tag:blogger.com,1999:blog-1697471610686007730.post4392820004602068544..comments2013-11-18T14:45:03.674-05:00Comments on f(t): Special Right TrianglesKate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.comBlogger34125tag:blogger.com,1999:blog-1697471610686007730.post-12554495915169383922013-02-14T10:17:16.979-05:002013-02-14T10:17:16.979-05:00I've basically done the equilateral triangle t...I've basically done the equilateral triangle thing exactly as you describe it, minus the intro with the dollar bill - Definitely makes things more interesting. I have not checked out "this page" yet, but I will. I think I need a couple of more hours in my day now that I've discovered this new world. Sherylhttps://www.blogger.com/profile/16595465854014581107noreply@blogger.comtag:blogger.com,1999:blog-1697471610686007730.post-50879482079218512342013-02-13T10:56:01.674-05:002013-02-13T10:56:01.674-05:00Hi Sheryl! Welcome! If you haven't already, ma...Hi Sheryl! Welcome! If you haven't already, make sure you see <a href="http://mathtwitterblogosphere.weebly.com/" rel="nofollow">this page.</a><br /><br />How you get them to find the ratio is kind of on you. :-) It depends on what class it is, what the kids already know, how comfortable they are with open-ended investigation...<br /><br />Personally, I think I'd do some version of <br /> - draw an equilateral triangle on the board<br /> - draw an altitude<br /> - ask them to write down everything we know for sure<br /> - ask them to articulate what it is we want to find <br /> - wait until someone remembers that they know the pythagorean theorem<br /> - troubleshoot their algebra mistakes<br /><br />Good luck! Let me know how it goes.<br /><br />Kate Nowakhttps://www.blogger.com/profile/14229054922453438248noreply@blogger.comtag:blogger.com,1999:blog-1697471610686007730.post-85355412242718401122013-02-13T10:47:16.722-05:002013-02-13T10:47:16.722-05:00I have been teaching for about 20 years and have o...I have been teaching for about 20 years and have only now discovered blogs. WOW - I am redoing a lot of my lessons - Thank you! I think my students will definitely like this as an intro. Just making sure I am not missing something - Once they unfold the bill, how do I get them to find the ratio? Sherylhttps://www.blogger.com/profile/16595465854014581107noreply@blogger.comtag:blogger.com,1999:blog-1697471610686007730.post-10740982212377437032012-11-07T15:56:47.130-05:002012-11-07T15:56:47.130-05:00Awesome intro to Special Right Triangles!! Just pl...Awesome intro to Special Right Triangles!! Just planned to use this dolla bill inquiry activity with my 8th grade pre-algebra kids next week. Love your blog!!mrsaitorohttps://www.blogger.com/profile/04492694849023177157noreply@blogger.comtag:blogger.com,1999:blog-1697471610686007730.post-76683683334734128422011-09-10T02:39:44.707-04:002011-09-10T02:39:44.707-04:00Very cool. Sadly, the ratio is different for Canad...Very cool. Sadly, the ratio is different for Canadian money. Of course, I will happily accept any of your donations.reflectionsinthewhyhttp://reflectionsinthewhy.wordpress.com/noreply@blogger.comtag:blogger.com,1999:blog-1697471610686007730.post-49481186991723705172011-02-25T23:06:51.758-05:002011-02-25T23:06:51.758-05:00Great teaching strategy! Every time I read one of ...Great teaching strategy! Every time I read one of your posts I'm impressed at how you come up with one of these. Engaging kids on a visual level is something that I want to employ! Thank you!donkeyboyhttps://www.blogger.com/profile/02630140841851130767noreply@blogger.comtag:blogger.com,1999:blog-1697471610686007730.post-23990548546398894012011-02-13T23:52:28.316-05:002011-02-13T23:52:28.316-05:00Addiction - no kidding. My whole weekend just van...Addiction - no kidding. My whole weekend just vanished. This is fantastic.Malcolm Eckelhttps://www.blogger.com/profile/01896314938716272887noreply@blogger.comtag:blogger.com,1999:blog-1697471610686007730.post-38586744299391831312011-02-13T20:40:27.077-05:002011-02-13T20:40:27.077-05:00Hi, Malcolm. Welcome to your new addiction. :)Hi, Malcolm. Welcome to your new addiction. :)Kate Nowakhttps://www.blogger.com/profile/14229054922453438248noreply@blogger.comtag:blogger.com,1999:blog-1697471610686007730.post-61514038093667882392011-02-13T19:26:30.637-05:002011-02-13T19:26:30.637-05:00I just found your blog (actually, I just found the...I just found your blog (actually, I just found the whole edublogsphere in general), and I love this activity. I'm using it tomorrow.