I overheard multiple conversations between students who noticed the rate of change both sped up and slowed down. They were convinced they were doing it wrong. That's the kind of dissonance I'm looking for!
I base all of the trigonometric functions/ratios in the unit circle. We use this activity as an anchor for our future discussions about all sorts of things: the graph of sine and cosine, how we create triangles within the circle, and even sine being the distance from the xaxis and cosine the distance from the yaxis. I overheard multiple conversations between students who noticed the rate of change both sped up and slowed down. They were convinced they were doing it wrong. That's the kind of dissonance I'm looking for! Here is a complete write up of the lesson.
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I didn't do a very good job posting about activities as I did them this year. I'm working through a new Algebra 1 curriculum (I'm not impressed with it) and I'm trying to rethink how I do formative assessment. My activities aren't changing much this year, the amount of time we spend doing activities and experiments feels about right. I guess that's a good thing. Here's an activity we do at the beginning of our unit on linearity: cup stacking. I took this idea hook line and sinker from Dan Meyer and Andrew Stadel. Rather than do this as a 3 ACT lesson, I use it as a chance to get students out of their seats and talking to one another.
We begin by giving 5 styrofoam cups to every pair of students. Then I show a picture of me getting my height measured at my doctor's office (in cm). They have to figure out how many cups tall I am. We also almost always need to come to an agreement about how to measure the cups. Quick setup and clean up and almost complete engagement. You can't ask for much more than that. This activity helps students solidify their understanding of the new vocabulary identifying the angles created by a transversal between two lines. To make it a weebit more interesting, we also look at transversals across nonparallel lines. I spent some time searching for a card matching exercise for exponential functions. I found a couple but none that did exactly what I wanted them to do. Rather than spending an hour searching for what I wanted, I spent an hour and made it. Although, to be real, I'm not sure what I made is what I wanted either. I plan to use it early next week. Students match percent growth and percent decay equations, graphs, descriptions, and tables. Not every function has a description or a table. I thought I'd cut down on the process of elimination solution method. I also included some function forms students would probably not be familiar with yet but I would like them to take a stab at looking at the form of the function and guessing what the matching graph might be like. I've included the files as both a pdf and word doc.
Update: I added an extra page with recursive definitions of most of the functions... just in case you want to add the extra step of complexity!
A few years ago I started introducing trigonometry using circles. It started in Algebra 2 and moved down into Geometry. By basing all trig off the unit circle, we make an easy transition from Geometry trig (finding missing sides and angles of triangles) to Algebra 2 trig (sine and cosine functions and transformations). After a couple of years, I'm never going back.
Q: If you were the person taking this video, what would the graph of your height look like? Sketch a graph. I get all sorts of mountain looking graphs or semicircles. It's what I want. We are now going to get some intellectual dissonance. After students committed to a graph, we break out the hula hoops. I borrow them for a couple of days from our PE teacher. As an aside, I borrow stuff from the PE teacher and the science teachers all the time. Believe it or not, they are excited that your doing fun stuff in math class. They want you to borrow their stuff. Not kidding. We give a twominute talk through regarding how we are going to collect data. Something like: Put some tape on the hula hoops and put a dot on the side of the hula hoop. This dot is you. Throughout the experiment, you will always measure the height of the dot. You will roll the hula hoop 10 cm and then measure the height of the dot. In cm. When you get to the end of your meter stick, keep your fingers on the bottom point and slide it back to the beginning of the meter stick. Do this for one complete revolution of the hula hoop/ferris wheel. Most groups finish with the data gathering in 30 minutes or less. As another aside, I have students start at the bottom, like they are loading onto a Ferris wheel. Technically, this will look like a cosine function with an avalue of 1. However, we're measuring height, which is a sine function. Whatever. In four years of doing this, I've never had it throw students off the scent of the math we're doing.
