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.

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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.

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The rules of the game were as follows:

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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.

Identity Reference Sheet, Simplifying Trig Expression Template


I hate homework

I hate everything about homework (I guess except for the idea of it).  I hate trying to figure out what is the best way to grade homework.  Do you grade for participation or correctness?   Participation obviously! Right? Homework is supposed to be a learning experience and you want students to be able to practice without the worry of doing something wrong.   But if it is just for participation, then why would a student work hard to try and figure out the right answer?  Maybe I should just give quality feedback instead.  That will motivate students to put in more effort, right?  I am sure it would, but who has time for that?!  How many problems do I  assign?   If I assign too many, kids won’t do them.  If I assign too little, is it even worth it for them to do their homework?  How do I even know the students are doing their own homework?  I don’t.  For all I know, last night’s homework was trending.  If I just scrap homework all together, then students, parents, and probably administrators will be on me for not giving enough practice and they are probably right.  I hate figuring out what percentage I should make homework in my grade book. 10% is the answer, right? Except for the fact that students think that 10% isn’t worth their time.  Which is kind of right if you are only trying to pull a D or C.  So make the homework worth more?  Then we just inflated grades and they don’t accurately represent what students know.  Then you have to answer when and how do you go over homework.  Not the first 10 minutes. Or do you?   Is it worth class time to go over the homework? Probably not.  So do you not go over homework?  Well, you probably should.  But just the problems that most students have issues with.  How do you figure out which problems most students have issues with?   Ask them.  Yeah but how?  Show of hands?  Tally chart?   But how much time is this all going to take?  Should I just post the answers?  Maybe the answers and the worked out solutions?   Yeah but then those worked out solution will be tweeted out and everyone in 6th hour is going to have perfect papers.  Also, if students are just checking for right answer, that’s not promoting a growth mindset. These are the thoughts that are rolling around in my head when I think about my homework policy.

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”.

1 Scoop .001Contain 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:

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

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Next, I had students pair up and find 2 different polynomials, of different degrees, using Desmos that satisfied the different Drake end behaviors.   Polys 2.0.008

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 .

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Then did some whiteboard practice.

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

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

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We then TPS over this slide:

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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.

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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.


Then they check  their answer.

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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.

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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.


Things I have been working on…

Math Recorded

This semester I have beScreen Shot 2013-05-24 at 3.17.41 PMen trying to encourage other teachers to start recording themselves teach.  I brought this idea to my assistant principal (who liked the idea), and he started randomly picking teachers to record.   Teachers were terrified and hated to get picked.  Which I get, it is scary to have your peers watch you teach.  We also ran into the issue of what do we use to record with?  Then, once we got something recorded, we never had time to watch the entire1-hour video. So what I am going to try this semester is start out by just recording myself and then sharing it with other teachers using this site.  Instead of recording or uploading the entire video, I am just going to put up small clips of different parts of the lesson.   That way, if a teacher wants to see how other teachers open their classes or how they facilitate a discussion, they can watch just clips of those videos. (Also, it is much easier to upload a 10 minute video then a 60 minute video.) Teachers can simply filter the Google Spreadsheet for whatever they want to see.

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Later on in the semester, I would like to have teacher start feeling more comfortable with recording themselves and then upload those videos to the website.   I think watching these clips could lead to great discussions at our PD meetings and could lead to teacher being real reflective about their teaching.   To record the videos I bought a small tripod to hold my iPhone.   I then simply edit the clip and upload the video to Vimeo.   I change the privacy settings to require a password to view in order to protect the privacy of our students.

Vertical Non-Permantent Surfaces

After seeing the slides from Peter Liljedahl’s presentation on whiteboards, I was intrgued on how to best hang my large whiteboards that I made out of panel boards.  I came up with drilling holes in the corner of the whiteboard and then using grommets to make sure the holes last.  Here is a picture of my room with all of my whiteboards hung up.



And here is an up close picture of the corner.

IMG_1871Now, I just have to figure out what I am going to do with them!!!

Quadratic Toolbelt

This year when I was teaching students about quadratic functions I wore a tool belt and told students we were going to learn how to use some tools in order to work with quadratics.  So, for example, when I taught students how to complete the square, I would say something like, “this “tool” is used for putting a quadratic function into vertex form.”   Or, “Sometimes we can use the “factoring tool” for solving quadratics equations, but if that tool doesn’t work, we can always get out the power tool, “the quadratic formula” and he will do the job.”   Then, I would belt out a Tim Allen grunt. Using the tool belt analogy helped students realize what all these things (completing square, factoring, quadratic formula, ect.) are best used for. When we were working on more open ended problems, I would hear students say things such as, “Let’s try a different tool.” or, “Isn’t the completing square tool good for finding the vertex?”    I am going to continue using this analogy when we look at other functions and see if it has the same effect.