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:
And my Algebra 2 INB posts here:
I am starting Algebra 1 again from the beginning as a semester class, so you can find my revised posts for that here:
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.
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.
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.