# Tag Archives: Mathematics

## [2016 Blogging Initiative] Week Four: A Lesson Introducing the Unit Circle

My Relationship with Textbooks: “It’s Complicated”

My first several years of teaching I avoided the math textbook as much as possible[0]. One year I even waited to hand out textbooks to students until the second quarter. I assumed (incorrectly) that using the textbook would make me a lazy, bad teacher. However, at the start of this year I decided to embrace the textbook for the good resource that it can be: a bank of practice problems[1] not a replacement for my teaching[2].

Background: My Classroom

One other thing I’m doing this year is flipping my classroom. The flip, however, isn’t just lecture. I’m trying to challenge my students do problem solving through the vidoes, and I hope to show how I’m trying to achieve that in this lesson. For one thing, I provide guided notes for the students to fill out as they watch the lesson. I also don’t do every problem: I ask them to pause the video and try some in the middle of the video. To that end, I’m also using EDpuzzle which pauses the video and asks them questions that I’ve created at a variety of levels.

When we get back together in class the following day, the students are randomly assigned into groups of 3 or 4. Students spend about 10 minutes going over the notes and making sure each students’ notes agree with one another and that students understand the topic. After that students work on practice problems, from the textbook, on the same topic. [3]

The Challenge

So we’re chugging along and we get to the Unit Circle. This is the first lesson that I disagree with how Blitzer (our textbook) approaches it. I’ve had success with students in the past by teaching special right triangles first because students see them in the Unit Circle. So I decided to create my own “chapter” and left the textbook, like old times.

The link below is a short (<13 minute) video so you can see what the students will do for HW prior to class. But you should watch it because that’s the interesting part of my lesson. 🙂

https://edpuzzle.com/media/56b43368fe5ccd81111fd654

Here’s the handout:

As you can tell from the video, I show students the special right triangles and where the values come from. My hope is that they use the Pythagorean theorem if they ever forget the shortcuts in the future, but most students will, unfortunately, probably forget that. I’m not sure how to share that with them differently.  However I only give students a few points from the Unit Circle, and ask them to “figure out the rest”. If they can figure it out on their own before coming to class, and they understand the special right triangles, then I think that it will be more likely that the Unit Circle will stick.

Since I’ve deviated from the textbook here, I had to find practice problems online, but that wasn’t too difficult. Students will go to my website and simply click on the worksheet links (complete with answers) to practice this in class. I’ll only print out the sheets for those students who want more practice beyond class and have no internet at home.

I’ve assigned the video (only 2 students have watched it so far), but we’ll meet in class Monday to see how well they did filling out the rest of the Unit Circle.[4]

Request for Feedback

How can I improve this approach? How can I teach special right triangles in the video so that they do more of the “heavy lifting”?

How are the quality of the questions in the EDpuzzle video? Are there others you thought of that I could do?

Is there a better way to approach the Unit Circle that you’ve seen/used other than special right triangles?

If you could answer any of the questions above, I’d greatly appreciate it. Thank you for reading! (and watching??)

[0] I still avoid it in Physics–I haven’t handed out a textbook in 3 years, with the exception of one student who begged for it. It didn’t help her.

[1] It’s also a good resource for ideas for 3-act lessons.

[2] I’ve seen some teachers teach how to read a textbook, which is a valuable skill, but one that I’ve decided pass on for now. I want my students to understand the math first and foremost. I’m still not sure how I feel about not teaching students to use a textbook effectively and efficiently.

[3] Because I believe that HW is practice, earlier this year (before I flipped), I don’t grade HW. Students also didn’t do the HW (with very few exceptions). Now, I still don’t grade that they watch the video, but I’m not afraid to email or call home if students are missing it chronically. Also when students get to class, they recognize that they’re responsible for learning the material at home, and so will work harder at the start of class to understand what they didn’t watch. It’s amazing how much more “HW” (practice) they’re doing now just because it’s happening during class.

