Marshmallows and Buggys and Wild Ideas!

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First week down and one of my goals for this year is to blog at least once a week, so here I go!

In physics, we did marshmallow towers and then Buggy Lab. I loved the way we did marshmallow towers this year because we did them twice. The first time was at the freshmen orientation, so it was a great activity to do with the new kiddos, meet a bunch of them and have some fun. Most towers failed.

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Pixel face here looks like he’s enjoying it nonetheless.

But that’s the point. They try out these amazing tall designs and realize too late that they aren’t sturdy enough to hold a marshmallow. Last year, that was the end of it. We just discussed what I wanted them to take away from the activity. But this year, they got to experience the takeaway. The second time we did the challenge, students were armed with a few new tricks:

1.) they knew of at least 3 designs that wouldn’t work because they all took pictures of their first fails

2.) they knew that the secret was to place the marshmallow on top first and build up from there

Almost every group had a successful tower the second time around. What a powerful visual to promote a growth mindset! A bunch of failed towers one day, a bunch of awesome sturdy structures the next! BAM!

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We also did our buggy lab this week and I think I’ve got the setup for this one down. But I’m never sure how much to feed the students about the structure. This year, I gave away a little more than usual. I told them to make sure to include a graph, equation, sentence, and motion map, and I helped them out with creating all of them as needed. Some might say “well, um, isn’t that your job?”. Yeah, but it still feels a little weird, like I’m robbing them of a good learning experience. But in the end I think I did the right thing here. My science dept’s big goal is improving scientific investigation skills. And one of the things I’ve been doing wrong is not providing enough scaffolding early on in the year. Here’s the handout we used:

And here’s a board ready to go into the discussion next class:

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Ok, so that’s it for marshmallows and buggys, what about the Wild Ideas I promised?! Well, in my Intro to Engineering and Design class (awesome right?!), we did a challenge where they had to get a HotWheels car to travel between two tables on a fishing line. One of the materials they had to accomplish this task was a balloon. So one group had the idea to put their car inside the balloon and then blow it up and tape a paper clip to it to attach to the line. It totally worked! Sorry no pictures, but take my word for it, it was awesome!

And that’s one of our first lessons for the class. Don’t throw out any wild ideas, sometimes they’re the best ones. Also, sometimes they suck and the boring-est idea turns out to be the best. But the ability to tell the difference and pick the winner makes for a good engineer. I’ll take pictures next time. 🙂

Modeling Quadratics: First Attempt

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I’ve had trouble with getting students to model a quadratic function since forever. Apparently I’m not the only one, as it comes up often in the MTBoS. Last year, I made a step in the right direction by starting with the vertex form of the function. Well, actually let me go back one more step. We started our quadratics unit with absolute value functions. Check it out:

Now that we have a good idea of how to shift around a parent function, it’s time to try building a quadratic from scratch. (Side note: try patterns 106 and 108 at visualpatterns.org when you start your quadratics unit, some of your students will amaze you, or your money back guaranteed!) Anyway, so here’s my plan:
Actually, first a quick word about my schemes for next year. I plan to run my class like a game, so there’ll be quests, XP, leveling up, no grades, and other stuff I haven’t figured out yet. And I think I’ve got my theme. Wait for it… “Hitchhiker’s Guide to Algebra” (not sure about the name, but you get the idea). Total credit for this idea goes to Jessica Anderson (@triscicurious). Check out her blog, especially if you’re thinking about gamifying too, since she’s already doing it.

With that in mind, I am trying to write tasks around that theme so that I’ll have some ready to go for next year. The more I get done before the year starts, the more self paced and non-linear my class game can be. So I tried to make this task fit with a Hitchhiker’s Guide theme. Hence the “Intergalactic Lower Appendage Ball League”. Get it?
Anyway, when students come in to class, they’ll get into groups and each kid will receive 16cm of string and this handout:

Inspiration credit for this idea goes to Scott Hills (@Planting_ideas). Thanks Scott!
As they wrap up measuring and recording, I’ll put the Desmos graph (woot images!) shown below up on the smart board and ask everyone to come up and enter their results. Go ahead, click it and put in your results and see what you think, I’ll wait!

If they haven’t already, they should notice (once we’ve got enough up on the board) that the points are forming a parabola. BAM!! Quadratic Modeled!! Well, almost. Each kid will then complete a summary report of their findings where they’ll have to write a function and think about key parts of the graph. Then they’ll share their thoughts with their group, and then each group will make a whiteboard to display for the class. If we have time, I’ll have them blog the results at our class blog.

What do you think? Got any ideas for improvements? Have you done something similar before, any tips?

Why Energy Pie Charts?

