Algebra 2 Unit 5 Interactive Notebooks: Polynomial Functions

This year I’ve committed to posting each unit of both my Algebra 1 and Algebra 2 INBs.

My district is moving to a standards based curriculum, and has identified priority standards for every course. These are the standards we are required to address and assess our students over, so they pretty much form our units.

You can find my Algebra 1 (year long class) INB posts here:

Unit 1 | Unit 2 | Unit 3 | Unit 4

And my Algebra 2 INB posts here:

Unit 1 | Unit 2 | Unit 3 | Unit 4

I am starting Algebra 1 again from the beginning as a semester class, so you can find my revised posts for that here:

Unit 1

 

Our fifth standard for Algebra 2 is F.IF.7c:

Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

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Ignore the proficiency log side of the index file – I decided about two days into this unit that I HATED the way I had set this up and wanted a new one, which I am now using successfully, so you’ll see more about that in my next posts! Also ignore the fact that skill 3 has been pasted in in the picture above – I realized we needed that skill to say something totally different from what it originally read after we had already put them in our notebooks. The change is in the uploaded files, though 🙂

Skill 1: I can name a polynomial based on its degree and number of terms

I got the naming polynomials chart from Sarah Carter here and I really like it. It’s to the point and does its job! I had a total mind blank moment when I was filling my copy out, as you can see in the picture, because I forgot about degree 0. I did not have this mind blank during class, thankfully! We also use a Frayer model as she does in that post to talk about the definition of a polynomial.

I throw in adding and subtracting polynomials here because it is actually an Algebra 1 standard, but I find that my Algebra 2 students need a refresher on combining like terms and putting polynomials into standard form before we start working with graphs and other manipulations of polynomials in our next few units. This also let us get some practice naming the resulting polynomials!

Skill 2: I can graph a polynomial function with technology and identify its key features

On the outside of this you see definitions of what each of these key features is – where do we look for them? My students already know where to find the intercepts, but the extrema and end behavior are new ideas to them. When you open up the flaps, you see instructions on how to find each one using a TI – 83 plus graphing calculator, which is what we have a class set of.  I need to find some better way to phrase the instructions on how to place the “left bound” and “right bound” when finding the x-intercepts and the extrema. I have some ideas from working with my students when we were practicing, but this is a tricky thing to communicate without individually showing each student!

Then I put two practice polynomials in the center, one with the graph given so they could make sure it looked okay before finding the key features, and one with just the equation. These two examples took us an entire 47 minute class period to discuss and get through together, so I’m glad I only put 2!

Skill 3: I can match a polynomial function to its graph and identify its increasing/decreasing intervals

I decided to separate increasing and decreasing intervals from the other key features because the other four that students have to be able to find (extrema, x-intercepts, y-intercept, and end behavior) all essentially involve looking in one single place on a graph. The increasing and decreasing are intervals, so they’re a bit different. I’m really glad I separated these this year because my students understood all of the key features a lot better than last year’s Algebra 2, when I tried to do all of those at the same time.

The inside of this needs to be edited to align more with our assessments. The main skill needed in our assessments is to be able to identify the minimum degree of the polynomial by visually inspecting the graph. I will change this page next year to just include that skill, because any other matching can be done by just graphing the equation in a calculator and matching the image that results.

I threw in two more examples that asked to find all possible key features, which gave us some good practice as a class manipulating the calculators and how to list each feature.

Skill 4: I can write a polynomial function based on its graph

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My students LOVED doing this. I was so surprised, but they kept asking me if we could do more practice with this skill because they just wanted more of this! That made it really fun to teach. I like the structure I used for these notes as well, I think it was really clear to students.

After we did these for practice, they played Match My Polynomial on Desmos, which they loved and was good practice with immediate feedback!

Skill 5: I can write a polynomial function based on a list of its key features

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This essentially is a preview of a skill we will go deeper into about two standards from now. It leads somewhat naturally from the previous skill and our sequence guide suggests that we include it here, but it is also included in that other standard in the future.  I think that I plan to just let it be in that other standard the next time I teach this and conclude this unit after skill 4, since this unit is so focused on the graphs of polynomials and this skill doesn’t particularly fit there.

 

You can find the files I created for this unit here, in Publisher and PDF versions. Any files that were not my own are linked within this post 🙂

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Algebra 1 Unit 4 Interactive Notebooks: Exponential Functions

This year I’ve committed to posting each unit of both my Algebra 1 and Algebra 2 INBs.

My district is moving to a standards based curriculum, and has identified priority standards for every course. These are the standards we are required to address and assess our students over, so they pretty much form our units.

You can find my Algebra 1 (year long class) INB posts here:

Unit 1 | Unit 2 | Unit 3

And my Algebra 2 INB posts here:

Unit 1 | Unit 2 | Unit 3 | Unit 4

I am starting Algebra 1 again from the beginning as a semester class, so you can find my revised posts for that here:

Unit 1

 

Our 4th Algebra 1 Standard is a repeat of our second (A.CED.2), but this time with the emphasis on exponential functions instead of linear functions.