[4] And if I’m on my blogging game, I’ll blog about how it went. Unfortunately it’s tennis season, so I probably won’t find time to soon.

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## [TMC15] Day 2 Reflections

3D Printers, from Heather Kohn, had me excited about 3D printers. I thought about writing a grant for one last year but never did. I think I want to now. Creole (sp?) takes a PDF and makes it into a 3D object. (Talk to John Stevens, too!)

In the Activity-Based Teaching (my morning session), we talked more about spiraling curriculum and the benefits. We also did a cool activity where he asked us all to remember 20 words and regurgitate them as best as possible. This typically makes a parabola, which can unload tons of questions from there, not only about math, but about studying and real-world applications, such as telling someone a list of objects to be sold, etc. I was comforted when Alex & Mary admitted that it is difficult for more higher-level math courses to do activity based teaching. However, Alex says he does this with his “Precalculus” class (the Canadian equivalent of it for his district).

During lunch, Julie, Sam, & Tina talked about the blogger initiation for this year and I got excited about the possibilities. The main idea they want to work around is having 2 strands: 1) for bloggers/tweeters who are brand new and 2) for bloggers who aren’t, but could use encouragement. They’re also talking about having mentors & mentees and grouping people, which I think would be HUGE for both the mentor and the mentee. I really want to mentor 3 or so teachers from across the country who want to “test out” blogging and tweeting but aren’t sure what to do. That would be fulfilling as well as motivating to get more involved in the community myself!

Christopher Danielson shared, at the keynote, how we need to find our favorite thing and do that, both in our classrooms and online, so our students & other teachers (and by extension other students) can benefit from what we love. He shared that what he loves is ambiguity (at it’s root) which is why he wrote the book “Which One Doesn’t Belong?” After hearing his talk on it, I’m definitely going to use the website that Mary Bourassa created: Which One Doesn’t Belong.  However, Christopher did explain that, while it’s great for everyone to get excited about ambiguity (or debate, as I’m going to use it to that end some), I still need to think about what I love and why I love it. I need to do some reflection on that later.

In the afternoon sessions, first I attended Meg Craig’s “Function Transformations without Tears”. I not only got some cool worksheets out of it (more posted on the TMC wiki), but she provided a neat, new approach to function transformations. In the past I had always done an exploration, which took way, way too long since it’s Precalculus and they should have seen much of this already. The new idea is to “move the origin (whether or not it’s on the function” and graph the parent function using the new origin. It gets a little tricky when talking about what I used to call “stretching”, but Jed explained how he sticks to the vocabulary “Input” and “output” and Jim kept going back to the question “what do you plug in to get the origin?” (Which, I just realized, doesn’t work if the function doesn’t go through the origin… but it’ll be a good starting point). I’m also going to use “h,k” notation and move the k to the other side so it’s actually physically next to the y! I also want to consider using the “Window Pane” method for graphing sin & cos, as well as look deeper into figuring out whether I can also divide both sides to get the other factor closer to the y for consistency. Would it be weird for me to ask my student to turn this:

$y=a(b(x-h))^2+k$

into this:

$y-x=a(b(x-h))^2$

or even into this:

$\frac{y-x}{a}=(b(x-h))^2$

or possibly/ultimately into this (if $c=1/b$):

$\frac{y-x}{a}=(\frac{(x-h)}{c})^2$

This will take more thinking.

Perhaps it would be better to instead define $z=1/a$ and have students do this:

$z(y-x) = (b(x-h))^2$

That not only makes it obvious what is affecting the input vs output, but it’s also consistent, so there’s no memorizing “oh yeah, inside the function means opposite of the other thing…”. I’m excited about this! Of course this applies to any function where we’re applying a transformation.

$z(y-x)=f(b(x-h))$

The final session was on SBG and I got into some great discussions with Anna Hester, Nathan Kraft, and others concerning when to grade (formative vs summative assessment), how much to grade (size of scope of standards), and how to connect standards (leave it alone & talk about it, or grade it somehow…). Great discussions and lots of food for thought.

