We're currently wrestling with the unit circle in trig class, so it's time to post pictures of all of our trig interactive notebook pages for unit 2. Unit 1 was a review of the algebra and geometry concepts necessary for trig. Unit 2 covered the three basic trig ratios and each of the pre-requisite concepts for the unit circle. You can find my pages for Unit 1 here.
Unit 2 started with our unit divider.
Here's a close-up of our SBG skills for the unit:
To start off the unit, each student had to cut a right triangle out of colored paper and glue it into their notebooks. We labeled one of the non-90 degree angles as theta. Then, we discussed how to identify the adjacent and opposite sides as well as the hypotenuse.
This is my students' first experience with these terms because the geometry teacher they had did not teach any right triangle trig. It's a long story, but this trig class is the first time my students have ever seen any trig.
My greek alphabet posters are perfect for showing my students where theta comes from.
I tried my hand at doing a discovery activity to help them discover sin/cos/tan by themselves. This failed MISERABLY. In the end, I just defined the three trig functions for them in a foldable. I had my students fold their colored paper in half and divide it into three doors.
They labeled the outside of each door with the abbreviation for a trig function.
On the inside of the foldable, they wrote the equations for the trig functions and drew triangles showign the given sides for each function.
Students used the next page in their notebooks to do problems from a pizzazz worksheet.
I have given my students word problems whenever possible on their quizzes this year. So many of the problems involve angle of elevation and angle of depression, so I printed an illustration off the internet of this for students to glue in their notebooks.
Next, I showed students the Gettin' Triggy Wit It video on YouTube. You can watch this video here. Because some of the lyrics are a bit hard to understand, I gave them a copy of the lyrics to glue in their notebooks.
Next, we used sin, cos, and tan to find missing angles of triangles.
My students were a bit miffed that they had to just write these equations. They thought I should have given them a foldable for these!
I did make them a practice book, though. These small practice books are one of my go-to ways to having them efficiently place practice problems in their notebooks.
Each book holds six practice problems. Here they are:
After practicing finding missing sides and angles of right triangles, I decided to give the class some hands-on practice by having them build their own clinometers.
I printed a clinometer template off of VirtualMaths.org for them to use.
|You can find this template online here:|
I mistakenly assumed that they would be able to put together their clinometers, take their measurements, and finding the missing heights all in one fifty minute class period. I was WRONG. Somehow, it took my trig students almost 40 minutes of the 50 minute period to build a single clinometer. With a template. I'm still not sure why this took that long.
Here's what the clinometers looked like when they were completed:
I created this template for students to organize their work in their interactive notebooks. They were required to use trig to measure the height of three different objects around the school grounds.
After covering basic trig ratios, it was time to delve into the exciting world of radians! We started out by discovering "What is a radian?" I blogged about this activity in detail last time I taught trig, so I'm going to send you to that post for the step-by-step details. I did make a tiny change this year. Instead of tracing jar lids to get a circle shape to cut out, I borrowed my husband's new safety compasses. I love that my husband and I get to teach in the same school and share resources!
If you didn't click the link above to read how this activity works, the pipe cleaner is cut to the length of the circle's radius. Then, the pipe cleaner is placed around the circumference of the circle to make wedges in the circle.
If everything is done correctly, you *should* end up with 6 full wedges and a partial wedge. This represents the fact that there are 2pi radians in a circle, or 360 degrees.
We set up a proportion to determine approximately how many degrees are in one radian.
After figuring out what in the world a radian was, we started converting between radians and degrees. At the beginning of the year, we spent a couple of days on unit conversions. You can read about that here. I was hoping that would pay off big time when we got to converting between radians and degrees, and it did!
We ended up playing radians and degrees war to practice our unit conversion even further. I blogged about that activity here.
To discuss sketching angles in standard position, we made a spinner. Since the initial side is always on the positive x-axis, we glued the initial side arrow to the positive x-axis. The terminal side of the angle was attached by a brad so it could easily spin.
My favorite thing to have students do after putting together their angle spinners is to ask them to make a 90 degree angle. Some students will place their terminal side on the positive y-axis. Other students will place their termal side on the negative y-axis. Which is right?!? This leads to a discussion of positive/negative angles. We draw arrows on our papers as a reminder.
We wrote definitions for standard position and quadrantal angles.
We finished this off by completing two practice books. We did the practice book for sketching angles in standard form with DEGREES first. I did this because students are most comfortable working with degrees.
Students were able to complete these relatively easily. There was a bit of confusion with some students on positive/negative angles and remembering which goes clockwise and which goes counter-clockwise.
In retrospect, I wish I would have given my students some angles that were larger than 360 degrees or smaller than -360 degrees. This would have led in to coterminal angles more perfectly!
When students finished this practice book, they assumed we were done with this skill and begged for their quiz. Instead, I passed out the next practice book: Sketching Angles in Standard Position in RADIANS.
Their first instinct, of course, was to use what they had learned about converting between degrees and radians to convert all of the radians to degrees. I told them to not do that YET. There was an easier way. They didn't fully believe me, but they did put away their aspirations to convert away all of the radians for at least a little bit.
I did tell them that if they didn't like the method I showed them that they could convert the radians to degrees on their quiz. I am happy to announce that not a single student chose to convert the radians to degrees on their quiz. They decided working with radians wasn't that bad after all. Yay :D
Here's how we did it.