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Note: This is my first year teaching freshmen physics and it’s also my first year teaching physics using modeling.  Lots to learn!  I’ve had to remind myself of this a lot:

Photo Mar 25, 7 23 27 PMAnyways, this post is about energy.  On with it then:

ENERGY UNIT

So after discussing the various forms of energy and then giving the students some time to play with Phet’s energy skate park (with some guiding questions), we jumped right in to a lab.  The groups were each given an inclined plane and a cart.  We discussed what we could measure, and decided on height of the cart as the independent variable and velocity as the dependent variable.  The velocity should increase as the height increased right?  Let’s see!

Problem: Why are we interested in this relationship?

They didn’t have a clue, so the lab was a total disaster.  Some groups had issues collecting good data (that’s another problem for another blog post) and the rest had no idea why they had collected good data.  I think the reason is because they didn’t start the lab with a good question in mind.  What is the velocity vs height graph supposed to tell us about energy anyways?

Back to the Drawing Board

As I told my students today, it’s just not good science to run an experiment with no purpose in mind.  So where should we start then?  As always, the blogosphere has the answer.  I found this post by Kelly O’Shea.  Starting with pie charts? Sounds good to me.  Maybe that will help us gain a better understanding of energy transfers and then we can head back to the lab.

When I discussed this with my colleague who teaches the same class, he asked why I decided to go with pie charts at all (instead of going with bar graphs).  This comment from Kelly’s blog post shows others have had the same question:

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You can check the comments on Kelly’s blog post for her answer.  But my answer was that I didn’t really know where else to start and Kelly’s lesson seemed to make sense.  But after doing the activity, I feel like I’ve got a much better answer.  First, here’s what we decided on for our three pies of a ball falling from a really tall building (I used Empire State):

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The discussion that came up based on these pies centered around “what should the proportion of kinetic to potential energy be in the second pie?”.   That was a question that everyone seemed to want answered and no one had a solid answer to.  “Well,” I asked, “what would we need to know in order to determine that proportion?”

BAM!!

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They asked those questions right away with exactly zero prompting from me.  All I did was write them down.  Then, it was super simple for them to see that right there on the board staring them in the face were the exact questions we were attempting to answer with our inclined plane set ups.  Now we have a purpose for our experiment!! We are trying to answer the question, “In what proportion are the gravitational and kinetic energies of the ball when it’s about halfway down?” (or perhaps anywhere on the way down? :).

Other Good Arguments That Came Up

bunny board 2In this whiteboard, you get a visual representation of the students arguing about two things.  First, should there be thermal energy in the first pie?  Someone had drawn it in and then someone else in the group erased it (after a board meeting with another group).  The first student stood by his original pie though.  His reasoning was sound, he explained that thermal energy is internal energy and that the bunny must have some internal energy to begin with because it’s got a certain temperature.  Then argument two is in the third pie, where once again students dueled over whether a wind up toy would be left with some elastic potential energy even after it stopped moving.  It seems the side of “nope” won out on that one.  This next group agreed on the third pie, but not the first:

bunny boardSorry, purple expo marker! You will probably not last long in the hands of this group.

Quadratics and modeling

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So, I’ve been trying to teach Algebra this year by modeling. I needed a good way to build the quadratic function model, so I started (about two weeks ago) by having the students analyze a basketball shot video. They used Logger Pro and found that vertical position vs. horizontal position made a nice curve. Like this:

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But then I was lost as to how to get them to come up with the equation for the curve. They could get Logger Pro to tell them the equation, but that wouldn’t really help them. So, how can they come up with the equation when they’ve never seen one like this before?

I decided to take a different approach. We started by examining the function y=x^2 and then examining different ways to shift the function around (using Desmos). The handout I used had them graph different shifts (like shift y=x^2 up 3 units). Then I asked them to find the equation for that shift using Desmos. They had to figure out which piece of the equation to change.

Then after doing the activity (and a bunch of practice with it, making whiteboards, etc) they knew how to build a quadratic function from scratch. I think now it’s time to go back and find the equation of the original basketball shot we analyzed (BTW, I also teach these kids physics, so we’ve been studying why the ball behaves this way as well).

When I started writing this blog post, it was going to be about a failure to stick to teaching with modeling. As I typed it though, I realized that this was still a modeling unit. The kids recognize that the shape of the basketball graph is a symmetrical curve. We just didn’t have the appropriate skills yet to write the equation for the curve. Now we do, so it’s time to complete the model. I’ll let you know how it goes!

Modeling in Algebra I: An example

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The modeling method I’ve been using to teach physics has really made me think about way I teach Algebra.  Here’s a good example:

The topic:  Exponential Functions

The way I taught it last year:

Step 1: Introduce the topic with some notes and some example problems.  I used a PPT presentation to show the students the equation y = a*b^x and explain what all the parts mean.  Then we looked at some example problems.