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## [TMC15] Day 1 Reflections

This should only be a short blog post to help me remember what I experienced today. It was super fun and it’s exciting, even if a bit overwhelming, to find that everywhere you turn there are people who have a very similar goal at their job: get students to learn & love math.

Here’s what I discovered & learned (in reverse chronological order):

• Arithmetic, specifically just adding and multiplying numbers, can be beautiful (Art Benjamin’s presentation was awesome!)
• Anything can be turned into a debatable activity (Chris Luz, @pispeak had a great presentation with tons of info!)
• Debate is great for students for many, many reasons. To name a few:
• Gets students thinking on both sides of an issue
• Students can see when it’s better to solve problems different ways (when you force them to choose one side and debate 1 on 1 with a classmate)
• Makes it exciting
• A bunch of other reasons I didn’t write down.
• Giving students a framework/vocabulary for debate makes “attacks” less personal and more appealing.
• Physics Educational Research–Physics teachers have already done the research in how to best teach Physics. Wow, how didn’t I know this already? Lots to look for here before I start class in 2 weeks. *gulp* (Thanks Eric!)
• Bree (@btwnthenumbers) shared how she uses Evernote (why haven’t I really used it before?!?) to organize her MTBoS material. I need to get better at sorting things right when I read them and Evernote can help do that as long as I’m diligent while I’m reading.
• I need to go through the 200+ bookmarks I’ve saved into Chrome but never looked at again. Why didn’t I realize that I don’t look at those?!?
• Alex Overwijk & Mary Bourassa shared how to get students to ask good questions: it’s by making them reflect on what makes a good question and putting a poster of what they realize on the wall (okay, it’s a little more than that, but I wrote the rest down in my notes, okay?)
• Trader Joe’s doesn’t open until 8am. Why?!? No idea.

I’ve also met someone who:

• Shares a last name with me.
• Taught with my father-in-law (Nicole Paris, @solvingforx)
• Teaches in my former “hometown” (Mary Brown, @marybachbrown)
• Went to my college (okay, so I already knew you two, Anna & Julie!)
• Shares a room with me (well, okay, Jamie, @jrykse, & I had planned that out ahead of time though we had never met in person)

Looking forward to more fun tomorrow!

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## Classkick — Cool New (Free) App

Dan Meyer usually rails against apps & website rather than for them[1], so when I read that he was saying how good an app is, I thought I should check it out.

And it looks awesome. I had high hopes of Nearpod, but the I didn’t like a few things: (1) The entire class had to stay on the same slide, (2) students could only write on some of the slides, (3) teachers couldn’t really give individual feedback, and (4) students (on our network) were often kicked out and had to log back in, which was an arduous process.

By comparison, Classkick is very fast to get students logged in: just a 6-digit/letter class code to type, which is unique to the class period and the lesson, so I guess you could have students in the same class on different lessons if necessary. To reinforce how fast and easy it is to setup, I heard about it during my planning period, and was using it in class (with a lesson I had created) 50 minutes later. Was it the best lesson in the world? Probably not, but then again I was just planning on lecturing/doing practice problems, so this was so much more interesting.

Students’ Interaction & Usability

I felt really comfortable using the interface and creating a lesson was super-quick. Dan has a good point about the writing taking up so much space, but since you can scroll down, as well as make one page per problem if necessary, space doesn’t seem to be too much of a problem. Space-management seems to be more of a problem as some students just thought to write/answer on top of the prompt (see below).

