Puzzle Sheet

I'm proud of this puzzle sheet I just made, and I like the self correcting aspect of these things. I think that if I had 20 more hours each day, I could spend my time making these types of things up. Notice that it basically has the exact same pictures from the previous sheet, but now the students have to place the vertices in order and determine what type of triangle congruence it is.

After we had our day of constructing triangles in geometry to see if they were all congruent or not, and after we discussed how to read information, then I had the kids learn about the various postulates and theorems (SAS, SSS, ...). THEN they're ready for this puzzle just to see if they can piece things together. If they complete the congruence statement with the triangle vertices in the right order, and if they correctly place them on the blanks at the bottom, then an interesting fact appears. (UPDATE: yeesh! There was a typo on the picture in #10, but it's fixed now in the download.)

Proving Triangles Congruent

We just started talking about congruent triangles in geometry. After a day of notation and vocabulary and such, I start up the class with asking them how many pieces of information a triangle has (6). Then I ask a series of questions: what if I asked everyone in here to draw a triangle. Is there a 100% probability that everyone would draw the same triangle? What if I told you one side had to be 5 cm? What if I told you that one side had to be 6 cm and one angle 40 degrees? And so on. Then they conjecture what would be the minimum amount I'd need to tell them. I get a range of answers, which is great.

Then we went through a series of carefully thought out constructions on my part of the 6 possibilities (AAA, SSS, ASS, SAS, ASA, AAS in order). I like that it gives them practice with protractors and compasses. There's also time for them to process (through the drawing), why some information doesn't give a unique triangle. We discuss how we know I gave them all the possibilities.

Then we barely had time (in a block class) for this side of the sheet:



In the past, students have had a hard time "reading the triangle information" in the right order. I think I found a fix. I have them lightly circle the sides or angles without information. Then I tell them that when they read the given information in order, they can't pass by 2 or more pieces of non-information. This seemed to help.

I assigned this side for homework:

Reaction Reality Check

Today was our last day of school before the Thanksgiving break. Woo Hoo for 2 day work weeks! As I'm scurrying along in my flustered crabbiness from my November funk and getting ready for the day, a geometry student stops by to inform me that she won't be in class today and to drop off her homework because her family is leaving early for their long drive to visit family for the holidays.

What's going through my mind: Great! She's going to miss our important activity of constructing various triangles to see if AAA, SSS, ASA, AAS, SAS, ASS can give unique triangles or not! She'll not be able to make it up! She won't know how to do the homework!

What comes out of my mouth: Aye! You're going to miss a vital activity that you can't do by yourself. Well, you'll have to look through the book and figure things out. Look through the index for "triangle congruence". Good luck. Here's the homework. .... and honestly, I don't remember what else I said.

She leaves. Hopefully, I wished her a happy holiday, but I don't remember in my rush to have myself a pity party.

Time passes, and then I realize what a jerk I was. I send her the following e-mail:

----------------------------------------
Hi ***,

I feel like this is how I sounded on Tuesday morning when you came by to mention you'd not be in class:

"What?! How DARE you miss math? Family?!?! What's more important - Math or Family? Think carefully! Your WHOLE school career depends on what you do next!"

What I really meant to say was:

"How fun you get to spend quality time with your family on this family vacation. Have fun. Thank you for coming by to let me know. How responsible of you. I know you will have no trouble making up the work because you're THAT good."

Oh and by the way:

We did an activity in class that you'll have to hear about, but you can get the gist of the importance of it by reading sections 4-4 and 4-5 in your geometry book.

See you after break,

Ms. ***
-------------------------------

Seriously, what goes through my mind sometimes? I teach math robots who live and breathe math?

Spherical Geometry

This past week I spent part of a class period on a spherical geometry activity, and it went well. Beforehand, I went to the store and bought some 4"-5" diameter soft/squishy balls (tennis balls will do, but not really large 12" diameter ones, I think, because of the rubber bands). I bought enough for one per table (4 kids). I also had 3 rubber bands per table.

As always we had a quick discussion about these things being "math tools" and not annoying distractions that you throw or bounce or snap people with, etc. MATH TOOLS. Then I had a brief introduction on how we've been studying Euclidean geometry where everything happens on a plane and everything is defined on a plane, and what if we defined things on a sphere where our objects had to all live on the surface of the sphere? With that I handed out this worksheet:



and told them to do the bold questions, 1,5,7,9,11,14. We discussed the answers, then I walked them through 2,3,4. Then I set them on their own to explore the remaining questions, one at a time, and making sure that everyone had a chance with the ball. And we discussed the answers after each problem.

There are a ton of resources on the web that have answers, and I love the surprising results. They're brain hurters. For their homework, they had to find at least 2 more cool facts about spherical geometry, and then they had to find 1 new fact about a geometry on another type of surface.

This all took about 45-50 minutes with my advanced class. They came back with some interesting facts, and it was a nice addition to our normal routine of "every day geometry".

Synopsis of the Day

This should sum up how my school day went ... that started with a faculty meeting wherein we:

* were informed about how our school district is thinking about cutting various things to meet their budget shortfall (which may include teachers giving up a prep period)
* were chastised for not willingly giving up class time for students for extra curricular things (because we were greedy with instructional time)
* were told of upcoming events that would further alter the schedule to allow less class time
* had 1 hour less of prep time in the morning because of said meeting ....

SO....... one of the highlights of my day (aside from numerous other student interactions), was a student who informed me at the start of geometry class that they had a new joke to share (which as a bonus sums up what's expected of teachers):

Q: How do you fit an elephant in a Safeway Bag?
A: Take away the "f" in "safe" and the "f" in way.

