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All you math lovers out there should know that tomorrow is Pi Day. But should we celebrate pi or tau? Don't know what I'm talking about? Take a look at these two videos.

 

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geometrydaily:

#275 Perpendicular – A new minimal geometric composition each day

This is my new favorite tumblr, via Jim Rugg, who apparently is the only place I get inspiration from lately. I love this bit from the “about” page:

“Why are you doing this?

I love it. I get a serious flow when I draw simple shapes, combine them and experiment until they start to “sing”. I’m a designer with all my heart. It’s an experiment. A journey into a world of possibilities.

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The other day, I received a call that sent my brain scurrying about in curious ways. You see, a lady needed information from Amsco in order to print Braille editions of Geometry and Preparing for the Regents Examination: Geometry. I gave her the information and went on with my day. I mean, it’s neat that our books are accessible to even more students. That should have been the end of it. . .

But then my brain kicked me in head with this question: How do you do constructions if you can’t see them? This was followed by: If you couldn’t see, would it be easier to draw a triangle according to a set of side and angle measures or to construct a triangle congruent to a given triangle?

I had to find out. During lunch, I looked up what measuring tools exist for the visually impaired. (For my purposes, I disregarded technology.) There are rulers with raised numbers and tick marks, and rulers with a sliding piece to keep track of your measurement. There are also Braille protractors with raised angle indicators at 10-degree intervals and a special pointer wand that you can use to lock in your angle while drawing the ray.

My own protractor and ruler happen to be tactile, so I closed my eyes and tried to measure out lengths and angles by feel to form a triangle. It was possible, but boy was it tedious! I’ll admit that having the right tools and the proper training probably make it easier. Still, once was enough for me.

Next, I drew a triangle on a piece of paper and flipped it over so I could feel it with my fingertips. Then I got out my compass and straightedge and tried to construct a sightless copy by the book (Geometry, pages 196 and 197, actually). After I got used to drawing my arcs heavy enough to feel, this was sooooo much easier. Neither compass nor straightedge have finicky numbers.

You know, I admire the visually impaired students who are going to be tackling geometry and wish them well. And if you happen to be a math student or teacher who has used any of the tools mentioned, I would be happy to hear about your experiences.

If any of you blog readers wish to learn more about the many Amsco books available in large print, Braille, or sound recordings, check out the American Printing House for the Blind at htmhttp://www.aph.org/index.htm.

--Jessica

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Our latest series of textbooks for high school math are Integrated Algebra 1, Geometry, and the upcoming Algebra 2 and Trigonometry. I've been getting calls from teachers who are not sure what the next book in the series is. We called it Geometry, not Integrated Geometry or Geometry (Revised).

I should also point out that there are review books that go along with our textbooks: Preparing for the Regents Examination Integrated Algebra 1 and Preparing for the Regents Examination Geometry.

As per overwhelming demand, our Integrated Algebra 1 and Geometry books are now available on CD in PDF form, with the purchase of the textbook.

Lastly, there is now an online tutorial for our Integrated Algebra 1 textbook through WinPossible. Teachers can track the students' progress. The Algebra online tutorial can be found here. A demo can be found here. (Note: that is NOT a demo of the Integrated Algebra 1 course.) A Geometry tutorial is in the works and will also be available on CD.

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As promised, the long awaited screencast is finally here. Yay! We use free and opensource software to make the screencast:

• Blender - 3D graphics software for computer animations, hijacked for use with solid geometry. (Again free!)
  
• CamStudio on Windows or record-my-desktop on Linux to make the screencast.
• I then uploaded the video to TeacherTube, the YouTube equivalent for educational videos.
  

What we'll show in the screencast is that through a line in space, there is only one plane perpendicular to the given line at a point. I decided to use 3D software for this presentation because building the actual model is somewhat tricky (or tedious). Note that the screencast is more of a proof of concept than a ready-to-use teaching aid.