Here's some of the resulting graphs. I take pictures of some of the student work and we have a conversation about it. Q: What do you like or dislike about this graph? Students tend to not like the "lumpy" portions of the graph. I put the following graph on the screen in response. Students like this one much better. As if almost on cue, some student always says we should connect the points. Then we get to have a good discussion about whether the graph should be continuous or discrete. Students generally settle on the following as their "ideal" graph. We then talk about rate of change in table groups. How do you talk about rate of change with this graph? After flailing about wildly, we usually are able to give a decent explanation about whey when the point we are measuring is at the bottom or the top of the circle it doesn't have much vertical change in distance as opposed to when it is on the sides. Day 1 usually ends here and students finish their 7 questions on the handout. This year, Day 1 was half eaten by questions over a previous assignment and we gave up half of Day 2 to finish gathering data, graphing, and having this discussion. Day 2 begins with a discussion about the unit circle. The following is the definition of "sine" from Wolphram Alpha. Notice the first part of the definition. Forget the "common schoolbook definition". As a class, we define 'sine' as the vertical distance of a point on the circle to the xaxis and the 'cosine' as the horizontal distance of a point on the circle to the yaxis. It's *expletive* amazing! We talk about what this distance looks like when we move the point around the circle. Here's some of the webbased visualizations I use: Sine and Cosine as vertical and horizontal distances in the unit circle http://www.analyzemath.com/unitcircle/unit_circle_applet.html https://www.mathsisfun.com/algebra/triginteractiveunitcircle.html https://www.desmos.com/calculator/v7x6br3w6a A visualization of Sine and Cosine in the Unit Circle: (If you want to moving one, click on the link below) http://i.stack.imgur.com/p8O4P.gif This is THE move that sets up everything! It sets up talking about sine and cosine as functions. It sets up a point on the unit circle being (cos, sin). It sets up the Law of Cosines not being some weird abnormality that's never really understood by students. It sets up that what we commonly refer to as trig ratios are lengths of lines in the unit circle. It also sets up any triangle with a hypotenuse other than 1 being a dilation of a triangle on the unit circle, with a scale factor of the length of the hypotenuse. Of course we end up deriving the shortcut methods for finding sine, cosine and tangent without having to go back to the unit circle every time. However, the fact that these are real lengths and not just abstractions goes a long way with my concrete learners. Sine is a vertical distance. Cosine is a horizontal distance. Tangent is the distance from the tangent to the point on the circle to the xaxis. Which is why tangent of 90 degrees and 270 degrees is undefined. This also helps discuss Cotangent, Secant, and Cosecant when they come up later in the year. They are all actual lengths in the unit circle!
If you're teaching Algebra 2, after this throw in a little Cosine Ferris Wheel from MARS and you've got yourself a decent start to a trig unit.
Oh, there's a field trip and this is the only section of Algebra 1 I'm teaching today? Okay kiddos, we're making some visual patterns and inundating Fawn Nguyen's email account? Ready? Go! Sadly, two of the groups weren't able to come up with equations for their patterns during class. Sadly, those groups also weren't interested in completing the equations outside of class if it wasn't an assignment. All in all, not too shabby.
The directions were simple: •Drop a ball from any initial height, measure in cm. •Measure the height for 5 bounces of a ball (I suggest using a table). •Graph the points (graph paper) •Create an equation that models/fits your data. •Do this for two different balls (don’t put both balls on the same graph) The results were fabulous!
Students had great conversations surrounding how to find the constant multiplier if it never ended up being the same between bounces. We also had heated arguments about whether connecting the data points with a curve made sense in the situation. (The students came to the consensus that they did not think the graph would be continuous.) This week we start a unit on modeling linear data. But that means we need data. As it turns out, that's the fun part! Exploring linear functions in Algebra 1: Bungee Apparatus: exploring the relationship between weight and the amount of stretch. Spaghetti Bridges: looking for ways to describe how the number of pieces of spaghetti affects how many glass beads a bridge can hold. I love the conversations we have around these tasks. The physicality of the task ends up being an excellent mental hook for students to come back to when talking about topics like: independent and dependent variables, interpolation and extrapolation, correlation, error, and regression. The lineup over the next two weeks:
water height based on how many cubes we drop into a container, how the pullback distance of a toy car relate to how far it goes forward, the gender pay gap in these United States, Barbie bungee jump I'll do my best to put up pictures as we go. This is an epic month for Algebra 1 if there ever was one!