<br /><br />I've been going two and a half years now in this job without thinking to start looking online for people sharing stuff like this. OOPSIE.<br /><br />Thanks.Malcolm Eckelhttps://www.blogger.com/profile/01896314938716272887noreply@blogger.comtag:blogger.com,1999:blog-1697471610686007730.post-25239889072783866472011-01-29T14:36:41.800-05:002011-01-29T14:36:41.800-05:00I definitely believe the kids need to have a thoro...I definitely believe the kids need to have a thorough understanding of the unit circle in order to fully understand the graphs and values of the trig functions. However, when it comes to actually finding the sine of 210 degrees, I am undecided. I do see a lot of kids who need to count their way around the circle to find that value and I'm not convinced that is best. I would much rather they know that it is the same (other than the sign)as the sine of 30 degrees and then find that by knowing it is opposite the short leg and is, therefore, 1/2. I did find an example or two of studies that compared students' retention and understanding of the trig functions given both methods and they seemed to say that the right triangle approach is better in the long run. In answer to your second question, I think it is great for the kids if they have been taught to come at a problem a variety of ways so, no, I don't think we all need to teach exactly the same way. That said, I also believe that while I am teaching trig, I need to be preparing my kids to handle the calculus course they will be taking. If I can teach some of the concepts similarly to the way they will be learning them in calculus, I think it will benefit them when they get to that course. Honestly, I have done so much research on the two methods of teaching the trig functions. I know that I am not a fan of "tricks" for memorization...I'm one of those teachers that asks the kids "why" all the time and wants them to know why they are doing something rather than memorizing a trick.Kathyhttps://www.blogger.com/profile/03192520514281357632noreply@blogger.comtag:blogger.com,1999:blog-1697471610686007730.post-28680104050398529242011-01-27T16:43:50.326-05:002011-01-27T16:43:50.326-05:00Hm, well, it kind of seems like micro-managing at ...Hm, well, it kind of seems like micro-managing at a level that I don't engage in. However, I understand the frustration with imperfect recall of memorized facts... but I don't get the sense that my students memorize coordinates on a unit circle, either. They tend to figure out which quadrant, draw the triangle, and use the sides of special right triangles. Do you have a preferred method, Kathy? Do you think it's better for teachers in the same school to standardize?Kate Nowakhttps://www.blogger.com/profile/14229054922453438248noreply@blogger.comtag:blogger.com,1999:blog-1697471610686007730.post-30921165020972573122011-01-27T16:07:00.095-05:002011-01-27T16:07:00.095-05:00Perhaps I didn't explain it well then. They d...Perhaps I didn't explain it well then. They definitely use the unit circle to visualize where 210 degrees is and we do use the unit circle often for determining the reference angle or the radian measure. However, for actually finding the sine of an angle, we are supposed to teach them how to find the sine of 30 degrees based on a triangle rather than having them memorize the coordinates that go around the circle. She tells them right away that they should not have to count around the circle to find the sine or cosine of an angle, rather, they