Google Genius

Am I the last person to know this? Today we were calculating all sorts of values related to polygons and regular polygons and interior and exterior angles of regular polygons and such. As I'm sending the kiddies off with their homework, I thought to mention that if they didn't have a calculator at home, they could just "google" free calculator, and they could use one of the options that showed up.

Then a students said, "well, you could just type in your math equation right into the google search bar". What? So I tried it:

1080/3. Yup.
sin(pi/4). Yup.
log(5.234). Yup.
arctan(1). Yup.
e^(-1). Yup. Yup.

Holy Moly! Genius-ness.

Blogging Topics....

Maybe every "anonymous" blogger reaches this point. I've been blogging for about 6 years (I had to just go check the first month shown on the front page), and to this date, I've never directly mentioned to anyone I know, husband, friends, teachers, that I blog. At first it was because I didn't know where I was going with it. Then later it was because .... who knows. Maybe it was a "once it's out, I can't take it back" type of thing, or a "if no one I know knows I blog, then I don't have to self edit" type of thing.

Last year, I got a rude wake up call. I had written about a student and her lack of class attendance and such. Then another student was joking with me and mentioned that this 1st student had seen that I'd blogged about her. What? First of all, how do you go searching for math teacher blogs as a student (why?), and 2nd of all, why this blog? So then that started me being more circumspect. Again, maybe everyone has to do this.

Then a month or so ago, a teacher in our district was let go because of her blog. She was revealing personal student information. Yeesh, more self editing. .... All this to say, there are a ton of funny things that happen at school, but I can't process in writing. So basically, now I feel limited to sharing teaching tips, which is not bad, but it sure would be nice to vent/laugh/process/write about various things.

Anyway, bla bla bla. Just "venting" (which apparently I can't do :)).

Here's one funny (safe) thing. For some reason, unbeknown to me, a student keeps associating me with llamas: "do you have a llama herd?", "llama lover", "llamas", "how are the llamas?". I just roll with it. It reached a pinnacle today, and I went online and did a "llama lovers" search.

Go on, I dare you to type that in. Apparently, there are full on societies of llama lovers. There are songs, pictures, blogs.... Then I got crazy and tried, "yak lovers". Sure enough. There's a "yak appreciation society", a "yakkin about yaks" site, .... platypus lovers? But of course! Oh my goodness.

Anyway, this student bought me a present today. Guess what it was. Yes, a small stuffed llama. Soft. Crazy. I think I can make it my evil twin and channel my evil teacher thoughts into it: "you! shut up!" What? That wasn't me, it was the llama speaking.

SAT questions

Great! New resource for problems. I bought an SAT math only workbook, and it's chock chock CHOCK full of geometry problems asked in all sorts of cool ways. There are a ton of triangle problems using "angles sum to 180" facts and parallel lines and transversal problems.

Here's an example: picture triangle ABC. Then picture point D on segment BC. So you have your larger triangle separated into 2 smaller triangles. Put variables representing 5 of the 6 triangle angles (a,b,c,x,y). Let the 6th angle be 50 degrees. The question asks find a+b+c+x+y.

Love it.

Angle Pair Nitty Gritty

As I was merrily going along teaching linear pairs and vertical angles and corresponding angles and such, it came to my attention that various students were not "there" yet with internalizing the pictures and definitions, so I came up with this activity:



I like the activities I've seen where sometimes the answer is "none" and sometimes there's more than one answer, and sometimes the answers are repeated in different questions. I've tried to incorporate that style into these problems. I think it worked pretty well.

I'm also giving them a quiz on 6 types of angle pairs for the next grading period. I told them they have to identify each angle pair correctly, and spelled correctly, and no abbreviations, and no doctor handwriting, and in a timed manner. If they miss ANY, they get a 0%, and they have as many tries in the next 6 weeks to get it 100% correct. I likened it to recognizing the letters of the alphabet. It sure would be a shame if they couldn't and/or it took them a long time to process the information. I restated the fact that geometry (math) is like a foreign language, and they have to have fluency with all the words.

In other funny news. A student whose grade is suffering and who came in for tutoring today and was actually grasping things said, "wow, I should really listen more in class because this makes so much sense."

More Memory Tools ...

My 10th grade geometry students keep trickling in to take their logic retests, and many of them still had a hard time remembering the laws of inference to use in logic proofs. They would confuse the following two:

p --> q
p
-------
therefore, q

(Law of Detachment)

p --> q
~q
-------
therefore, ~p

(Law of Contrapositive Inference)

I tried 2 or 3 or 1000 different ways of explaining it, and nothing stuck. I thought I hit gold with the following:

If the Dog is a Terrier, then the CHihuahua likes MENTos.
The Dog is a Terrier. (I thought this would help them connect it to detachment).

and

If the CON is a TRAitor, then I POSsibly like TV.
I hate TV. (for contrapositive inference).

So that helped some kids, but wasn't the raving success I thought it would be. Then just the other day, we were going over problems in tutoring, and I was circling things to make a point, and saw this:



Do you see it? The Detachment scribbles look like "D", and the Contrapositive scribbles look like "C". Well. They thought that was SO cool, and then they started accusing me, "why didn't you just show us this in the FIRST PLACE!". Like I was secretly storing away the "good stuff" and doing things the "hard way" until they finally wore me down to giving up the goods. Is this on ALL their minds: teachers always do things the hard way and hold back on us. Phwueey. I told them I just thought of it, and there was nothing to prevent them from figuring things out on their own and stop bothering me kid. .... They also passed their retests with flying colors.