I was missing a good chunk of my class to extracurricular activities, so I decided to not tell my students how to do stuff and instead see if they could figure out little things, like being able to ask for help, help each other, hide teacher/helping student comments, scroll down, etc. These are “advanced” juniors in math class so I was hopeful, but ultimately some things weren’t as intuitive as I thought they would be for the students and this disappointed me. Many of the students didn’t take time on each slide as I thought they would and just went “fast-forward” to go to the next one. I had to ask them “why are you skipping slides” to which they responded “oh, you wanted us to do each one?”  *Face-palm*

Here are some specific examples of students interacting with the features:

• The “scroll down with two fingers” function isn’t intuitive, but it does allow for much more white-space on which to work (certainly not infinite, though).
• The “ask for help” is a great idea, and I love that students can help each other, but with this small class (6 students at any one time because of extracurriculars), nobody really helped anyone else. I could see this working if the class was larger or if students had more time to get use to the program (which I plan on using again!).
• One student changed the colors of her pen, which I didn’t notice that you could do the first time around (I played with this for maybe 15 minutes before using it in class!). I thought that was awesome that she found out something that I hadn’t discovered yet. She even wisely used color coding.

The question at the bottom that you can’t see asks “What do you notice?”

Pros and Cons

Writing on iPads is a pain, as you can see from my horrible handwriting, and my students’ worse handwriting. But I would say that hand-writing for equations for students is infinitely better than typing into notes or something foolish like that on the iPad.

I like how easy it is to insert images, though I had to really blow up some images and their was quite a loss in quality of the image when I did that (I blame the iPad more than the app for low resolution screenshots).

Students being able to move at their own pace is a huge plus.

Suggestions

Each page could only scroll down so far. One student decided to combat horrible handwriting by using huge handwriting, which I was fine with, but he ran out of space before finishing his answer to “What do you notice?”, which is a bummer. If pages could go farther (or on “forever”) that would be nice.

It would be nice to give teachers an option to type prompts. Yes, I can record my voice (which would be annoying to hear 20-30 times in a classroom), and yes I can type, save it in Dropbox, screen shot it, and then use it, but that seems a bit excessive for typing a simple sentence like “What do you notice?”

Along with the last one, it would be nice to be able to create content through the computer. I much prefer using a computer over an iPad, and I was frankly surprised that you had to create the content on the iPad. Fortunately the experience was much better than I feared it would be, but in the end I’d still rather create content on the computer.

I’ve gotten used to Doceri on the iPad for content creation, so I kept finding myself wanting to zoom in and out so my handwriting would look better. I think if this would be added, it would overcome a lot of the problems mentioned above. I know you want to strike a delicate balance between simplicity for students’ sake and power of an application, but I think this is one item that could add tremendous value to this app.

Summary

This is an app I would highly recommend that you check out–it’s the kind of thing I had hoped Nearpod was when I first found it. I think students will use the features more fluently as we doing this a few more times, and next time I’ll go ahead and jump in and explain features I think that are important. Also, I’ll spend more time preparing the assignment so it’s more interesting to students. I’m excited to think of the possibilities! I will definitely report more on using this in the future.

[1] In his defense, most apps & websites are garbage and I agree with the vast majority of his assessments of them.

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## Race Across the Room! Activity

Yeah, a lame name, I know, I gotta find a better one.  But all the students were surprisingly really into it!

Okay, so I started the day with the following Dr. Seuss “Sneetches” video. (It was Dr. Seuss’s birthday, so yeah, this was a few weeks ago.)

Not at all related to what we’re doing in math, and only vaguely related to something we did earlier in the year (functions & inverses), but it’s got a good moral to the story, and got them out of the Monday Doldrums (even if it took a little longer than I wanted to at about 12 minutes…).

Then, after a brief discussion of how the video is related to math (see powerpoint below), I explained the rules of the “game”.  Basically, it involves students working on a problem, and when they get it, they bring their answer up to me.  If they got it right, then they can go to the next “station” (aka a folder with a small sheet of paper inside).  If they got it wrong, then they basically gain the ability to ask others who are struggling with the problem.  The first question started on the powerpoint (2nd to last slide), and then they moved to the folders and “stations”.

