# Simplifying Trig Expressions

This is a concept that I have always struggled to teach.   I believe in starting a concept by doing very informal math and slowly build up to the final product.   But with something like simplifying trig expressions I have never found a way to do that.   This year I tried something different and it turned out to be a lot better than anything I have tried in the past.

I started without giving any direction on what our final answer should be.   My initial goal was to get students comfortable with changing the expression using different identities and see what identities were on their reference sheet.     I gave each student a template with rows and 2 columns and a trig identity refence sheet that I numbered.

One column was for showing their work and the other column was for writing the number of the identity they used.  I also had students highlight the term they were changing.   I demonstrated the first problem.  They did the next example on their own.  I then had students do one step, then change papers with someone sitting around them, do one more step and so one.   For the first 30 minutes we just changed expressions using the identities.   The students had no idea what the final answer should be and thus, there was no way to get a wrong answer.  I felt like this helped with student engagement.  There was some moaning.  For example, “What is the point of this?”, “How is this going to be a test question?”, but nothing too drastic.  Once I felt like they had a good feeling for how to change things we switched to, The Trigonometric Identity Game.

The rules of the game were as follows:

I put students in teams of 4.  The first 2 minutes there was no talking with their team members.  After that, I allowed them to discuss with their team and come to consensus on their best answer.  I would walk around and check students’ answers and then award points.   Then I put a new problem on the board and repeated the process.

The minor switch of making simplifying a game where all answer are acceptable, just some are slightly better than others, made a huge difference.   Students didn’t get discouraged.  I never got the question, “How do I know I have the most simplified answer?”  Instead, students were actively trying to find a simpler expression.   Adding the -1 bonus point for solving the problem in the least number of steps had students reworking problems in different ways trying to find a more efficient method.  My job in between rounds became sharing the different methods the teams used,  as well as show other possible solutions.   This felt 100 times better than anything I did in the past which mostly was me doing a ton of example problems and kids doing a ton of practice problems.  I am curious to see if this can be used with other simplifying concepts.

# I hate homework

Here are the questions I want my homework policy to solve:

1. How do I grade homework so that it promotes students to strive for the correct answer, without penalizes them for incorrect answers?  Students attempting the homework is priority number 1, but I want students to strive for the right answer as well.

2. How many questions do I assign that give students enough practice, but does not overwhelm them?

3.  Do I give problems on just what we worked on in class or from previous concepts or a little bit of both?

4. How do I check and go over the homework so all students are engaged and it is worthwhile use of class time?

5. How do I decrease the amount of copying?

And here is my current idea for next semester:

Don’t make homework worth any part of their grade.   This should decrease copying significantly.  Now the question is, how do I ensure students do their homework?  Well I think this is where using SBG helps.   After we have learned a specific concept or part of a concept, I would give out homework with the caveat that the problems on this homework are similar to problems you might see on your Concept Quiz.   If you feel like you need more practice on this concept,  do this homework tonight.  If you don’t need more  practice, then don’t do the homework.  If you think later in the week you do need more practice, the homework can be found over there on the wall.   Oh, you bombed concept #12?  Look, there is homework for concept #12 over there on the wall, why don’t you grab it and try some problems (This is very similar to MathyMcMatherson’s Wall of Remediation).  The number of problems on the homework is totally up to the students.   There will be more problems on the worksheet then students should need.  I might circle different types of problems on the homework and tell students, these are the problems worth trying.  These would be like the bare minimum set of homework.  If they feel they need more practice, there are extra ones to try.  As an added motivational piece, every time students finish  two homework assignments they are given a reassessment ticket.    My next thought was how do I go over the homework?  Do I even go over the homework?   I think I should because I do want students to do the homework nightly.   Do I walk around and check everyone’s homework to get an idea of how many students are doing it? Does me checking their homework really motivate students to do the optional homework the same night it was given? Maybe.   My original thought was to have the answer key on the back of the homework.   Maybe they don’t get the answer key until the next day?  Now, do I go over the homework?  Well, I know I am not supposed to according to a lot of people.  The beginning of class is when students are the most focussed and that time should be spent on something more important.  So, maybe we do some type of opener to start.   I walk around room and check homework.  I have students circle or point out the problems they want me to go over. Then, just like this guy said, instead of working out the problem that most students asked me for, create a similar problem for the entire class to try instead.  We go over that problem together.  I am thinking there could be some TPS going on during this time. Maybe this is way to elaborate of a  plan for homework.

I still don’t know if I love this plan.   A large part of this homework plan is relying on high school students to be self motivated and do their homework even though it is not worth a grade.  Can I really expect teenagers to do this?  But is making it worth 10% really going to change that?  Or will that just increase the number of students copy someonelse’s homework. Will giving students a reassessment ticket really increase motivation?  I think in teaching you have to come to the realization that there isn’t a perfect solution; there are just better solution. Anyways, here is another better solution to homework from David R. Johnson’s book Every Minute Counts. The talk about homework starts on page 18.

