Sometimes It's an Accident

We are learning about angles in grade 7.  Yesterday, I asked students to draw 10 different angles (at least 3 acute and at least 3 obtuse) and measure them with a protractor.  Historically, I've been really bad at teaching students how to use a protractor, but they made their best pass at it.

Today, I drew random angles around the classroom on our whiteboards and was going to ask for volunteers to walk up with their protractors and measure them at the board.  But before doing so, I went around the classroom asking the class to classify the angles as acute, obtuse or right.  When we came to a right-is angle, the class was divided; some said acute, some obtuse and a few said right.

"Ok, so what do we do?"

Sam picks up his protractor, holds it out in front of his face, closes one eye and peers through the hole at the bottom of the protractor.  I stepped back and watched what he was doing.  He was peeking at the vertex into the hole, while lining up one of the rays with the guides at the bottom of the protractor.  He then says, "Mr. Cox, it's pretty much 90 degrees."

Now, I loved this for reasons.  1) This kid invented a hack for doing a little better than estimating 2) The entire class understood what he was doing and started using his hack and 3) I had never thought of doing this before.

But that isn't even the good part.

I had drawn three straight angles that had a second ray breaking it into two supplementary angles.  We argued a bit about whether there were two or three angles shown.  Everyone eventually agreed that there was an acute, obtuse and straight angle represented.  Then we got to the measures.  Glad to say, the pairs they measured were all supplementary.  Then came the two students who made mistakes (on purpose).  We discussed how some students use the wrong numbers on the protractor, but if they classify the angle first as acute or obtuse, that helps them know which number to use.

That wasn't the good part either.

I wanted to start a conversation on supplementary pairs and I was going to use the drawings of straight angles broken into two supplementary angles that were on the board.  But then I thought about Sam and his protractor hack.  Change of plans.

"Ok, take a look at your protractors and look at the pairs of numbers.  What do you notice?"

We created a list of numbers.

170   10

160   20

150   30

140   40

130   50

120   60

110   70

100   80

 90    90

Then I added one more entry:

34   ?

"Without using your protractor, make your best guess about the number that should be paired with 34."

We had three answers. 

154   146   156

Argument 1:  "I think it's 154 because 150 and 30 are paired together.  Since we added 4 to the 30 to get 34, we need to add 4 to 150 to get 154."

Quick check of the class to see who understood the argument. I was careful to let them know that saying they understood the argument was different than agreeing with it.  They understood. 

Argument 2:  "I disagree with 154 because on one side the numbers are increasing and on the other they're decreasing.  34 is between 30 and 40, so our answer needs to be between 150 and 140.  So, I think it's 146."

Homework:  Who do you agree with and why?  If you think the answer is something different, make an argument.

Can't wait until tomorrow. 

Exterior Angles

"Hey, Mr. Cox.  Is there a theorem that says this?"

I'm not sure.  Why do you think that's true?

"Well if a triangle has 180^o and I know two of them, then the third one has to be whatever's left over from 180^o.  But that third angle and x make a straight line so they have to add up to 180^o too."

If it's true, what would you call it?

"I remember seeing something about 'exterior angle' in the index one time, so that's taken.  Maybe I'll look it up and see what that one means."

*goes and looks up Exterior Angle Theorem*

"Ah, man!  Someone already discovered it."

If Only It Was Always Like This

Instead of working through the problems in the geometry book, we decided it would be fun to try to prove all the theorems as we come to them.  A couple of weeks ago he proved the midsegment theorem.  Now we are on to trying to prove that the centroid is 2/3 of the way down the median.  So we sit around the white board easel and discuss how we might go about this.  At this point, I don't even know how we're gonna prove this thing.  We learned from the midsegment theorem that defining the vertices using (x₁, y₁),  (x₂, y₂), (x₃, y₃) helped out greatly.  So, what the heck, let's try it again. 

We know we can define the midpoints generally as well, so that's what we do.  Then it hits! We can define the vector from the vertex to the midtpoint of the opposite side, use a scalar of 2/3 to determine the vector from the vertex to what's supposed to be the centroid and then translate the vertex to said point. 

Ta-freakin'-da

 Turns out that the centroid is 1/3(x₁+ x₂+x₃, y₁+ y₂+y₃). 

I didn't know that.  If this is what it's like to become a co-learner with your students, then sign me the heck up.

Triangle Centers Lab

The other day I made up a triangle centers lab for my 8th graders. 

Here is how it went:

Day 0: Homework for tonight is to make a triangle larger than your hand out of some material heavier than paper.  Cardstock or cardboard are ideal.

Day 1: Open GeoGebra and get to work.  Kids got after it.  Some slowed themselves down by not reading directions very well.  The nice thing about GeoGebra is that it's easy to erase. 

Day 2: Most finished the lab and went onto extension activity.  Those who didn't finish had a difficult time managing time.  They could do the work, but staying focused was the issue. 

Extension:  Now that you know how to find the circumcenter and incenter, construct an inscribed and circumscribed circle using only compass and straight edge.  These students haven't done anything with a compass, so I offered a 6th point (assignment was worth 5) for those who could figure out how to do the constructions on their own.  If they chose to look up the "how to" of constructions, they then would have to prove that the method of angle bisecting works. 

David came up with his own extension.  He asked, "why does the centroid allow you to balance the triangle?" 

"Nice question.  Now go away and come back with an answer."  He's figured out that the three medians divide the triangles into six smaller triangles with equal area and that would account for equal weight distribution from the centroid.  He can see it in GeoGebra, but is working on a formal proof.  Brandon tried to backdoor me with a proof by contradiction, "well the six triangles have to have equivalent areas because if they weren't, the large triangle wouldn't balance."

Don't try to beat me at my own game, son. 

Chris' reflection:  "Hey Mr. Cox, you just kinda gave us a test without giving us instructions."

"Yeeeeaah kinda, huh?

For Next Time:  Stamp each page after students have demonstrated the correct constructions.  Then allow them to go to the next page.  Take a little more time discussing the difference between "drawing" and "constructing."

What's the point?

One of my favorite activities is to have students draw a point on a paper and see how many distinct lines they can draw through the point.  I usually set it up as a competition to see who can get the most lines inside of 15 seconds or so.  On your mark, get set, GO! Pencils start flying.

Then to bring the lesson home, I say, "Alright, flip the paper over and put two points on the page.  Now we're gonna see who can get the most lines through both points." 

Ready, set, GO!

They get the first line fast.  Then they panic as they move the ruler and pencil searching for that elusive second line.  Most of 'em end up looking something like this:

I tease them a bit and we all get a good chuck out of it. I know, I know.  It's not nice to take advantage of these trusting impressionable children.  But I don't care who you are, that thar's funny!

 

And, they never forget it.