Trig Hand

This is not new. You can read about it here, here, here, and here, just to name a few. 

As of a few months ago, it was new to me. How did I miss this?! I found it when I was searching for unit circle resources, and I thought it was pretty cool. . . Evaluate trig ratios for special angles, quick-and-dirty. I taught my students to use it. That went well.

Here's the summary, followed by the brain dump:

Use your left hand, palm facing you.



Fold over the finger that corresponds to a reference angle. For 45 degrees, there are 2 fingers below and 2 fingers above the bent finger. For 30 degrees, there is one lower finger and three uppers. And so on. Also, I have freakishly small hands. We'll talk about that later.



Then use these three rules:



Other thoughts, in the order they came out of my head:

The trig hand works for angles in other quadrants if you identify the reference angle and know whether the sign should be positive or negative.

It also works if you count the thumb as 90 degrees (four lower fingers and zero upper fingers) and the pinky as 0 degrees (zero lower fingers and four upper fingers), but doesn't extend to the other quadrantal angles. This go-round, I taught these angles separately and used the hand for all four quadrants.

Wait a minute . . . What if the pinky represents the x-axis and the thumb represents the y-axis? Now doesn't the rule also apply to 180 & 270? You would have to add the negative, but we already did that for the other quadrants. Hmmmm.

For finding reference angles my students observed this pattern:
45, 135, 225, and 315 all have 45 degree reference angles and all end in 5.
60, 120, 240, and 300 all have 60 degree reference angles and are all divisible by 60.
30, 150, 210, and 330 all have 30 degree reference angles and are all divisible by JUST 30.
Similar patterns can be found using radians.

My biggest concern, as we all know, is that memorized tricks don't stick. I spent a lot of time on conceptual understanding before I introduced this but still . . .

For next time, I am thinking this is a better fit for my Calculus class. In Algebra 2, students need to understand the why. In calc, we would expect that the understanding is already there. Meanwhile, it would be really convenient to have a quick-and-dirty way to find these values when they pop up.

Finally, someone in almost every class pointed out that, "Mrs. Gruen, I think you have the world's smallest pinky".

Squirrel!



Two nickels. They might be right.

Unit Circle Philosophy

This year, we decided to expand the number of trig topics we teach in Algebra 2 to include the unit circle, the graphs of sine/cosine, and modeling of periodic motion. These are topics I haven't taught in quite a while, so I am getting to take a fresh look.

Here's how I started out:

1.  Review pre-requisite skills:  Angle measures, trig ratios, special triangles.

2.  Develop the idea of a reference angle.

3.  Given a special angle, draw a triangle in the appropriate quadrant, identify its reference angle, label its sides, and find the values for sine, cosine, and tangent.

4.  Replace old special triangles with new ones where hypotenuse = 1.

5.  Cut out special triangles where the hypotenuse equals one unit. Label their sides and glue them onto a unit circle.

6.  Label the points on the circle. Use the circle to evaluate sine, cosine, and tangent for all the special angles. Look for patterns.

7.  Extend the pattern to the axis angles. Use the circle to evaluate sine, cosine, and tangent for 0, 90, 180, 270.

At this point, I feel like my students have a pretty good conceptual understanding of the unit circle. What now? This is where I am stuck.

I generally don't believe in telling students to just memorize something, but I also cringe when I see a calculus student reach for their calculator when they encounter something like sin pi/2 or tan 3pi/4. 

Don't students need to be able to recall these values later on without a circle or a table in front of them?

What is the best way to tell them to remember these?

I did some searching and I decided on a mnemonic device. 

I am sort of ashamed to be using it.

So far, it is working.

My next post will be the procedure I decided to use, but I am wondering if anyone has any thoughts on memorization and the unit circle? How do you approach this in your classroom?

Sending My Friends to NCTM

First of all, I feel like I only blinked and there went three months. I subscribe to the "blog when you feel like it" philosophy, but I still find myself feeling guilty for such a long absence. 

To break my silence, I thought I would share something fun.

My two math teacher colleagues are going to NCTM this week. I am not. I am very sad about this, but I  am sure that I will get over it this summer, as I asked to be sent to #TMC13 instead. Yay!

I made my friends an activity book, to help pass the time during their very long drive from Small Town, KS to Denver.

They can earn 50 points for solving puzzles like this:

They can earn points for taking pictures of yellow cars, license plates, gas station/restaurant combos, and more.

There are games and drawing assignments:

A few activities are for my own entertainment:

And a few activities are designed to help me live vicariously through their experiences:

And finally, I am not above putting them up to stalker-ish behavior.

Have fun, friends! Learn a lot. I will miss you!

How I Sextupled the Time It Takes To Teach End Behavior

In Algebra 2 . . . I used these cards:

Hi Students! Here's a deck of cards. Each one has a polynomial function and its graph.

The degree of each polynomial is determined by its highest exponent. Talk with your group and agree on the degree of each one.

The leading coefficient is the number in front of that term with the highest exponent. Talk with your group and agree on the leading coefficient for each card.

Now we are going to focus on the ends of the graphs. Look at the left side. Does it point up or down? What about the right side? What you are looking at is called the "end behavior". Group the cards according to end behavior. Show me when you're done.

