Tag Archives: Worksheets

Scaffolding the Trig Identities

Last year my students struggled so much with proving trig identities.  You know, this kind of thing:

Prove:    \csc{\theta}\cos{\theta}\tan{\theta}=1

Part of it is that my students never did proofs in Geometry, so they have no former experience with the idea of “proofs” (they don’t get it in any other classes either :/  ).  Another part of it is that they flip out when they see a bunch of letters and numbers and have no idea where to begin (mostly a confidence issue).  But another big part of it is that my Precalculus students have forgotten (1) how to manipulate fractions (any operations with them), (2) how to factor, and (3) how to distribute or multiply polynomials (I dislike using the word FOIL, but I find myself repeating it over and over).  And if they remember how to, they have no conception of checking whether their work is accurate or not and more often than not make mistakes which throw off their entire equation.

SO I had the idea of starting with what they know (or should know), e.g. \frac{1}{5} + \frac{2}{5} = \frac{1+2}{5} = \frac{3}{5} and moving slowly into progressively more and more complicated equations, moving from just numbers, to variables, then trig functions.  I then begin another thread where the least common denominator is less than multiplying both denominators together (i.e. relatively prime), and work through “numbers –> variables –> trig functions”.  I think this is what my education grad teacher meant when she kept repeating the word “scaffold”.

Well, here are the results of my efforts from last year, and when looking through my previous lesson plans, I re-discovered it and thought “hey, I actually had a decent idea last year!” so I thought I’d share that here.  I don’t remember this helping a whole lot–mostly because I had the larger issue of students simply not doing the work, but I’ve already had students exclaim “I needed this, because I was always bad at it!” which makes me feel good.  (Even if it’s not helping, they think it’s helping and that’s a step…)

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An Open-ended Function Problem

So in Precalculus today, I decided to start with something (after the warm-up) I’ve wanted to do for a little while.  It was good that I waited until now because there were things that we’ve learned in class that they were able to use.  I am certainly going to do this again, although I need to think about how to scaffold so that they do not take up the entire hour and a half of the block period.

I told them “I’ll give you five minutes to start on this” and an hour and a half later they were still working.  Here’s the problem I gave them:

Stove Heating Element Problem: When you turn on the heating element of an electric stove, the temperature increases rapidly at first, then levels off. Sketch a reasonable graph showing temperature as a function of time. Show the horizontal asymptote. Indicate on the graph the domain and range.

I embedded the problem into a link sheet, as shown below, so they had to come up with an equation, graph, and table, not necessarily in that order (I warned them to try it in an order that made the most sense).  Here’s the LINK sheet:

I handed out the worksheets, divided them into groups of 3, gave each group a whiteboard and a few markers (I didn’t have enough for every student to have one), and offered laptops from the laptop cart to them.  There were enough laptops for every group of 3 to have at least 2 laptops, but very few of the students went for the laptops first.  They had learned how to use Geogebra the previous week using this worksheet on sliding/shifting functions that I created last year.  Being able to slide/shift functions up/down and left/right really came in handy for many of them.  About half of the groups realized that Geogebra would be much, much easier than typing in equation after equation into the “y=” of their graphing calculator while the other half of the groups needed my prompting to grab a laptop.

I was excited by how much they discussed and the kinds of discussions they were having.  One by one the groups figured out various little important things by studying the graph.  Things such as (1) “hey, that looks like a square root graph, but can’t be because square root graphs don’t have asymptotes!” (2) when time is zero, temperature shouldn’t be zero, (3) which units they should use (seconds or minutes; Celsius, Fahrenheit, or Kelvin), (4) how “quickly” the graph should get close to the asymptote, (5) where to put the sliders into the function when using Geogebra, (6) which parent function to use, and (7) how to move a graph up/down and left/right.  Most of these topics were covered in most groups without prompting, which was wonderful!

The students really struggled, and I absolutely refused to help most of them–especially when I was the first one they turned to in order to see if something was “correct” or not.  Another great thing about this exercise was the number of good answers: there were at least 3 different types of functions used, all of which could have modeled the situation correctly.  I told the students “there is no one correct answer–there are several right answers!”  I also warned them that there were even more wrong answers, so they couldn’t just put down whatever they wanted and call it “right”–they had to justify their answer.

One of the downsides to this activity was the amount of time it took.  I didn’t realize that authentic discussion took so long, or that they would be so painstakingly slow at figuring out a good-looking solution to an open-ended problem.  That probably means that they have been “force-fed” too often in their math careers, although I know I did that last year and there will be times I’ll have to do that this year, too.  My hope is that the more they do this, the more they’ll appreciate it and the faster they’ll get.  I perhaps should have done this Geogebra activity on stretching functions before because that would have also come in handy, but I didn’t want them to have forgotten the shifting/sliding function activity.

