Lesson #2 - Augsburg College

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Geometry: A Shapely Approach to Math
What part of the protractor would you expect to use to measure an obtuse angle? And an acute one?
(The left side? The right side? These answers will differ depending on which direction the angle is
facing.)
What does it mean for one angle to be ‘bigger’ than another? (Its measurement, in degrees, is larger.)
Does an angle measurement tell us anything about the length of the lines which form the angle?
(No!)
Draw two angles with the same measurement (as best you can!) but make one with lines only an inch
long, and the other with the lines extended to seven or eight inches. Are these two angles different?
Now you’re ready to explore how much the degrees in a triangle add up to. Before doing any measuring,
ask students to comment on how much the three angles of a triangle add up to. (More than 90! Isn’t it 360?
Wait, if they were all obtuse angles than it would be have to be bigger than 3 x 90 because there are three
angles in a triangle. That’s crazy, how can you make a triangle out of three obtuse angles? Can you make
one with three acute angles?) Keep in mind that we want students to learn through discovery, so be sure
never to say the answer is 180 degrees. Maybe suggest that you bet there’s one magic number all triangles
have in common. Draw five or six triangles on the white board and ask the group: Don’t you think if you
add up the angles in each triangle they will all be about the same?
Draw a triangle on the white board and tell students the goal is to find that magic number. Ask for suggestions
on how to find it. Continue with questions or suggestions until students articulate that we could do
a bunch of examples and collect data. They should each make a table on their own paper to record the angle
sum measurements of a number of triangles. Draw a bunch of different triangles, one at a time, on the white
board. Let each student have a turn at using the protractor to measure the angles inside the triangle, write
them down and add them up. Along the way, look for opportunities to guide students toward articulating
facts they observe about triangles and angles and, if and when they make these statements, validate them and
write them down. For example, one of my students exclaimed, “It’s impossible for a triangle to have three
obtuse angles!” while his classmate noticed early on that “an angle that’s 180 degrees looks like a straight
line.”
When there are about ten minutes left stop the data collection. Now your group’s job is to find the magic
number, if there is one. Is the data similar? What do you notice about it? (All the numbers are bigger
than 100! This one doesn’t really fit because it’s 155 and all the others are between 170 and 190!) Discuss
and debate what the magic number might be. If students are frustrated by the fact that their data includes
a bunch of slightly different numbers and only a couple that are the same, you may want to talk about the
human error involved in measuring angles. Convince your students that it would be impossible to measure
each angle with complete accuracy. You can do so by referencing a time during the lesson that two students
looked at the same angle with the same protractor and disagreed about what number to use as the measurement.
When time is up, if students haven’t come to consensus about the magic number, tell them that they
should ask everyone they know between now and the next lesson. They can also do their own research online
or in an older sibling’s math book to confirm or deny their hypothesis. The magic n umber is 180; every
triangle has three angles which add up to 180 degrees. If the students discover this during the lesson and are
confident of it by the end, tell them you bet they’re right and the best way to check is to ask everyone they
know and see if anyone else thinks it’s 180.
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