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For a change, I was reading the Sunday Times on paper instead of online, when I stumbled upon an article in the Magazine that sent me to the computer seeking more information. Gomboc! I had to see it. I needed video.

I was reading the 7th Annual Year in Ideas issue of the Sunday Magazine. The article was “Self-Righting Object, The.” Clive Thompson wrote,

The Gomboc is a roundish piece of clear synthetic material with gently peaked, organic curves. It looks like a piece of modern art. But if you tip it over, something unusual happens: it rights itself.

Twelve years in the making, Gomboc is the work of scientists Gabor Domokos and Peter Varkonyi of the Budapest University of Technology and Economics.

The Gomboc, which, naturally, has its own Web site, strikes me as an excellent subject for research, reading, and writing for mathematics students. Here are some Gomboc-related assignments for your students:

1. Have students read the New York Times article (under 400 words). They will enjoy reading that the two scientists spent “a few years doing the math”; and also that they tested 2,000 pebbles on the beach to see if they could right themselves, before concluding that they could not.


2. Have students view video of the Gomboc righting itself and write one or more of the following:
   a. a short expository description of how the Gomboc moves, breaking the process into steps.
   b. a creative monologue of the Gomoc’s thoughts as it struggles to right itself.
   c. a brief poem, such as a haiku, that evokes the qualities of the Gomboc.
   
3. Have students seek information online about the tortoises and beetles mentioned in the article. Ask students to write a short essay comparing their self-righting movement and that of the Gomboc.


4. Have students seek information and video online (or in a toy box) about Weebles, the pre-school toy made popular in the 1970s. As the jingle goes, “Weebles wobble but they don’t fall down.” Ask students to write a short essay contrasting the self-righting of Weebles with the self-righting of the Gomboc.

Your advanced math stars can also explore the mathematical background, definition, and interesting properties of the Gomboc.

Students might ask, "Will this be on the test?" Generally, no. Students are not asked to read and write about mathematical objects on standardized math tests. Reading and writing about mathematical topics will, however, challenge students’ conceptual understanding of mathematics. Students will have to explain steps, develop arguments, and express opinions. Reading and writing activities encourage students to ask their own questions about mathematics, increasing their investment in the learning process. And importantly, these assignments will give students whose verbal skills exceed their mathematical skills a chance to shine in math class…which can be motivational. Who knows, they may even self-right.

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Who is that dork with the gloves and the sailboat? Ouch, ease up on the name calling! That handsome gentleman is actually showing off the geometry manipulative of the previous post.

Recall, that we were going to prove that if a line l is perpendicular to two lines (AP and BP) in a plane p in the plane at a given point P, then the line is perpendicular to every line in the plane at that point.
To begin the proof, choose points R and S so that P is the midpoint of segment RS. Let line PT be any other line in the plane through the point P and let Q be the point where segment AB intersects line PT. We'll use a sequence of congruent triangles in order to prove that line l is perpendicular to PT.

Step 1. Since RP is congruent to PS (under the cardboard), and angles APR and APS (under the cardboard) are both right angles, triangles APR and APS are congruent by SAS. Therefore, the orange strings are congruent (AR and AS, under the cardboard).	Step 2. Similarly, we can show that the green strings (BR and BS, under the cardboard) are congruent.
Step 3. We then have that the triangles with the orange and green sides are congruent (ABR and ABS, under the cardboard) by SSS. In particular, the angles of the triangles at A are congruent. Step 4. By SAS, triangles AQR and AQS are congruent.	Step 5. Finally, since the triangles ARQ and AQS are congruent, the blue strings are congruent (QR and QS). Then, triangles QPR and QPS are congruent by SSS. In particular, the angles at P (QPR and QPS) are congruent.

Since angles QPR and QPS are congruent and form a linear pair, they are both right angles. Therefore, line l is perpendicular to line PT, which is what we wanted to prove!

Yay! My apologies for the bad photos--this activity would have made more sense if it was a video. Hmm, maybe a screencast for next time... Read the rest of this post

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