A.CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Technically our curriculum has the standards involving exponent and radical properties before this one, but I decided to make the switch and put this first because 1) I think this content is accessible without understanding all of the exponent properties and gives students some exposure with calculating exponents on a calculator to base some of those properties on, 2) I think this naturally leads from what we’ve just been working on with linear functions instead of seeming totally foreign and different, and 3) I think the end of the exponent and radicals standard, which involves simplifying radicals, leads perfectly into quadratic functions, which comes after that.

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I have been having an internal crisis about the left hand/back side of my index pages (which are adapted from Sarah Carter‘s. You can find more info about mine in my first Algebra 1 post from this year.) I wanted to give students a place to track their progress and to record something when we do grade checks weekly. I used to hand students a separate log for this purpose each week where they recorded their grade, whether they were happy with it, and an action they were going to take that week to improve their grade, and then I would collect and hand them back each week. Since moving to standards based instruction and grading this year, I wanted this process to focus less on their end grade and more on what they understand (which is the whole point of SBG itself). The version of the proficiency log in this index page I do NOT like, and I quickly realized it was not what I wanted. The next unit, that we are currently in, already has a different proficiency log page. I think I like it better but want to tweak the way I’m using it a little bit. I’m chalking the rest of this school year up to experimentation with this proficiency log and hoping to settle on something I’m happy with by the start of next school year so I can try being consistent throughout the whole year!

Anyways, all of that is to say that I no longer like this proficiency log page and you should ignore it 🙂

Skill 1: I can identify when a relationship is exponential

I started out this unit by reminding students that all of the functions we had been working with so far this school year have been linear and that we would now be looking at a new type of function. I gave them a set of 15 functions – 5 graphs, 5 tables, and 5 equations, and asked them to categorize them however they felt appropriate BESIDES grouping them as tables, graphs, and equations. They were prompted to look for similarities and differences. Some students did not do well with this little instruction – they really wanted to know how many groups there should be and how many in each group and what I would call the groups!

We discussed how students sorted them – many students just grouped them into linear and nonlinear (there were some functions that were not linear or exponential), and many students grouped the equations into the ones that were solved for y and the ones that were not. They noticed a lot of things about the representations that I had not planned, which was a good discussion!

After our discussion, we reviewed what they remembered about linear functions, which was a lot! We formalized the words and gave some examples along the left side of their foldable. As we did this, I thought I was very clear about where the students should be writing each piece of information and each example, but apparently I was not. When I next teach this, I will be splitting up the inside of the foldable into boxes for them to write the information in, and I will probably type something small in each corner saying if the example or the information should go there. My students ended up with their linear and exponential information mixed in together, which made this foldable basically useless for them to look back on! Bummer. Anyways, after we reviewed linear, I introduced the definition of exponential and how to identify exponential graphs, tables, and equations.

Then we went back to their 15 representations and I asked them to split them into 3 groups: linear, exponential, and neither. We recycled the “neither” group and glued the linear and exponential examples onto opposite sides of a page in their notebook. This was a good way to introduce this because several of my students were super proud that they had sorted the functions “correctly” the first time, and it gave them confidence about this new exponential thing!

Skill 2: I can graph an exponential function

I printed this on one-sided graph paper so that the inside page would already have grid lines and I wouldn’t have to worry about printed graphs not copying well, which is a frequent issue with our copier. I’ve been using this trick a lot this year, which you guys seemed to love when I tweeted about it! A bonus of this trick is that it forces students to draw their own x and y axes, a skill which I was previously unaware that they did not possess and now realize that it is important to talk about!

We made x/y tables in order to graph ordered pairs here, and I made sure to include a linear graph so that they wouldn’t get complacent with assuming a certain pattern. We also discussed graphing the equations using a graphing calculator to check the work they did by hand – I did have one student who just used their calculator and then vaguely sketched the graph on their paper throughout the unit, so maybe I need to place more emphasis on clearly plotting at least 3 specific points in the future!

Skill 3: I can write an equation to represent an exponential function

This page was short, sweet, and effective! (It has several examples, including one linear example, inside). We made notes of transforming patterns that appear to be division into multiplication by the reciprocal throughout the examples, and how to “backtrack” to find the y-intercept if it was not listed. I threw one graph into the examples, even though writing exponential equations from given graphs is a skill in our Algebra 2 curriculum, just so that they would see the possibility.

After we took these notes, we did a Match My Exponential activity on Desmos, which I do for linear, exponential, quadratic AND polynomial functions when we first start writing them because it gives instant feedback and allows students to self correct their mistakes.

