Showing posts with label radical equations. Show all posts
Showing posts with label radical equations. Show all posts

Friday, December 6, 2013

The square root of x+1 equals x+1... A Common Core Investigation

OVERVIEW
Fairly straightforward radical equation in the title but there is so much hidden potential here for students in Alg 2/Precalculus.
REFLECTIONS
• The solutions to the equation above are -1 and 0. No big deal, right? The usual algorithm --- just square both sides and solve the resulting quadratic by any one of several methods. Done. Cheerio. But wait...
• We can encourage students to "make it simpler" by substituting 'a' for x+1 obtaining a^(1/2)=a, square both sides yielding a=a^2 which gives 2 easy solutions 0,1 and then x+1=0,1 producing the final result. Not that big a deal though except...
• A graphical interpretation of these equations is illuminating and illustrates multiple representations/The Rule of 4. You could demo this with the graphing calculator displayed on your smart board or have the students graph by hand or on their device. The graphs of y=x^(1/2) and y=x intersect at x=0 and x=1 then, by translation, the graphs of y=(x+1)^(1/2) and y=x+1 will intersect at x=-1 and x=0. Students should be asked for this conclusion BEFORE checking the graphs to verify!
• Is that all there is? Hardly! The current trend on assessments and hopefully in texts is to have students analyze a family of equations using a parameter. But first we can generalize numerically:
Solve
(i) (x+4)^(1/2)=x+2
(ii) (x+9)^(1/2)=x+3
Are there still 2 solutions for each of these? Solving just a couple of these and recognizing extraneous or apparent solutions would traditionally have been the WHOLE lesson! Not any more...
By the way - why the "4" and "9"? Did I change the pattern from the original equation?
• Now for the parametric form:
(x+k^2)^0.5=x+k
What questions should STUDENTS be asking themselves BEFORE WE ASK THEM?
• Students can certainly be asked to solve the latter equation for x in terms of k. Some will struggle with the procedure/algorithm. Hopefully someone in each group (or the whole class!) will obtain x=0 and x=1-2k. BUT WILL THEY CHECK THE 2nd SOLUTION! The use of a parameter goes beyond making a better standardized test question. Now the student has to recognize that, in order for there to be 2 solutions, k must be less than or equal to 1 which was suggested by the numerical examples above.
• Of course I'm anticipating most teachers' reactions to an exploration like this. I've provided much more than can reasonably fit in a 40 min lesson. Use it as you see fit or just ignore it. It will go away or will it?

Monday, January 19, 2009

MLK, Inauguration, Math Contest and A Radical Investigation!

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...

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