The carrot in front of them were the participation points (as you can see from the powerpoint above).  First place earns 40 points (they need 100 each week, so that’s a sizeable chunk!), and each place you drop, the people earn 2 points fewer at each interval.  My idea was that the difference was small enough that they wouldn’t strongly want to cheat to get ahead (or pester their friends for help too much), but it’s just enough to motivate every student to start out the problems on their own in the hopes that they can solve them before some, if not all, of their peers.  And I’ve got to say, it actually worked in both of my classes!  I don’t know if it is because they really wanted the participation points, or if they are just naturally good students & hard workers (mostly true…), but they were definitely hurrying around the classroom, trying to get the answers and understand what they had wrong when I told them “nope, try again!”  I was definitely unhelpful, and I was impressed with how little they complained.  Perhaps they’re used to that from me by now? 🙂

This activity didn’t work nearly as well on the second day, where some of them were stuck on the same problem and, with no-one finished yet, they had no-one but unhelpful Mr. Newman to “help” them.  Of course, those students were also on the right track and just making a few mistakes here and there where, if they looked carefully, they easily could have seen what they were doing wrong (for example reading the problem I gave them instead of assuming what it was saying!).  But hopefully they learned their lesson.  Overall, I’d say it was a good review activity, as long as I didn’t burn 20 or so minutes explaining an unrelated activity (for some reason, I forget what I was telling them, but I feel like we didn’t get a full 50 minutes to work on it).

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## Kickstarter Project: The Number Hunter Promo

Kickstarter is such a great idea: let “the people” choose what should be funded, and let people with great ideas get those ideas out there and funded.

One great idea that was shared with me was a sort of “Bill Nye” meets “Crocodile Hunter” meets math series to spark student interest in math.  I’ll share the project below, as well as a short video explaining it.  As of right now they’ve got 12 days to go and about \$2,000 left to raise, so I do hope that they make their goal.

One good thing about Kickstarter is that is the goal is not met, nobody has to pay, so you can support projects that may or may not make it, and not be worried about the worst-case scenario: you pay money to a project that doesn’t happen.  So go out and support the team behind “The Number Hunter”!

Kickstarter Page:

http://www.kickstarter.com/projects/564889170/the-number-hunter-promo

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## Make Students Show Their Work

All math & science teachers want their students to show their work, but so often students forget/are being lazy/think they don’t need to show their work, especially if they haven’t had a test in a long time.

Just Innocently Grading Quizzes One Weekend…

This weekend I was grading Chemistry tests, and was so frustrated at either (a) students not attempting problems when they know the first few steps, or (b) only writing down their answer to a very long and complicated problem (Stoichiometry).  So, I started class today with the following warm-up:

Students pointed out how “6th grader #1” got 1 wrong, but deserves the most credit because they showed the right mistakes, and so showed understanding whereas #2 deserves more credit than #3, even though #2 didn’t get very far on the problem.  I even went so far as to claim that #1, in some of my quizzes, and if they demonstrated an ability to do that correctly elsewhere, could still get a 100 because we all agreed #1 understood how to do the problem.

Of course, throughout this discussion, I still had the one or two smart-butts who were convinced that student #3 knew it the best (I’m 99% sure they were just joking), but I don’t think I convinced them to show their work because they “always” get it right (they really don’t…).

Easier to Talk about an Easier Problem

One thing that allowed us to have a discussion about understanding is that the math was easy enough for all the students (juniors in HS) to solve, so if you were trying to convince 6th graders to show their work, this might not be the first choice.  If I had chose a Stoichiometry problem (complicated Chemistry problem) to demonstrate “why you should show your work” then I think my students would have gotten lost in the mechanics of the problem and not seen the bigger picture.  Now, I hope that my students will not forget to show their work in the future.

How Do Get Your Students to Show their Work?

So I guess this took 5 minutes out of class in place of a warm-up (I had another warm-up after that one), so it did not take a lot of class time.  However, it helped tremendously that my students were used to my quizzes (which I think they recognize as testing understanding and not “how many you got right”), and it helped to have the discussion right on the  heels of a test where students can think about what they did wrong and learn from their mistakes.  Would this discussion have the same effect at the beginning of the year?  How do you convince your students that work is important?  Does it work?  Or is it a constant battle?