Another cool tool that I showed them and a few groups were brave enough to use was the Desmos Graphing Calculator.  My students had not used this yet, but I think that making sliders and editing the function is a little easier to do there than Geogebra (just a little easier).  One problem was that we were using the school’s laptops, and they only had IE 8 (no Chrome, no Firefox, no other good browser…) so the latest Desmos did not work well at all for the students.  Geogebra is a little more versatile and I think I’ll keep teaching the students how to use it because of that.

Overall, I think this was a good exercise for the students and I’ll certainly try this again, hopefully with more scaffolding and more direction.  The students did not appreciate the fact that I would not “help” them early on, but they felt a sense of accomplishment at the end of the class period as their group began to get closer and closer to a good-looking graph.  Many of the students left class still feeling frustrated even though they were successful in the end.  I suppose they thought that they wasted an hour of their life because they were just frustrated, but I really hope they learned something from it.

Questions I have for you, if you care to respond:

  1. What can I do to shorten the amount of time it takes them to solve this?  I gave them the “parent” functions a week ago, so they had the hyperbola, inverse square, exponential, and power function, and of which would have worked (with tweaking).
  2. How would you change the problem?  Along with that, what extension questions can you think of?
  3. Are there any other tools (program online) that might help students use this?  I’m looking for free thing–yes, I know Geometer’s Sketchpad is a wonderful tool, but when all these other tools are free, I find it difficult to request money for programs that are so similar to free programs.

Thank you for your help!


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Tiered Assessment in Physics

So I want to try SBG (Standards Based Grading), but I got interested in it after I already handed out my syllabus for this year.  What can I do?  Well, I came across this cool idea from Steve Grossberg called Tiered Assessments.  As he explains, it’s not better than SBG, just different, but with a similar idea–you are trying to figure out how well each student understands a very specific set of skills.  Steve used Bloom’s taxonomy to create increasingly conceptually difficult questions.

So I took this idea and tried it on a single quiz for my motion unit.  I did not have Bloom’s taxonomy in front of me, and although I’m only 2 years out from education grad school, I don’t remember it that well, but I do remember “creating” being at the top and I think my quiz moves up the taxonomy as you move up in difficulty.  I also did not warn my Physics students that we would have a quiz, so perhaps it wasn’t the best time to ask them “so, did you like this type of assessment better?!?”, but I did like being able to see where each student sat on the spectrum–and it did really create a spectrum!

Here is the quiz on motion maps (not the most precise name, I know…) if you care to use it, whether as a template or copy it verbatim.  Unfortunately, I realized that each question is tackling slightly different skills as well, so although I like the idea, I would probably have to spend more than just 30 minutes on it to really have it match up as well as what Steve and his colleague put together, but it’s a start!

Please let me know what I can do to improve it, or if you have suggestions for future quizzes such as this (call them “Knowledge Demonstration Opportunities”… I know, I know), or if you’re upset with me for even comparing this to SBG at all, please let me know.  I think that this is a good intermediate step for those of us who may not have the tools to go all out on SBG, although I saw that Frank Noschese has a neat way to do this as well.

Edit: I wanted to let you know that I hadn’t yet taught my students how to solve the “A” type of problem, so while it may seem like a plug and chug–these students actually had to create their own way to solve.  And although a low percentage of them got it, nearly all of the students had good strategies and most of the students make very close guesses based on their models! (Oh, I’m trying out modeling for the first time this year in Physics… I’ll be sure to blog more on that later, too!)


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An Assignment of Which I am (was) Proud

So I now teach Precalculus, Physics, and Chemistry, and this is the beginning of year 2.  I wish I had more assignments I was proud of, but last year when I had three preps, I really just stole a lot of lesson ideas and adapted them, so there were very few, if any, that were original.  However, the year before I started teaching all these subject, I created a lot more of my own lessons.  Incidentally, that was also my first year of teaching (causation, or just a correlation?).  I still remember my best activity throughout that whole year.

So I was not a very good middle school teacher.  I had the patience, but I just didn’t have the discipline, and that combination only prolongs the amount of time before the classes reaches a level chaos where learning is impossible.  I tried so many different things, and because the students knew I didn’t start the year being strict, they ate me up alive and nothing worked.  However when we did this activity, the students were so silent, you could hear a pin drop.  We played Coordinate Battleship!