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Skill 4: I can write an equation to represent exponential growth and decay

This is almost the same set of notes I use for the same skill in Algebra 2, which I should have realized was not a good idea. We took a test after practicing this skill and it was not good at all…so I told my students we would pretend it didn’t happen and go back and practice more and retake it. I think that there are just not enough examples here, especially of finding that growth or decay rate from the scenarios (instead of just “12% growth). So, more examples next time! I also think I need to add more structured notes on graphing growth and decay functions – in my head, it was the same thing as graphing an exponential function like y=3(2)^x…but it was not the same in my students’ heads! Large numbers that meant they had to decide a scale to use and tiny growth rates that meant their curves almost looked straight threw them super off their game.

When we went back into our extra practice before retaking the test, I also had them do a World Population Growth project, which I blogged about here. I think this really, really helped them get experience writing the equations, evaluating them, and graphing them. I’ll plan to do this project before giving a test next time!

All files can be found here, in PDF and Publisher (or Word) formats.

World Population Project (Algebra 1 Exponential Functions Resources)

My students recently took their end of unit test over exponential functions (writing them and graphing them)…and it was a disaster. They all went into a panic during the test, and their results were about as bad as they feared they would be. I clearly had not prepared them well enough.

I went into a panic of my own, decided that I could not put that test in the gradebook, went back to the drawing board, and came up with some review activities for us to do over the next few days. I also decided that engaging with real data in a way that would be pretty difficult would potentially prepare them to engage with the more sanitized numbers and scenarios of test problems more confidently. I employ this strategy frequently – putting problems in notes or practice that are more complex than any that will be on their tests, so that then the tests seem easy in comparison.

“What’s exponential in real life?” I thought to myself. “And what will I be able to get data for that is readily available and I can put together…in the next hour?” (I was trying to get something that we could use the next day in class). Many of our example problems involved populations, so my brain came up with world populations! “I can probably get a list of the populations of every country,” I thought. I searched country population growth rates and was led to this Wikipedia article (I know, Wikipedia as a source, cringe, but I was on a time crunch and THEIR data sources seemed pretty trustworthy). I copied the growth rates table into a spreadsheet, and then decided that I would only use the earliest year from each source, so deleted two of the columns.

Then I set out to find populations of every country for 2009, 2010, and 2012, the years the data came from. This was more difficult than I expected it to be, and I do not remember where I finally ended up finding the source I used that gave me downloadable or copyable data to put into my spreadsheet. I put this information into another tab on my spreadsheet, but I wish I had put each year into a separate tab because so many of my students ended up being confused by the fact that there weren’t the exact same countries on each list and so the countries didn’t line up in the same row for all 3 years. Some of them ended up with the population for the wrong country for one year and not realizing until they went to make their predictions and they were way different! (At least they realized it then, that’s great analyzing your solution for reasonableness!)

I gave the students a template to organize their information on. I chose to do this mostly because I only wanted to spend two class periods on this project, and if I had just given them a checklist of information to include and had them make their own posters, it would have extended the project to at least a week, and included so many “where should I write this?” “What should I title this section on the poster?” “Should these things go together?” questions, when I wanted them to focus on the math.

I gave them access to the spreadsheet via our Google Classroom, and told them they could select any country they wanted, as long as the dataset listed a growth rate for each year and a population for each year. This mainly excluded the tiny little countries, territories owned by other countries, and countries that stopped existing sometime during that time period. Some of my students found the weirdest name they could find, and some of them just went with Canada “because it was easy”. Several students chose Japan because my school has been really into anime and Yu-Gi-Oh!

After they chose a country, they started to fill the information in on their template. I got to conference with each student as they worked and help them transform the growth rate into a multiplier for their equation, help them set up some tables to find points for graphing, and help them make predictions using their equations. I think my written instructions were too wordy for my students’ taste, so I’d like to revise them in the future to be more bullet point-esque. I’m a very wordy person, as I’ve realized in grad school when I spend half of my time editing projects to be shorter so they fit the length requirements! (And also when I read my blog posts. I really need to figure out how to say things using fewer words)

I think my students got a lot of great practice writing and graphing exponential growth (and occasionally decay, some countries shrank in population!) functions through this project. They were also invested in their chosen countries – when it came time to think about why their predictions didn’t match the actual 2017 population of the country, so many of them came up with specific reasons that tied to their specific country and its culture. This could be a really cool project to tie in with your social studies teacher in that regard – if they could have researched more about their countries at the same time or beforehand, there could have been so much interesting knowledge for them to bring to their analysis of the population numbers!

I hung all of their projects on the glass wall that separates my classroom from the hallway, and one of my students excitedly showed his project and a bunch of his friends’ projects to his parents as they came in for conferences last week. I’m happy they took a lot of interest in this project, but I’m even happier that when they retook the test, most of them showed proficiency!

Here are all the resources I used for this project. I also found this site to be a good place to direct students to find current country populations. It still had 2017 populations when we did this project, but now appears to show 2018 populations, so you could adjust the project template to reflect that it is now officially 2018!