Step 2: Explore the topic with an activity.  We did the classic M&M’s activity to explore a “real world” example of exponential growth.

Step 3: Practice.  We used problems from the text, most of which did not have a context.

Step 4: Assess.  We had a test after about a week or so.  Done.

The way I taught it this year (modeling):

Step 1:  Explore the topic starting with this clip from the movie Contagion.  I got the idea from Leslie Macfarlane on 101 Questions.  I used this handout with it.  We started by discussing what factors would effect the spread of a virus.  We came up with a bunch, but most importantly: the type of virus (R-0 in the video clip) and the number of people who currently carry it.  Here’s what their handouts looked like filled in:

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Step 2:  The kids figured out on their own (in groups) that the flu virus doubles each cycle and the smallpox virus quadruples each cycle.  They also were able (with a little help) to figure out that the cycle # (x) needed to be the exponent in the function.  Then, it wasn’t hard to get them to write the function like this:

(total # of people infected) = (starting number of people) * (mutiplier)^x

Step 3:  Generalize it!  We then discussed a general form for the equation.  They needed to make a connection between the R-0 number from the video and the multiplier they used to generate their charts.  As soon as I brought up the question, they noticed that it was just (1 + R).  At this point, my head almost exploded from awesomeness.  Now, they understood that the “multiplier” they used was given by one plus the growth rate.  How did that happen?!  I didn’t “teach” it to them!!  So now they’ve got y = a * b^x, they know what the “b” means, and since we graphed each initial scenario, it was simple for them to see that “a” is the y-intercept.

Step 4:  Expand the generalization to new scenarios (exponential decay).  We used this handout to figure out that if you are losing a certain percentage each cycle, then it’s just (1 – R) instead.

Step 5: Practice.  I gave them 3 or 4 scenarios of each type to practice.  For each scenario, they made a chart and a graph (using Desmos or by hand).  The answer to any questions asked was secondary to showing me that they could analyze the situation using the exponential model.

Step 6: Assess.  Two short multiple choice quizzes, some questions with contexts, some without.  Then an open ended task in which they had to explain to me the differences between two functions.  Retakes are available on all three assessments as needed.  They also take the assessments when they are ready.

Why is this new way better?

I have students now who are willing to try figuring things out on their own.  They see that it can be done.  Especially in Step 4, I saw a bunch of students working to find a function that defined an exponential decay scenario.  They didn’t say “Mr. Owen, what’s the formula?” or “I don’t know how to do this because you didn’t teach us this yet.”  Check out the example below.  This girl decided that (hmw * .10) = y  and that the next value in the table was given by (original amount – y).  I just had to suggest using substitution to combine those two expressions.  From there, it wasn’t too hard for her to see that to do that 10 times, you just raise the expression to the tenth power.

Photo Jan 10, 9 26 28 AMHere’s an example of a practice quiz done by a student (this one has no context):

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Algebra Physics Connections

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I just remembered today that I wrote a post about how the constant velocity lab in physics matched up nicely with the linear equations stuff in algebra, but then I never followed up on the results.  Here goes:

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I started the unit on systems of equations with this problem.  It was awesome that they took to it so quickly.  With very little help from me, each group came up with a testable solution to the problem.  Some used the chart, some took it a step further and graphed the lines.  “Interesting that they were able to come up with such specific solutions then” you might say.  Well, what they did was approximate the solution with the graph or chart.  Then some groups used guess and check to find the number, and some groups used technology to find the intersection (Desmos or TI-84).

Next time, I need to make sure to let them test their solution with the actual cars.  That was the plan, but I never went back and did it, d’oh!

Anyways, in the past the sequence of learning happened this way:

1. Learn different techniques to solve a system of equations.  Graphing first, then substitution, then elimination.

2. Practice those techniques on a bunch of problems.

3. Apply the techniques to a “real world” problem.

Even as I type this, it seems like a logical way to do it.  You can’t apply a skill until you master the skill right?  I think that’s why this continues to be the way that so many math teachers teach this unit.  But my experience starting with the real world context showed me that they can and will find solutions to problems even if they didn’t start out with all the right skills.  There seemed to be two types of students in general:

1. Students who understood the problem already and came up with a solution fairly quickly.  These students should be allowed to use their class time for extending the problem into different contexts and doing whatever assessment I’ve planned for this skill so that they can move on or tutor other students.

