Modeling Exponential Equations Continued

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Yesterday I talked about our introduction to exponential equations, which had the students create patterns and find the connections between the different representations of the pattern. The next lesson was the M&M lab. I have a problem which seems exactly opposite of most teachers, but I have too few students in this class. There are about 15 enrolled, but day to day I have often have about 3 show up. (I teach at a public, alternative school serving at risk youth. We have lots of challenges, but attendance is up there with the best of them.) I had them each do the task alone so we’d have a few different models to talk about. The low numbers are great for self guided learning and lots of hands on time, but they really kill some cool group projects that benefit from lots of voices. I’ve had to rethink a lot of my favorite lessons to try and make up for the low numbers. Again, this is the first year I find myself wishing for rich student mistakes.

Even though the students knew we were studying exponential equations, they all seemed delightfully surprised when the scatter plot had the same shape they’d found the day before. We did one round of growth (Start with 2, add one for every M) and one round of decay (Start with total, take out every M) per student.

S1: It looks like the graph from yesterday, but the numbers don’t have a pattern.

S2: I bet they do.

S1: Then why are our numbers not the same?

Enter, find a pattern time! As well as experimental versus theoretical models. Students started to talk about how they could check for a pattern.  They all wanted to jump to slope, but at least one remembered the division pattern in the table from yesterday and they were off to find the growth rates.

Today we are going to pick up where we left off. They each have a growth rate and a decay rate. Warm up will be answering some questions about how there graphs behave as well as writing an equation for their model. Then we are going to use Desmos to input all our points to find a group pattern and model equation o see how it compares with the individual models and the theoretical model.

Resources: I used the first page of a M&M lab I found here. I only used the first page because I didn’t want to give the students the percent change formula to calculate the growth factor. We had been doing patterns (Multiply by 3 for example) the day before so I wanted them to use that knowledge to come up with a way to find the growth factor. That way there is no equation to memorize and growth and decay are found the same way as opposed to remembering 1+r and 1-r. The students used the first sheet to create there own recording table and graph for the decay part of the experiment. And our class discussion from yesterday was my guide for their warm up reflections today.

Side Note: Spring colored M&Ms are pretty, but they are really hard to read the Ms on. Especially the light yellow ones. Either that, or my eyes are old. But… they also added to our discussion on experimental trials. Did all the M&Ms really have an M on one side or might some have rubbed off? Would that skew our models at all? Could we have skipped some of the lighter ones? Where else does error show up?

Exponential Representations

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Algebra 1B is working on exponential equations. We introduced the unit with the idea of paper folding to the moon. This is a well used example, but I used the TedEd lesson here for inspiration. I didn’t show the video or anything, we just started to compare our numbers to real life objects so they could conceptualize the numbers.

Students folded paper in half to create a number of folds vs layer chart and make predictions about folds they couldn’t complete. Since the model only work for awhile, we talked about how to move from a model using the pattern. I also challenged them to break the world record , but no luck there.

After discussing how the record holder managed to get to 12, I slid this in my “also try with geometry” file since it seems like a lot of deep math to explore with similarity and different sized papers versus folding ability.

I asked the students come up with there own patterns that followed a similar rule.  They drew boxes on graph paper and nicely color coded the information we saw repeated in the pattern, table and graph. I “accidentally” read a students paper backwards and my graph decayed! So they all made new decaying patterns too. Spent some time discussing the surprise that some students felt with the shape of the graph and how quickly it was off the paper. This was also a good spot to revisit scales on graphs.

Anyhow, a few examples:

  

Note for Self (or Others): Since we didn’t do any transformations, the reflection/write ups seemed to implied an exponential could never cross the x-axis as opposed to the actual idea about their being a limit. We talked about limits, and we’ll revisit when I we start doing the transformations, but I wished I had challenged their reflections right there. Even without going into a transformation, I should have pushed them to explain it not as “can’t cross” but instead the idea of the value approaching zero and why it is the case.

Filling the Void

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Time to meet all the standards seems fleeting, yet the students also run into empty time during the day. I don’t like empty time. One example of this would be teacher absences. My site has 4 teachers total, so if one is out and a sub doesn’t come we all absorb the extra students. Since I’m busy with my regularly scheduled math class, I like to have challenges for the interlopers to work on. My goal is for these to build number sense or estimation skills, any other vital skill we just don’t hit often enough in the curriculum. Since all of the said students also have me for math sometime, I have a semester long tracking sheet where students log their attempts at said challenges. There goal is to make good headway on at least 15 over the course of the semester. Any empty time can be a challenge time. I have a collection of them so they can work anytime they have spare moments.

Today is one of those days. A teacher is out on a field trip so I have extras. So I threw up this:

1+ 23 – 4+ 56 + 7 + 8 + 9 = 100 . There are other representations of 100 with the 9 digits in the right order and math operations in between. Find some.

I don’t know that this is an especially exciting problem, but the students are working and talking about math. And I can still run my class. I suppose that’s a win for today. They show me the work and write about their thoughts. I make them do that often, write about math. I think its something we often don’t ask enough as math teachers.

Any resource suggestions for self guided math play? I rely on Estimation 180 a lot. Doing a full series would count as a challenge. Five triangles is another resource I use. But I would love any and all other great math resources.

Conditional Statements

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As math blog people may already know, Sam Shah has a great lesson up here to get students exploring and talking about conditional statements are truth values without just naming and assigning them. We tackled a modified version of the second part of the lesson in my Geo B class last week.

Modification includes: Mixing up the Truth Values. The original let students know the given statement was true. I took away that requirement and purposely added in some false ones. I also added in some that would be true regardless and false regardless. The result was we had posters with all the following patterns: TFFT, FTTF, TTTT, FFFF. This allowed us to have a deeper conversation on what finding truth value patterns might mean.

For our Gallery Walk, Students had a few minutes to walk around and read/think about the posters. Then I gave each student 5 (or more) post-its and they had to leave comments, corrections, or insights. When they finished, I handed them a recap sheet with 2 questions about the posters and a final Statement to practice the conditionals. We meant to have a whole class discussion afterwards, but this took the whole period. We started the next day with the discussion. This would have been better if it could have been same day, but it was still powerful. I got to play pestering questioner a lot. I like this. Asking students “Why?” and to defend themselves is fun. Its even more fun when they start doing it automatically.

Key Insights: Our students are amazing people. Even the ones that think they don’t know math have such powerful thoughts when we listen. So LISTEN! I wasn’t expecting the post-its to capture all of their thoughts, but just sitting back and listening to them talk among themselves and debate the post-its led to the deepest insights. I jotted notes, but I wish I had captured audio of the class.

Math Games

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I was listening to a great episode of Reply All (a podcast from Gimlet Media) about Swatting last night and in doing so spent a lot of timing trying to understand the world of professional video game playing. Then I turned to the real experts, my high schoolers. I had them explain to me the world of online gaming. Talk about turning on a faucet, they were all about talking strategy with me. Enter Play With Your Math and Space Race. They were in a competitive mood. We had to have a time out when the battle got intense — math can be brutal! But they did it. They wrote their reflections on the game and how math can help them win and I had to throw them out when class ended today. A few are still here. Apparently math is awesome.

This picture looks like a scribble, but it was the result of a math fight. What is allowed? Can you cheat at math? 



Math Gets Another Blogger

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I’m going to throw my hat into the math blogging ring. I’ve been a MTBoS and TMC lurker for awhile, and figured it was time to try and give back something. No guarantees it will be anything good! But welcome to the wild ride.

I am a math teacher at a public, alternative high school school. I’m the only math teacher at my site, so I get to teach it all. This is good. The bad? I have 7 different preps!