Scatter Plots and Trends

I had high hopes for today’s Algebra 1A lesson on scatter plots and introduction to trends and correlation. I was going to use the Movie Compatibility idea found here, but I chose to use the top 10 songs on the Billboard chart. The students¬†and I have had lots of discussions about music. They always want to know about what I think of some song I’ve never heard of and I in turn put on my music. I’m an old school jazz fan, so I get lots of groans. I was looking forward to the delight when I had an assignment with the cool kid new music. Dun dun dun…. that was my mistake. Apparently the Billboard Top 10 is a terrible representation of the teenagers I work with. I assumed since I hadn’t heard of the songs, my students must have ūüôā One student had heard only 2 of the songs, and I think the most was someone who had heard of 7. We had to scrap that plan and move on.

Instead, we did a sorting activity where students took two variables at a time and decided whether they would have positive, negative, or no correlation. I found the activity here, and since it was an in the middle of class change, I didn’t update or change it. I had the students match the graphs first and then share ideas of things that would fit in each category. ¬†Then we added the notes in blue. then in pairs or alone, the cut out and sorted the different scenarios and taped them into the correct flap. One of the better debates was around age and height.

My most common line of the day was “If you can defend it…” I like that even in a card sort, there is not necessarily a right answer, and a seemingly true answer is no good if you can’t justify reasoning. I’m going to look for some crazy correlations to share tomorrow. We touched on correlation/causation today, but I think I can drill home the need for proof with a few example like: “The per capita consumption of cheese, people who die by being tangled in bed sheets”¬†.

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Side Note: I tasked the students with going home and listening to the 10 songs that we tried to rank today so that they could rank them tomorrow. I like the idea of generating data, especially without the typical measuring labs. I tried to play the songs in class today, but they failed the internet safety filter. I never assign homework, so I scared a few today when I told them they had homework. When they figured out it was listening to the songs, they were excited though. Hopefully tomorrow should be smoother.

 

MSP – Attending to Precision with WODB

My main focus in class is on the Math Practice Standards. I try to model and expect students to live up to them in everything we do, but once a week we do a task to specifically focus on one of them. We teach content in that manner, so I think the MPSs deserve equal if not more  careful consideration.

Today’s Algebra A class (first semester algebra, but it offered every semester) was looking at MPS #6 : Attend to Precision. I wanted them to think about how they communicate their understanding with clear definitions and labeling work.

Thanks to the MTBoS, I have been frequenting the website Which One Doesn’t Belong? I chose shape #4 by Chris Hunter for the activity. Each student got a printed copy of the picture and a letter A-D as they walked in the door. They were tasked with finding out why that letter didn’t belong and then writing a definition for the group containing the other three. Next, they each drew another shape that would be in the group and one that would not be in the group. This took about 10-15 minutes.

We got back together as a class. Each student would read his or her definition and the rest of the students had to say which shape didn’t belong according to that definition. Using the feedback from the other students and based on the clarity and success of choosing the right shape, the writer was asked to redraft the definition.

As a close out, they were each tasked with deigning their own WODB and then writing precise definitions for each of the four groups. We’ll trade student generated tasks next time and see if the original writer’s definitions are similar to the one from student who is doing the task.

Exponential Equations Take 3

We finished up our M&M Lab today with the help of Desmos. We don’t have many graphing calculators or other technology in the classroom, but I have an iPad and we can project its screen up on the wall.

If you are not using Desmos, you should check it out. All we did is use it for a scatter plot and the regression line (which any graphing calculator can do) but there is so much more out there that it can do! Start here and explore.

But even not using the tech to its potential, the students still got a kick out of entering all their data and coming up with a class model. It came out with a growth factor of 1.42 for the growth function and 0.55 for the decay function. Students compared there personal model with the class model and then made predictions for future trials and well as explained the discrepancy between the model and the experiment.

Then we moved on to determining linear versus exponential patterns. Each student had to make up a table of values and present it. The rest of the class would call out linear or exponential and the “teacher” would choose one of them to write the equation. We’ll cement that as a warm up tomorrow and then move on to bigger things!

Modeling Exponential Equations Continued

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

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

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 Р4+ 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

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.