We already knew that pi was equivalent to 180 degrees. So, when we're given something like -2pi/3, we can take a -180 degree angle and divide it into three pieces. We want two of those pieces. And, there we have an angle. If you look at my angle, I did not divide my -180 degree angle in to exactly three equal sections. Oops. I told my students that we were "SKETCHING" angles. It was okay if our sketches are not entirely precise. If I wanted them to have precise angles, I would make them get out the protractors.
For pi/6, we divided our positive 180 degree angle into 6 sections. This meant each quadrant was divided into 3 sections. The invisible 1 in front of the pi tells us that we need exactly one of these sections. Bam. Pi/6. No degrees necessary. After sketching each angle, we did quickly convert the radians to degrees to see if our answers made sense. I did this to show students that this method actually was working!
After a few of these, they decided radians actually aren't all that bad. I'm so impressed with how my students have taken to working with radians this year. It's been so much better than the last time I taught trig. I believe the reason this year is going so much smoother is because I'm making my students more responsible for understanding the "why" behind things. Maybe it's because I, myself, understand the "why" behind the things we are doing better.
After sketching our angles in standard position, we began our discussion of coterminal angles.
Due to a lack of time to create a new activity, I decided to use the same coterminal angle card sort that I created the last time I taught trig. This was not my original plan because this card sort went HORRIBLY last time I used it.
Here's what I wrote about this activity on my blog last time I used it:
"I had the bright idea to make a card sort for my students over coterminal angles. They were going to write the definition of coterminal angles. Then, I was going to give them a page with all kinds of angle measures. They would sort them into groups that were coterminal with one another. Then, they would take a blank square and write an additional angle that was coterminal with the other angles in the group.
This. Did. Not. Go. Well.
And, I'm not really sure why. "
So, why did I decide to use the same activity that did not go well? Because I'm a different teacher now than I was two years ago. Two years ago, I had my students write notes for how to find coterminal angles. Then, I gave them the card sort and set them free.
That was not my approach. Instead, I gave my students the card sort. I didn't have them cut apart the pieces YET. Instead, I had them get into groups and get one of our new giant, group sized white boards that I got courtesy of my schools' foundation which awards grants to teachers.
I asked each group to take the 9 degree measures in the card sort and to sketch the angles on their dry erase boards.
Their work was messy and beautiful at the same time.
This student decided he wanted a whiteboard all to himself.
After sketching angles for a while, I asked students if they had noticed any similarities between the angles. They really hadn't because different students were sketching their angles a bit differently. I gave them the definition for coterminal angles to help them out a bit. Then, I asked them to look at their sketches and determine which angles were coterminal.
I did give them a pretty big hint. There are 9 angle measures. They can be sorted into three groups of three coterminal angles.
With this new information, students kept working. Some groups began to get a bit frustrated when they couldn't fit their 9 angles into 3 groups.
I took hold of this frustration and used it to encourage my students to think about what coterminal angles would have in common. I started by sketching a 20 degree angle on the dry erase board. Then, I asked students to come up with two coterminal angles of that 20 degree angle. They came up with -340 degrees and 380 degrees. Upon this discovery, I asked them how they could tell if two angles were coterminal based on this. One student piped up that coterminal angles were always 360 degrees apart. I challenged this. Are they always exactly 360 degrees apart? Then, a student corrected the statement. Coterminal angles are always multiples of 360 degrees apart.
And, that's what is making the difference this year. I am not giving my trig students facts to memorize. I am giving them experiences that allow them to discover things for themselves. And, it is making ALL the difference!
When I directed my students to look at the radian measures and find the coterminal angles, they again wanted to convert all of the radians to degrees first. Again, I encouraged them to think about how we could deal with the radians without turning to degrees.
We already know that coterminal angles are multiples of 360 degrees apart. So, how many radians are they apart. 360 degrees is equivalent to 2pi, so coterminal angles are also multiples of 2pi apart. So, we did some review of adding and subtracting fractions, and we were well on our way. No degrees needed!
Up next: Reference Angles!
The first image I gave students as a reference was from TutorVista.com. The second image I gave my students was from RegentsPrep.org.
On the inside of this booklet foldable, I had students sketch the angles (given in both degrees and radians!) and find the reference angle. I am happy to report that my students did not ask to convert the radians to degrees!!!
In preparation to move toward the unit circle in the next unit, I ended this unit by having students find the sine, cosine, and tangent of a point on the coordinate plane. This put together so many of the concepts we had learned in this unit and the previous unit. It's so exciting to see all of the pieces begin to fall together.
Here are the step-by-step directions I gave my students
I gave them a practice booklet to, well, give them some practice.
My students were quick to point out that the graphs for our last two examples were upside down in the book. Oops...
After students took their quiz over finding the sine/cosine/tangent of a point, I assigned them this task:
Create a foldable that summarizes whether sine, cosine, and tangent are positive or negative in each quadrant. They were quick to ask how they should do this? I simply told them to pick a point in that quadrant and find the sine/cosine/tangent value. I didn't give them any more guidance than that.
When I studied trig in high school, I was taught the "All Students Take Calculus" mnemonic device for remembering which trig functions were positive in which quadrant. I am trying my hardest to not use a mnenomic device for this at all this year. I want my students to know why certain trig functions have certain signs. In the past, I taught it to my students as a thing to memorize. It's not something that has to be memorized. It's something that can be understood.
To my trig students of two years ago, I am so, so sorry for taking away so many chances for you to do some thinking on your own.
And, that wraps up Unit 2 of Trigonometry. You can find the files for these interactive notebook pages here. (I have a few files on my school computer that I will upload on Monday!)