should understand what sine is and be able to instantly know it from a triangle. I will also add that she is an amazing teacher and her AP scores are phenomenal. We have one trig teacher at our school who uses the unit circle and it frustrates the calculus teachers that the kids need to draw a circle or count around it to find the sine of 210 degrees!Kathyhttps://www.blogger.com/profile/03192520514281357632noreply@blogger.comtag:blogger.com,1999:blog-1697471610686007730.post-54943855140244175622011-01-26T22:28:31.695-05:002011-01-26T22:28:31.695-05:00Hi Kathy - I'm not sure how you could understa...Hi Kathy - I'm not sure how you could understand that 210 in standard position gets you a 30 degree reference angle without sketching/understanding the unit circle. Also, I don't see how it is possible for a teacher to forbid a student to reason a certain way. But maybe I am missing something.Kate Nowakhttps://www.blogger.com/profile/14229054922453438248noreply@blogger.comtag:blogger.com,1999:blog-1697471610686007730.post-55634346802251367252011-01-26T04:52:29.156-05:002011-01-26T04:52:29.156-05:00I am not sure where the best place to post this on...I am not sure where the best place to post this on your blog would be so I am choosing your most currect trig entry. I love to read all of the blogs and find yours to be one of the best concerning trig so I'm hoping you can help. I teach trig at a high school where our AP Calculus teacher believes in the right triangle method for teaching how to find the sine of an angle, such as 210 degrees. Although we absolutely use the unit circle a great deal for many things (visualizing radians compared to degrees, finding sin and cos of quadrantal angles, knowing the signs of the quadrants, seeing how the graphs are formed), when it comes to finding the sine of something like 210 degrees, the students are absolutely not allowed to do that on a unit circle. They need to know that it has the same sine as a 30 degree angle, other than the positive or negative, and then they need to know the sine of 30 degrees by use of a 30-60-90 triangle or memorization. She strongly believes that we are doing the kids a disservice when we go through how a unit circle is made with triangles once but then allow them to memorize their way around with various tricks and mnemonics. The students are actually forbidden to sketch the unit circle for a case like this as she does not believe it leads to understanding, just memorization. Therefore, my students know all about the unit circle except for actually memorizing the points for the common angles. Do you have any thoughts on this? I've searched for a discussion on this topic and can't seem to find a good one!Kathyhttps://www.blogger.com/profile/03192520514281357632noreply@blogger.comtag:blogger.com,1999:blog-1697471610686007730.post-51961520510504500842011-01-19T19:51:46.439-05:002011-01-19T19:51:46.439-05:00I found this site today and thought of you and thi...I found this site today and thought of you and this assignment if you are interested. Festisite.com lets you make your own personalized money.Christyhttps://www.blogger.com/profile/05004248371200090884noreply@blogger.comtag:blogger.com,1999:blog-1697471610686007730.post-32168609084955813282010-11-13T18:01:14.436-05:002010-11-13T18:01:14.436-05:00How incredibly cool! I just found this blog, and ...How incredibly cool! I just found this blog, and have enjoyed your passion for teaching mathematics so much. I don't know if you might be interested but many math teachers seems to need a better way of drawing and writing mathematics. I know I did so my husband wrote a program that embeds in Word that has helped me tremendously. It's free and can be downloaded at electricabacus.com. Let me know if you like it!Debbiehttps://www.blogger.com/profile/13785974709475065965noreply@blogger.comtag:blogger.com,1999:blog-1697471610686007730.post-82140062957017618432010-11-09T05:15:12.398-05:002010-11-09T05:15:12.398-05:00Look at the top of the outlined