y1? What do the graphs suggest about the domain of each function? Explain the ERR messages!
Note: I used the word "suggest" because we want our students to understand that graphs do not prove mathematical truth.

When is it appropriate to use this approach? After you've taught the algebraic procedures of solving radical equations? Of course, part (c) of the activity asks for the algebraic explanation, but I've often used the graphical and numerical approach BEFORE teaching the procedure. I believe that it developed meaning for the traditional procedure but, in no way, did it replace the need for carefully explained instruction with a variety of examples! (The "balanced" approach!).

Further, the common reaction I've heard to this kind of instruction is that it is too time-consuming and appropriate only for the honors students. I couldn't disagree more. Developing meaning does take time and is absolutely worth it. It's all part of the "less is more" philosophy and, that, if the foundation is properly put into place, students can develop both the skills of solving radical equations and an understanding of the underlying mathematics. Enough preaching to the choir...


I hope you find this useful when building your next exploration in mathematics! Let me know...

Posted by Dave Marain at 9:54 AM 0 comments

Labels: graphing calculator, investigations, multiple representation, radical equations, radicals, Rule of Four

When a calculator displays zero as a result should students assume that is exact or only accurate to the precision the machine can store and/or display?

The next time students ask you why we use the conjugate method to rationalize denominators, here's an example of why we sometimes use the method in "reverse". This happens more frequently in calculus but the following is an apparently trivial numerical computation your students can try on their graphing calculators. The results of this computation depend heavily on the specific technology used (e.g., expect different results between the TI-89 and the TI-84), but hopefully they will get the idea. This numerical issue came up as I was solving an applied problem which required finding the difference between two very large numbers (the difference between distances from the center of the earth to a point slightly above its surface and the radius of the earth). This numerical issue has come up more before on this blog. Look here if you want to see another application.

Here's the computation:
Let R = 2.0916 x 10^7
We need to compute the following expression (denoted by **)

\sqrt{R^2+1.5^2}-R

For the Student
(a) Do the calculation directly on your calculator. You will want to store this value of R as a variable for later use:
2.0916x10^7 STO> ALPHA R
Does your calculator display zero? If so, explain this "error."
Note: This display depends on the calculator being used. I experimented with the -84 and -83. Let me know how the display appears on other machines. Of course, one would expect a very different outcome if using Mathematica!

(b) Rewrite the above expression ** by multiplying the numerator and denominator by the conjugate of the expression. (Hint: Put the original expression "over 1").

(c) Recalculate the value of ** using the modified but equivalent form from part (b).
What result do you see this time? Can you explain what may be going on?

(d) Find other numerical expressions that produce an incorrectly displayed result on your calculator! Post these in the comments section pls!

Posted by Dave Marain at 9:27 AM 2 comments

Labels: advanced algebra, algebra 2, approximation, conjugates, limitations of technology, numerical analysis, radicals

[As always, don't forget to give proper attribution when using this in the classroom or elsewhere as indicated in the sidebar]

In a standard trig unit, students learn those wonderful formulas for the sin and cos of the sum and difference of angles. Many creative methods have been developed to derive these formulas and, depending on the ability of the group and teacher preference, these are demonstrated or not. Students are typically shown various mnemonics for recalling them on the big test, but, in this investigation, students will derive sin 15° using only 30-60-90 triangle ratios and the Pythagorean Theorem. We will then compare the result to that obtained by the traditional formulas for sin(45°-30°) or sin(60°-45°) and show equivalence by algebraic methods using radicals. This is not an attempt to develop a general approach to deriving sum/difference formulas, although readers are invited to try a generalization. You may recall other posts on this blog of a similar nature.

THE PROBLEM/INVESTIGATION
Refer to the triangle above. If the print is too small, click on the image to magnify.
∠A = 75° and ∠B = 15°

(a) In the triangle above, locate point D on side BC such that ∠CAD = 60° . Express the lengths of the sides of triangle CAD in terms of a.
[Note: We could avoid the variable a altogether and assign a value of 1 since this is a ratio problem.]
(b) Show that CB = a √3 +2a.
(c) Use the Pythagorean Theorem to show that AB = a √(8+4 √ 3)
(d) Verify the identity (√ 6 + √ 2)² = 8+4 √ 3. Use this to rewrite AB.
(e) Use above results to obtain an expression for sin 15°.
(f) Use the standard trig formula for sin(45°-30°) to obtain an expression for sin 15°.
(g) Show your results in (e) and (f) are equivalent.

An Instructional Aside
When introducing the formula for sin(A+B), for example, teachers sometimes motivate it effectively using numerical values or considering the special case A=B. Here's an alternative:
Consider the special case A+B = 90°
Ask students to verify the formula for sin(A+B) in this special case. Simple, but at least it's something slightly different to pique their curiosity.

Standard Disclaimer:
This investigation is not copied from some other source. As it is original and has not been edited by others, there's always the possibility of error. Please feel free to suggest corrections/edits/extensions...
You know I welcome your comments!

Posted by Dave Marain at 6:32 AM 12 comments

Labels: 30-60-90, algebra 2, precalculus, radicals, sum-difference identities, trigonometry

Note: Many of the ideas below came from the excellent note in the Reader Reflections of the March 2007 Mathematics Teacher, contributed by Warren Groskreutz. One of his students derived and proved his own 'theorem' about certain radical expressions. I decided to develop these ideas into an activity. The last part is a bit different from 'Nathan's Theorem.' Mr. Groskreutz is to be commended for creating an environment in which such 'discoveries' can be made.

[Click on the small image below to enlarge.]

Posted by Dave Marain at 6:51 AM 0 comments

Labels: advanced algebra, radicals

The problem below (designed for Algebra 2 and beyond) was part of a weekly online challenge I established a little over a year ago. It ran for 2 1/2 months and the response from students was overwhelming. I posted the problem on our dept web page at precisely 6 PM on Mondays. Students had until 6 PM on Wed to email me their detailed solutions. One student would always wait until 5:59 PM! I would post the answers, comments and the best 2-3 student solutions. I rarely had to give my solutions - theirs were usually better! I may do this on weekends for a Monday challenge here and give you time to play with it for a day. Of course, I know some of you will have it solved in a heartbeat, but I'm hoping you will see why I created this for the students -- an opportunity to delve more deeply and learn how to deal with more sophisticated problems having several layers.

This week's problem is algebraic and not that bad. If you think you have all the answers, you can post them and your comments but hold off on detailed solutions to give others a chance! I will respond to the answers submitted at first. Sorry, it's still in hard to read format...

Posted by Dave Marain at 10:03 PM 8 comments

Labels: algebra, radicals