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.
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.
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I'm struggling to find time to write thoughtful posts. But I am taking pictures. Here's some fun shots of our Ball Drop experiment in Algebra 1.
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!
With systems of equations, I tried something new for review. Instead of doing the normal routine of making a review assignment and then having students solve it, and then having the students who don't do the assignment be the ones who need to do it the mostI had students make their own review assignment. Hey, why not? Here are the directions I gave to the class (link to Google Doc here):
Here are the sections we covered this chapter: 61: Graphing Systems of Equations 62: Substitution 63: Elimination using Addition and Subtraction 64: Elimination using multiplication 65: Applying systems of Linear Equations 66: Systems of Inequalities At first, there where the obligatory moans from the classroom when I introduced the assignment. But, when I told them they only had to create 6 problemone from each of the sections for the chapterthe mood changed. Don't get me wrong, they didn't cheer or anything, but I did have buyin from most of the class. When students asked me what a certain section was about, I directed them back to their textbook. I have both an inclass set and we have online pdf's so access should be an issue. Students checked their 'correct' solutions using Desmos. I appreciated how much this pushed students to understand the mechanics of solving systems of equations on a deeper level. It's one thing to use an algorithm. It's another to intentionally break the algorithm and see what you get when you break it. Sadly, this assignment didn't fix the problem of students who really really need to work on the assignment not doing the assignment. I guess if I had the magic fairy dust to fix that, I would be a very rich man. If I had to change one thing about this task next year, it would be that students didn't really know what I expected of them; there was some confusion as to what the end product might look like. So, I took pictures of this year's work to show to next year's students. I can usually go through the pictures without saying a word. Students start to get a pretty good intuition about what 'good work' versus 'bad work' feels like. This looks nice. This... doesn't look as nice. Here's some examples of studentcreated practice tests. I've included four different levels: the "Oh my goodness, can I frame this?" the "Good attempt but struggles with organization" the "I appreciate how much effort you put into this but it took me some time to figure it out", and the "Can you walk me through this, please?" Excellent Work: Good attempt but can use a little more organization to help the reader understand what's happening: Solid Effort: Can you walk me through this, please?: Thanks to Dan Greenberg for the idea!
What to do in Geometry after finishing up our circles unit with one day left before Spring break? Algebra review in the form of art class!
Here's some of the work my students elected to turn in. I gave them one class period to work on it and asked them to turn in whatever they had. If students wanted to do more, they could but it was not required. Hence the varying degrees of completeness. I think they did brilliant! Let's be real for a secondalternative assessments are really hard in secondary math. I have like 87 objectives and you want my kids to make a poster? For real? I've tried doing them a couple of times over the years, but they were usually such bad initial experiences I've stuffed all memories of them deep into my mental closet. This project is the exception to the burnthelessonplantogetridofallevidence I've come to expect from my attempts at alternative assessments. This is my 3rd year doing this project. Apparently, the 3rd time is the charm. I posted about my experience last year here. You can smell the fear in that post. At this point in quadratics, we've covered: graphing, factoring, the quadratic equation, vertex form, and transformations. What I really want is for students to see the connections between multiple representations. I want students to see that you can find the roots from the graph in Desmos. Those roots are the same as the roots in the factored form of the equation. Those roots are also the solutions to the quadratic formula. Slowly, we're making progress. Here's the rough outline of the project: 1) Find a "realworld" example which you can fit a parabola to. If you want to fit a parabola to Payton Manning's forehead, or Donald Trump's smile, or some other hilarious thing, that's fine with meyou'll just lose the "realworld" point. 2) Fit a parabola to the image using vertex form. 3) Show all points of interest on the Desmos graph (roots, yintercept, vertex, axis of symmetry) 4) Demonstrate you know how to calculate the avalue for the vertex form of the equation using the vertex and a point. 5) Show me you know how to find the factored form of the equation from your Desmos graph. 6) From either vertex or factored form, calculate standard form of the equation. 7) From the standard form, show me you know how to calculate the vertex of the parabola. It should be the same vertex as your graph. 8) From the standard form, use the quadratic formula to calculate the roots of the equation. These should be the same roots as your graph. 9) Put your name on the back of the poster so I can take pictures of your work and put them online without worrying about student identities (that's one of the reasons at least). Let's talk pacing for a moment. I give two work days in class for this project. On the first day, we intro the project, work on finding a super cool picture, and fit a curve to it (#13 above). On the second day, we work on doing the math and I hand out poster board to students. I have the project due several days after our second work day. There is no time to work on the project on the due date. Students waltz into class, I collect the posters, and we delve into radical functions. Next time, I will strongly suggest to students that it's worth it to do the work before starting on the poster board. I made a template for next year (below this paragraph). I'll probably import it into a Google Doc to encourage students to type out their work. Typed posters are usually much nicer to look at than the scrawl of most of my students.
If you're interested, here's my files for the project. The directions on the left are what I hand out to students. The checklist, which is included in the directions, gets paper clipped to the front of the poster so I can evaluate student understanding without writing all over their beautiful projects.
And now for the gallery. I'm going to include the good, the bad, and the ugly so you get a realistic picture of what to expect. I make it very clear to students that it is possible to get full credit without making these display worthy. I'm most interested in the math. However, I do offer bonus points for making things pretty. First, a couple of pretty ones. The math isn't perfect, but I'm not going to highlight the mistakes for you. Here are some more student examples to give you an idea of the range of quality to expect. Not everything is super pretty all of the time. 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.

Andrew Busch
I teach Math and Programming at Summit Middle School in Boulder, CO. Categories
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