triangle. The E is...Look at the top of the outlined triangle. The E is directly in the center on the top. This E stands for EYE and is the THIRD letter 'the'.Pez V1https://www.blogger.com/profile/10576771081662061346noreply@blogger.comtag:blogger.com,1999:blog-1697471610686007730.post-2143230192736067422010-11-07T13:25:36.889-05:002010-11-07T13:25:36.889-05:00If you know you're starting with two eq triang...If you know you're starting with two eq triangles and drawing a rectangle around it, of course it's no surprise at all. But if you're wondering about the rectangle and then notice the triangle thing, it's surprising. At least, I was surprised, and a little delighted. I guess one person's weird is another person's ho-hum.Kate Nowakhttps://www.blogger.com/profile/14229054922453438248noreply@blogger.comtag:blogger.com,1999:blog-1697471610686007730.post-81616454073031426172010-11-06T10:04:39.683-04:002010-11-06T10:04:39.683-04:00okay. i like it. but how is
"put one equilat...okay. i like it. but how is<br />"put one equilateral triangle<br />next to another with their<br />bases along a line; inscribe<br />the whole thing in a rectangle"<br />anything like "supremely weird"?<br />square roots are just *built into*<br />plane geometry (and our allegedly<br />intuitive notions of "distance"). <br />one gets used to it after a spell.<br />(n.b. the whole eye-in-the-pyramid<br />thing is indeed quite creepy;<br />another story though...)owen thomashttps://www.blogger.com/profile/09249915192605437832noreply@blogger.comtag:blogger.com,1999:blog-1697471610686007730.post-69299254025180754512010-11-05T21:53:01.516-04:002010-11-05T21:53:01.516-04:00George Washington was born in 1.732... and there a...George Washington was born in 1.732... and there are FOUR SYLLABLES IN "GEORGE WASHINGTON!!!"Rileyhttps://www.blogger.com/profile/14218029087264300691noreply@blogger.comtag:blogger.com,1999:blog-1697471610686007730.post-4570398174520859682010-11-05T16:14:50.010-04:002010-11-05T16:14:50.010-04:00Pro tip: Math teacher bloggers obsessively check ...Pro tip: Math teacher bloggers obsessively check their work on math posts before posting.<br /><br />@Kate: How many times larger would a dollar bill t-shirt need to be to fit on a person?Mr. Sweeneyhttps://www.blogger.com/profile/09576574228194571537noreply@blogger.comtag:blogger.com,1999:blog-1697471610686007730.post-38429337468860242732010-11-05T15:25:30.038-04:002010-11-05T15:25:30.038-04:00Washington was born in 1732. 1.732 is an extremel...Washington was born in 1732. 1.732 is an extremely close approximation of √ 3.Johnhttps://www.blogger.com/profile/15094759008523343946noreply@blogger.comtag:blogger.com,1999:blog-1697471610686007730.post-23903951331031364232010-11-05T15:25:07.211-04:002010-11-05T15:25:07.211-04:00@Sean I showed the kids the dollar bill shirt you ...@Sean I showed the kids the dollar bill shirt you made, too!Kate Nowakhttps://www.blogger.com/profile/14229054922453438248noreply@blogger.comtag:blogger.com,1999:blog-1697471610686007730.post-2785336677578936922010-11-05T15:23:34.091-04:002010-11-05T15:23:34.091-04:00@thomasdav I'm not sure what to say except, no...@thomasdav I'm not sure what to say except, no they're not. Think about how you can make a 30-60-90 by dropping an altitude in an equilateral triangle. Or, think about how the hypotenuse has to be the longest side.<br /><br />@Justyn squaring two numbers doesn't keep them in the same ratio. Since you are multiplying them each by different quantities.Kate Nowakhttps://www.blogger.com/profile/14229054922453438248noreply@blogger.comtag:blogger.com,1999:blog-1697471610686007730.post-84797576880920587732010-11-05T15:21:07.719-04:002010-11-05T15:21:07.719-04:00Well, except that squaring both sides of a ratio d...Well, except that squaring both sides of a ratio doesn't leave you with the same ratio...16/3 doesn't equal 4/root(3).<br /><br />Other than that, Justyn, you'd be right.Infinite EMFhttps://www.blogger.com/profile/07528418467944228869noreply@blogger.com