Next I want you to look for similarities in each group. What do the degrees have in common? What about the leading coefficients?

In 5 minutes or so, all of the groups had identified what the groups had in common according to even/odd degree and positive/negative leading coefficient.

We did some practice once they had this figured out.

Oh, and then I played some music and showed them some polynomials one at a time and they showed end behavior with their arms. And there was dancing.

This used to take me 4 minutes. I would write four rules on the board and they would copy them down.

Today it took 24.

But it was fun. And later I saw somebody end-behavior dancing in the hallway.

This approach is better than just telling them, right?

Most Read in 2012

My blogging slowed down a lot this year. I had half as many posts in 2012. This fall has been especially rough. I added some new responsibilities at school and in general have been all "How can I survive today?" instead of "Let's reflect on teaching practice and write about it" or "I tried this cool new thing I'd love to share".

I guess that is okay. I mean, I never wanted blogging to be this thing that stresses me out. I just thought it would be a way for me to stretch myself as a teacher, and it has. Just taking the time to think about what I am doing that is worth posting forces me to reflect and improve. If someone else can benefit from what I share, then that is a bonus!

I don't want to stay in survival mode, though. Here's hoping I can find the time to be more creative and adventurous in 2013!

Anyhoo . . . Last year I started the tradition of compiling my most-read posts at the end of the year, so here's this year's list (a bit shorter than the last one):

Green Pen is the New High Five:  I still use this on a regular basis in my classroom. A correct answer earns a green star and a green pen and the responsibility of coaching someone else near you.

Visualizing Volumes:  I used party decorations to show volumes of rotation in calculus.

Pieces Final Project:  Using function families and restricted domains to create a picture. Posting this resulted in a lot of great feedback which will improve the project for next time. This totally solidified my love for blogging!

2 PIzza Boxes and a Hot Glue Gun:  I glued together two pizza boxes to create privacy for test taking without moving desks. The truth: I discovered two incidences of cheating while using these. While they  will prevent accidental glancing around, they can't prevent deliberate cheating.

Factoring Before You Know How:  An activity to introduce (or practice) factoring trinomials.

Dry Erase Practice Folders:  Use recycled file folders and dry-erase contact paper (or laminating?) to create re-useable skills practice.

And one more post that wasn't read a lot, but I am just proud of it:

Teaching Students to Coach:  A nice little video that paints a perfect picture of how I want my students to work together.

Here's 2011's Top 10 List.

I also want to take a minute to thank all the very kind people out there in the math blogosphere. I started this blog thinking that I have received so much from all of you that I would try and give something back. The truth is that I could never compete with the generosity and kindness of the math teachers whose blogs I read, who also take the time to comment here and help me improve. I don't want to mention names because I know I will forget someone, but I am really grateful to be a little part of this community. Thanks to all, and happy 2013!

Dry Erase Practice Folders: Made 4 Math

A while ago, my calculus students were working on derivative shortcuts. I was trying to think of a good way for them to practice and self-check independently. Assigning problems out of the textbook to be checked in the back doesn't work because those answers are simplified. I really want students to just practice the rules without getting distracted by algebra.

I ended up using recycled file folders and dry-erase contact paper to make these re-useable dry erase folders. They were a huge hit.

To make the folders, I cut off about 2/3 of the front flap of a file folder. The inside of the back of the folder is now exposed. I covered it with a small sheet of dry-erase contact paper. I found a big roll online for about $20. There is enough in the roll for 60 folders, so it will last a long time.

I cut the remaining portion of the front flap into tabs, one for each problem. I lifted up each tab and wrote the solution underneath. This took a long time. Next time, I will try to use a set of problems that I already have typed up. On the up side, it is a one-time thing. These can be re-used again and again and again.

Now students can work out a problem in the dry-erase area. 

When they are done, they can flip up the tab and check the answer.

Then erase and move on to the next problem.

My students really loved these. Since I had made multiple versions, they asked if they could take home an extra one for practice. 

In the future, I can envision a file crate full of these . . . labeled by skill . . . so that students can just grab  and work on whatever as needed. It hate to think about making all of those, but I it just might be worth it.

A Story

Today is national adoption day. I would like to share a personal story. If you are not into that kind of thing, feel free to move on. I will be back to my mediocre math-ed musings in no time.

This is a story of a math teacher (that's me) and her husband, a principal of a small K-12 school. These two were struggling with infertility. There were four years of wanting to be parents, and no babies appeared.

This is the story of a courageous 16-year-old. She had an unplanned pregnancy, and she didn't know what to do. Others told her to end the pregnancy. She didn't want to. Instead, she carried that baby for nine months. She endured the judgmental stares of strangers as her stomach grew and her discomfort increased. She said no to the usual high school fun while she stayed home, ate healthy foods, and made regular doctor appointments.

She chose the principal of her little K-12 school and his math teacher wife to be adoptive parents. She went through hours (and hours) of labor. And then, she put her baby in that math teacher's arms.

Today, I am just so thankful for the gift that she gave us.
I am thinking about the sacrifices she made for the baby she wouldn't take home.

That baby is now three years old.
She's beautiful. She's smart. She's vivacious. She's hilarious.

We are pretty much head over heels.

And we were given the honor and privilege of guiding her through life.

I am feeling pretty grateful today.