This past Friday I had the opportunity to present at my state math conference (#CCTM16). I think I was the only presenter that needed a wagon to bring in all my supplies. Not kidding.
I presented on one of the few things I'm good atgetting students out of their seats and getting students talking to each other. Over the past 45 years I've been on a journey to involve more more active learning in my math classes. It's a bit more than just 3act math or problembased learning. Don't get me wrong, I love me some Math Assessment Project or Emergent Math. There are all sorts of wonderful things happening in the world of online math teachers that wasn't happening 10 years ago. I totally dig it. But that's not really what floats my boat. I'm interested in students modeling the world around them. All the better if they can also be the ones to run the experiment and collect the data. Basically, as close as my class can get to a physics classthe better. It's probably because I married a physics girl.
So, I presented on what I know; doing experiments in math class. We started out with a quick intro about how this isn't meant as one more thing you can add to your classes. This is meant as a replacement for something you already do in your classes. Math classes in the U.S. already try to do too much. Adding one more thing isn't helpful. Because it would be super awkward to spend 50 minutes talking about active learning without actually doing something we started out with an experiment: ball drop. I had the participants arrange themselves in groups of two or three. After a sudden death round of rockpaperscissors, the loser came up to get supplies for the group. Here's the slide I put up for directions.
I didn't give much more help than that. Some groups had issues counting the 4th or 5th bounce. For those groups who struggled I made a observation that catching the ball at the top of the bounce and then dropping it would not change the data. I think we were the only session of the day to have participants on the floor or actively using the hallway. Epic!
While groups were busy trying to model the data, I went around the room asking questions and taking pictures of solutions strategies I wanted to discuss as a larger group. When I called the groups back together for a whole group discussion, this happened:
My Reflector2 app stopped working. I have this app that allows my phone to mirror to my computer screen. Normally, it works well. It had worked well the 10 minutes prior to this moment. Curses. Time to do what you would do in a classroomswitch strategies.
We did our best to talk about solution strategies without looking at pictures. The room was very gracious about it. A classroom of students would have revolted.
Then we hobbled through talking about how running a classroom experiment naturally flows into covering the CCSS math practice standards and the Principles to Actions math teaching standards. Again, super hard to make seamless connections without the major component of those connectionsstudent work. But, this was a room full of sympathetic math teachers/coaches and they helped me out.
That's when we finally got to talk about some of the wicked cool experiments we can do in math class without needing a physics lab. Well, at least for most of them. I've got the presentation linked below, so I won't go into details here. The gist of what I wanted to say to them was, it's not too hard. You can do it. Ask the science teacher down the hall for help. For real. Ask her. She will LOVE it that you want to do an experiment and will help you gather supplies. Not joking. The physics teacher will probably even set up a lab for you more than once. Again, not joking.
I've already hear back from one person: "I was inspired enough to try it! I'm going to try to do a lesson with perimeter, area, and volume next week. I stole some boxes, meter sticks, and string from my science teacher next door. You were right, he was excited to help. He already had the boxes made for a lab they used to do. I'm going to try to use different balls for my students who need some more complexity and boxes for the students who need more straightforward data collection."  Mike That, my friends, is awesome!
All in all, it was a good experience. I didn't die like I thought I was going to. I might even do it again sometime. Except it would have to be on the exact same topic. I feel a bit like a onetrick pony.
Here's the slide deck for the presentation:
AND, you should be super jealous, I ate lunch with this wonderful #MTBoS crew! From the left: @MathEdnet, @lisabej_manitou, @pwharris, @0mod3, and me. 
Andrew Busch
I teach Math and Programming at Summit Middle School in Boulder, CO. Categories
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