# Infinite Ice Cream

I just created a 3-ACT task for infinite geometric series.  Here is ACT 1:

The question I want students to answer is, “How much ice cream will I need/eat if I continue this process for the rest of time?”  I didn’t want to make it obvious that the answer would be finite.   For example, if I did something like I get half a carton of ice cream, then you get half of what is left and then that person gets half of what is left of that and so on and so on, students are likely to see right away that the answer is not infinity; with this video I don’t think it is that obvious.

ACT 2 I just have two pictures.  One is the weight of one ice cream scoop in grams.  The other picture shows the serving size (which works out to be about 1 scoop) and the total number of servings in one container.  I could see students not even using this information and just sticking to “scoops of ice cream”.

ACT 3 is the best I could do to show infinite time elapsing.

I like this task because it starts informal.  We aren’t talking about any formulas or weird notation; that stuff can make its way in at the end of the lesson. It has a low floor and decent height of a ceiling.   As a sequel for students who finish early, you could ask, “What if I started with 9 scoops, then 3, then 1,ect?”  or “What if I started with a 0.5 scoops, then 1, then 2, 4, ect.?”    All of the files are on 101Qs.

# MTFSO–Graphing Polynomials

I recently tried Kate’s lesson on the discriminant.   Like most of Kate’s stuff, it worked great, and so I tried to do something similar with graphing polynomials.   Instead of just telling students how to find the end behavior and what the multiplicity tells us about the graph. Instead, I tried to have students figure something out own their own.   I did this lesson over 2 days.  The first day we looked at the end behavior.  I wanted to develop an informal way to talk about the ends of the graph.   So I showed my students this slide:

After gaining so street cred, I had students try a few examples on their own.

Next, I had students pair up and find 2 different polynomials, of different degrees, using Desmos that satisfied the different Drake end behaviors.

After about 15 minutes, we compiled our different polynomials on the board and I threw the “What do you notice/wonder?” question at them. Both classes had great discussions.   One class got all the important information.  The other classes got a lot of the important information. Now, I don’t know if this will make students remember how to find the end behavior of a polynomial better than if I just told them.  But just the skill of figuring something out on your is worth spending time prating.   And worse come to worse, if students don’t figure anything out, I can still tell them at the end of the lesson and all I would have lost is 15-20 minutes.

We then filled this slide out .

Then did some whiteboard practice.

Day 2 was on to multiplicity.   Here was the opener I used.

In pairs, students used Desmos to complete the following handout:

We then TPS over this slide:

Again, some classes figured out more than others.  Some students figure out more than others.  But at least they figured something out first, which is way better then me just telling them.

Here are the handouts and Keynote/PowerPoint.

# Estimations and Wagers

So I create a game and it is a total rip off of Wits and Wagers and Estiomation180.  The basic idea is the students are given some sort of situation to estimate.

They each right their estimation on their card.

Students place the cards in order from least to greatest.

They wager on which answer they think is correct.

The person closest to the right answer without going over wins. One of the students is designated the banker and will pay out the winnings based on the odds.   There is really endless possibilities for this game.  I created ones where students estimate logs, angle measures, equations, fractions, square roots, ect.  Here is the link to all of my games so far. I have a classroom set of iPads, so I just have students open the file in iBooks and then flip through each slide.   You could easily project the slides on the board and have everyone go at the same pace or just print them out.  I created the game board using a manila folder and just laminating all of the pieces so students can use dry erase markers. I think the game works best with 5-6 per group, this way you get a wide range of answers.  It might be useful to limit the time a student has to answer so their responses are actual estimations and not a precise calculation.  Here are the files to all the game pieces:Game Board, Game Pieces 1, Game Pieces 2, Game Tokens

# Function Wall

I sometimes think about what things I can put on my walls that are not your traditional math posters.

Word walls seem to be a strategy that work for some teachers.   I tried something similar this year with a Function Wall.   I teach Algebra 2 and what I did this year was every time we learned a new function, I hung up that function’s poster.

I thought this wall worked well this year.   First off, it reminded students all of the different types of functions we learned.  Also, just them seeing the different functions helps them remember the shapes of the graphs.   The thing I liked the most was if we were trying to model a situation with a function, I could point to the Function Wall and ask student which one of those functions do you think would best fit this situation.   Anyways, here are all of the files for the posters.  I had all of them printed for \$3/poster from this site.

# Self-Evaluation Checklist for Questioning

Some of the favorite books I have ever read about math education are the Every Minute Counts books by David R. Johnson.  These books give great ideas about arranging your desks, checking homework, direct instruction, the classroom routine, notebooks, etc.    One of the topics that shows up in all 3 books is questioning.  In Motivation Counts, David suggests using a Self-Evaluation Checklist on questioning.

I thought this was a good way to hold myself accountable for different questioning techniques throughout the year.    To save time, I threw all these questions in a Google Form and tried to answer the question every week or so.   I didn’t start this until the end of the school year so I don’t have a lot of responses.  Fee free to checkout the Form and the Responses*.

*On Sheet 2 of the response Google sheet, I keep track of the percent of time a mark ‘Yes’ for each question.