2. Students who don’t really understand the problem or how to solve it.  I think that the standard way of teaching this unit tries to cater to these kids.  But this way is better for them too.  First, they benefit from just watching a student (#1) solve the problem (as they work in groups to create the whiteboards).  They also are (hopefully) intrinsically motivated to learn the skills because they see how they can be used to solve the problem. These students can use their class time learning the basic skills of using an equation to generate an input/output table, and then plotting those points to make a graph.

Note:  They were the ones that wrote those equations on the boards we started with.  In physics class (also with me), they started with just the cars, some plastic markers, and some questions.  The ladder of abstraction was climbed from there.

This ActivePrompt Thing is Pretty Sweet

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Holy Moly!!  I haven’t used that phrase in a while, but it seems to be making a comeback through the Math Twitterverse.  I’m on board.

It perfectly describes my thoughts after my first block classes Thursday and Friday this week.  We used a new app from Riley Lark (the ActiveGrade dude) called ActivePrompt (which is apparently a working title, real name TBA) and it works like this.

Here’s what we did with it:

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I had absolutely no idea what to expect from this.  It was one of those lessons that could be a rock star moment or a complete failure.  I have awesome kids, so my hopes were high.  Here’s how it went:

Both classes started by making two vertical lines.  They split themselves into two groups and each group decided on a “column” (the word they used at the time).  My favorite question from this part (directed at another student): “Does it matter how far down I go?”  That question led them to make sure that no ones dots were on the same point (the “rows” had to be different).  After they got it, I asked them how they did it.  Here’s what they said:

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Since that was too easy, I next told them to create two more parallel lines that were neither vertical nor horizontal.  They pretty quickly realized that it was all about slope.  Their idea was to assign everyone a number and then have them go up and over by that amount (1,1) (2,2) (3,3).  The real genius idea (only one class did it this way) was to translate that first line up using the rule (x , y) –> (x , y+1).  Yeah, ok they didn’t write it that way, but that’s what they did, we’ll talk about notation later.  Also interesting was how they decided on an origin without even really discussing it.  It just sort of happened.  They started talking about going “up and over”  and pointing at each others screens and there you go, an origin is born.  One class chose the top left and the other chose the bottom left.  It was actually hard to get them to realize that they had even done this task of choosing an origin.  That’s why it’s last on our list:

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This video clip shows one student explaining the slope concept to another.  He can’t do it without a starting point!!

One of the classes realized that it would have been quicker to simply give each group an equation (same slope different y-intercept) and then generate the points to graph the equation by giving each group member a number.  HA!!  BUSTED!!  Inventing algebra to solve a problem that doesn’t necessarily require it, huh?!  Who are you and what have you done with my 9th graders?!  Here’s how they explained it to me:

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My favorite thing about this activity was that there were so few rules.  Just make parallel lines.  But they ended up having real discussions about how changing the slope or the y-intercept affects the way the graph looks.  Our current learning targets are: “Graph an equation using the slope and y-intercept” and “Describe the effects of changing m or b on the graph of y=mx+b”.  I could’ve had them watch me do a few examples (in class or in a video) and then practice similar problems.  But it seems like that’s asking them to start with the method and end with the problem.  Shouldn’t the method exist only because of the problem??  And yes, I know this one’s not “real world”, in the sense that it didn’t have a context like “the city planners need to make two parallel roads to connect blah blah blah”, but isn’t that context unnecessary here?  As long as the problem comes before the solution, it shouldn’t matter how “real world” it is.

As an extension I had them try to make a star shape with their points.  We only had time for this in one of the classes.  They decided which points would best describe the star, then made a sketch to help everyone place their dots.  It was interesting that they decided to base their dot locations relative to one of the star points instead of the origin they had previously used.  Anyway, we kinda ran out of time, so it didn’t come out looking great, but their plan was good and would’ve worked.  Here’s what their sketch looked like:

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How I know it worked:

Everyone was getting it.  Not just the genius group that came up with the idea to write the equations first, but everyone.  They all were engaged to some degree in the task and could see how the location of their individual dot was decided upon and how it fit with the other dots.  One student in particular had been struggling with graphing in a major way.  I was kinda stuck on how to help her.  But after participating in the activity, using a starting point and then going “up and over”, something clicked and she was able to see that the y-intercept was the starting point, and the slope was the “up and over”.  Just telling her that wasn’t making her understand it, she had to do it.  I used this Khan exercise as a quick formative assessment, and she got it right away. BAM!!

One final thought: technology was integral to this activity.  Without the laptops (or some other connected device), I wouldn’t have been able to use ActivePrompt.  And the Khan exercise, while not essential, did make a great formative assessment (all my kids have accounts, so I can see how they did on the exercise instantly).  For some other ways to use ActivePrompt, as described by the creator, check out this post on his blog.