IF THE ALTITUDE ON THE HYPOTENUSE OF A RIGHT TRIANGLE DIVIDES THE HYP INTO A 1:3 RATIO, PROVE THAT THE TRIANGLE IS 30-60-90.
ASK YOUR STUDENTS TO
FIRST VERIFY THE CONCLUSION BY CONSTRUCTING A TRIANGLE WITH GIVEN CONDITIONS
(1) by construction with mechanical tools
(2) using electronic tools (e.g., Geogebra or Geom SketchPad)
THEN HAVE STUDENTS
(A) FiND AT LEAST TWO SYNTHETIC (DEDUCTIVE) METHODS WHICH DO NOT INVOLVE TRIG.
Reflections...
(1) So are given conditions both necessary and sufficient? Of course in geometry we usually write "if and only if" or "iff".
(2) I know most educators are annoyed that I ask lots of questions but don't ever answer them. Like, "who has time for this!" My children, grandkids and my students have always thought I was annoying too. I do pride myself on consistency!
(3) So who's going to call my bluff and ask me to show at least TWO methods of proof?
[Alt on hyp theorems same as similar triangle methods?]
If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175 problems divided into 35 quizzes with answers at back and DETAILED SOLUTIONS/STRATEGIES for the 1st 8 quizzes. Suitable for SAT I, Math I/II Subject Tests, Common Core Assessments, Math Contest practice and Daily/Weekly Problems of the Day. Includes multiple choice, case I/II/III type and constructed response items.
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Friday, January 25, 2013
A CCSSM GEOMETRY ACTIVITY -- IS IT NECESSARY OR SUFFICIENT
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5 comments:
1)a) Draw line segment AB; bisect with midpoint C. Bisect AC, with midpoint D (this is the required hypotenuse).
b) Construct perpendicular line at D (this is required altitude).
c) Construct circle with center C. Where circle intersects altitude is third vertex,E, for triangle (assuring triangle ABE is right).
d) Triangle AEC is isosceles (because radii), and because D is midpoint of AC, ADE is 30-60-90, which means AEC is also; likewise AEB.
2a) Draw segment AB, construct midpoint C and draw circle with center C and radius CB.
b) Construct circle with center B and radius BC.
c) Where circles intersect, D and E, connect line segment DE, which intersects AB at F.
d) ADB (or AEB) is your required triangle; DF is an altitude by construction, and it cuts the hypotenuse 3:1 by construction. Triangle ADB is right because D is on circle of center C, and because triangle BCD is equilateral, it is 30-60-90.
Thanks, 'e'! Long time since I've seen your distinctive nickname. How are you? (apologies if you're not the 'e' I remember! You are Euler, right?!?)
Thoughts about your highly 'constructive' comments:
1) I sensed that you really enjoyed figuring it out. This is what makes mathematics so satisfying for me.
2) Who needs 2-column proofs when you can 'construct' the proof! The essential ideas behind why it must be 30-60-90 unfolded before my eyes as I followed your instructions.
3) I often asked my students why 30-60-90 and 45-45-90 occur so frequently. The light goes on when they realize that they come naturally from equilateral triangles and squares respectively. The keys to your two construction methods were building equilateral triangles IMO.
4) I 'built' the 1:3 ratio by duplicating a segment (in effect multiplying its length by 2), then repeating this with the new segment (i,e., multiplying orig segment by 4. Didn't feel like using a midpoint construction!
5) Again, how many students are exposed to the beauty of a deductive 'proof by construction'. Your students don't know how fortunate they are (blush, blush!).
6) How do you feel about formal proofs and hands-on classical constructions becoming obsolete? Technology must be better, right?
Thanks again. I get discouraged when the only comments I receive are from students in Indonesia who seem strangely interested in my tweets! I know people are viewing but I must be doing something wrong if my attempts at social networking are all ONE DIRECTION (sorry, had to get in a reference to my daughter's favorite group!). Then again, without appearing egotistical, I do believe it's always hard to be a 'prophet in one's own land'. I have thought many times of moving to a country which might understand me better, but then I'd probably be ignored there too! By the way, I'm not really that needy for attention but I do miss having dialog with people like you.
Hello Dave,
I'm Eddi Vulic, long-time follower of your blog. I love all your competition and SAT questions, which I pass on to my math team and work out on my own. As you've probably guessed, my favorite math subject is geometry, and yes, when I read this challenge, I went immediately to Sektchpad to see how I would build the picture. I much prefer construction to two-column proofs, because you are actually working with the "physical" ideas (whereas I feel 2-column proofs are artificial). A construction should feel like a well-written recipe ,with the how and the why both important. On the other hand, I do really like Sketchpad and Geogebra and Desmos, so that I and students can get our hands dirty with the math, which I think is crucial to understanding and persistence and creativity.
On a slightly different note, I bought your quiz book last year and wanted to know if the one you're currently advertising is different; if it is, I'd definitely buy it, too.
I for one am glad you are back at blogging, and I'll be around.
Thanks for all your work.
Cheers
Thank you, Eddi! I need an infusion of your enthusiasm every few months to remind me why I should not abandon my blog!
Other than writing detailed solutions to the first 8 quizzes, there are no changes to my book. I need the self-discipline to write solutions to a few quizzes each day and be done with it, but I find that challenging. Email me if you need a free update.
You and I share similar sentiments about geom proofs. Sadly, what we know helps students learn better cocneptually often falls on deaf ears. Your construction methods should be on YouTube and required viewing by all prospective math teachers. Seriously!
Thank you for your praise; I'll think about posting as an antidote to Khan...
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