Meanwhile, today on her Mathematics Calendar 2018, Theoni Pappas writes:
(drawing)
Unfortunately, today's question is completely driven by the drawing of a triangle with various interior and exterior angle measures given. None of the points are labeled. For the purpose of describing the problem on the blog, let me label the points as follows:
Triangle ABC,
D between A and C,
A between B and E,
Angle BDC = 59
Angle DAE = 127
Angle ABD = x
Clearly, our task is to find the value of x. We proceed as follows:
Angles BDC and BDA form a linear pair. Since Angle BDC = 59, Angle BDA = 180 - 59 = 121.
Angles DAE and DAB form a linear pair. Since Angle DAE = 127, Angle DAB = 180 - 127 = 53.
Angles ABD, BDA, DAB are the angles of a triangle. So by Triangle Sum:
x + 121 + 53 = 180
x + 174 = 180
x = 6
It's also possible to save a step by using the Triangle Exterior Angle Equality (TEAE). After we find Angle BDA to be 121, we note that Angle DAE is an exterior angle of Triangle ABD. Hence it is the sum of the remote interior angles ABD and BDA:
x + 121 = 127
x = 6
Of course, it's also possible to find Angle DAB first and then use Angle BDC as an exterior angle:
x + 53 = 59
x = 6
Our students, unfortunately, can't do this because they haven't seen TEAE in Lesson 13-7 yet. At any rate, we find that x = 6 -- and of course, today's date is the sixth.
This is what I wrote last year about today's lesson (and yesterday's activity):
https://mylifeasmissblog.wordpress.com/2016/01/17/5/
Here's what the author Olivia writes about her activity:
At the beginning of the week, I assigned a dilation project in geometry. Students were to pick a picture from the internet, draw a grid over top of it, then redraw the picture following the grid on a larger piece of paper. My district does not have an art program, so many of the students are definitely not comfortable when it comes to art. I heard a lot of negative comments that day from students saying they sucked at drawing, it was going to turn out horrible, and many pleas of students asking me to “please not hang them up!” They were especially adamant that they WOULD NOT be putting their names on their pictures. I told them it would be okay and they would turn out great. I said that if I could do it, anyone could do it! Well I gave them 2 full days of class time to work on their posters. I hung up a couple of the posters after the first day, because two of my students finished theirs by working on it in study hall. The next day, my other classes were all asking who drew what. I said, sorry guys this class wanted to remain anonymous. Well, my geometry class came in later in the day. I told them all that people kept asking about who drew what but I did not rat anyone out about who drew what. Then, a very exciting thing happened! Students started saying, “Mine looks so good, I’m definitely putting my name on mine” or “I want everyone to know who drew mine.” They told me that it wasn’t as terrible as they thought it would be and it was actually fun. I loved seeing students so excited and proud of their own work. It always makes my day brighter when students realize just what they can accomplish. Everyone ended up putting their names on their finished products, and I was left as one happy math teacher.
Well, there's nothing stopping us from assigning this activity today. I obtained the pictures simply by performing a Google image search for "cartoon character" -- those just happened to be the ones that came up.
There's one thing about activity -- it works better on a coordinate plane. But note that Lesson 12-2 of the U of Chicago text uses Slope and Distance Formulas just to prove that the mapping from (x, y) to (kx, ky) is a dilation with scale factor k.
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