This activity occurred soon after we learned about the coordinate system.  I split the class in half, and pitted one side against the other.  I passed out top-secret battleship maps (Battleship Map A and Battleship Map B are two examples) so that one side received one set of ships and the other side received the other.  The students would then take turns “firing” on each others’ ships, but they had to be very precise with their wording: “Parenthasis, three, comma, negative four, parenthasis”, otherwise there was a “miscommunication” and the missile didn’t launch property.  Teams alternated turns and students went on down the row.  Students had to listen because they didn’t want to “re-shoot” where their teammates had already fired.  They were also required to listen to the other team’s shots because if they couldn’t (as a team) correctly answer whether it was a “hit” or a “miss”, then they automatically lost a ship!  My favorite rule was that while someone was firing, on either team, if you spoke and it was not your turn (or you didn’t raise your hand), then you caused a “misscommunication” on your team (because the military is very strict) and you lost a turn!  This kids ate it up and it was great!!  Everyone was focused and students really wanted each other to succeed.  They would silently help each other figure out how the coordinate system worked and where other members on their team fired.

Most things about middle school I don’t miss, but getting to capture their attention through a silly little game where they are learning mathematics and don’t even know it is one of the little things that I do miss.  Of course, that simply did not happen enough for me to be a super-successful middle school teacher, and so here I am in high school and very glad of it (for now).

Here is a powerpoint that includes all the rules of battleship, the way I ran it; and here is another, shorter version of the battleship instructions (we played it a few times throughout the year).  Later on, when we learned about lines, the students would give me the formula for a line (point-slope form: y=mx+b), and then they would get a torpedo which would hit all the points that it intersected (I figured this out for them, but more advanced students could do this on their own!).  To make it fair, every 5th person on each team would get a torpedo.  They never figured out that if the slope is zero, you get to hit a ton of points…


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Link Sheets

So today was the first day of school, and immediately after school let out, I sat down in my classroom and make some “Link” sheets for Precalculus.  I actually got this idea from a conference in Albuquerque we went to, and easily the best talk had several great ideas concerning teaching and making more connections.

For these link sheets, you take one problem, typically an algebra problem such as graphing a quadratic, and you break the sheet into four sections.  These 4 sections can vary, but I usually use “Equation”, “Written Description”, “Table”, and “Graph”.  The students are given one or more of these four sections, and they have to figure out what goes in the other three sections.  I am thinking that once they get the hang of it, they will be making more connections than usual, but the problems can be very open-ended, with multiple correct answers for the equation, if only given the graph or description, for example.  We’ll see how the students do, but I am certain they will really struggle with it initially.  Here’s an example of a LINK sheet:

The students also seem interested in the Participation Points, especially because they get to choose how they want to earn their points.  I tried to be very careful about the way I worded this to them, to make them as excited as possible.

I also just checked my e-mail in the middle of this blog post and received something from the “Blogging Initiation Team”–a group of great teachers who are also great bloggers, and are trying to get other teachers involved in blogging.  They’re the reason I started this blog, so I want to respond to one of their 6 questions and join in on what they are doing.  (Thank you!!)

I’ve decided to respond to question 2 and 6 simultaneously which ask:

2. Where does the name of your blog originate? Why did you choose that? (Bonus follow up: Why did you decide to blog?)

6. One of my favorite topics/units/concepts to teach is_____. Why is it your favorite? (Alterna-question: change “favorite” to “least favorite”.)

So the name of my blog comes from a though experiment I learned about in Math Grad School (this is before I realized that I wanted to be a teacher!).  The thought experiment is designed to show how infinity really isn’t a number in the normal sense of the word, and strange things happen if you treat it like a normal number.  Here’s a link to the story of Hilbert’s Hotel, although my professor told it to us a little differently.  (You have to read partway through the article to get to the story.)  So I decided to “photoshop” (I really used GIMP) and created an endless hotel hallway as well as an infinitely long school bus–both of which are in the story, and I changed the “bus” to be a school bus since I’m a teacher and all.  I suppose I could have changed the hallway to an endlessly long school hallway, but it was hard enough finding a hallway that could be replicated like that and was in the creative commons photo database (and yes, I am still a new teacher so I’m afraid of getting sued for the silliest little things).

So, one of my favorite things to teach are lessons which “blow the students minds”.  I was a philosophy minor in college and even considered going to philosophy graduate school (well, “considered”… nobody accepted me, however), so I really enjoy getting to teach things that make students go “wow, I never thought of that!”  Any kind of math, science, or philosophical thought experiment are so much fun to show students for the first time, and although that’s not in the curriculum, I do find time at the end of some classes to share some of the stories such as Hilbert’s Hotel.


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