Exponential Functions

After our Linear Function unit (which you can find here and here), we jumped right into Exponential functions.  

Our beautiful tabs!

Our learning targets

We started by describing exponential functions algebraically, graphically, and numerically.  Here is the foldable we did.  It is very similar to the one we did with linear functions:

Foldable closed

Foldable opened

Then we did a card sort.  We did exponential vs. not exponential.

I asked them to write a reason for why each card was on the side it was on.  Also after our notes on the base and y-intercept, we went back and found the base and y-intercept for each of our exponential functions.

Next we worked on finding the base and the y-intercept.  Notes with a graphic organizer, which is very similar to what we did with linear functions:

I gave them a practice worksheet.  I thought I was really clever and designed it so that they could easily complete, tape in their notebooks, and view later...

...however I then proceeded to copy it upside-down.  :S

Then we graphed exponential functions.  This is where it because crucial to have studied negative exponents prior to this unit.

I had to do another practice worksheet like the one before so that I could prove that I was capable of copying correctly.

Practice WS outside

Practice WS inside

Our last learning target was on writing equations to model exponential situations.  I gave minimal notes that tied to our learning of linear functions (the y-intercept is the start) and the learning we had done with exponential functions.  We had been wondering and noticing that some exponential functions increase and some decrease, but today I FINALLY made it clear what causes that difference.

I did a stations activity with them.  I had a handout for students to use to facilitate the process and our discussion.  Basically I just had students go to nine stations and write the start and the change.  Then we had a class discussion over the equations.  They seemed to think it was pretty easy, which I didn't really expect...I guess I'm just an amazing teacher :)

Handout that they used to go from station to station

NOTE:  To view files click the appropriate link.  It will open in Google drive; it will not show correctly in drive.  Choose to download file.  It will download the word doc/ppt for you with all of the correct formatting.

That's all I have for now.  I don't feel much like reflecting on the unit right now, so I guess this post is finished :)  Sequences are coming up next!  (Well, actually we've started them.  Post to come after the 20th!)

-Kathryn

Units for Intervention Class

I have been teaching an additional intervention class all year.  This is part of the Tiered Algebra project that our local AEA has been working with schools in our area to implement.  We had been given some guidelines, but not much, for what to do in that time.

• Connect it to what is happening in the Algebra class
• Don't just reteach/review what you are doing in Algebra
• Don't just have "do your Algebra homework" time
• Do more hands-on stuff
• Do more explicit instruction
• Use appropriate scaffolding
• And others, but that's what I could think of off the top of my head

So here I am, a general Algebra teacher, with very little understanding of how to differentiate in my regular classroom, now seeing 24 of my students twice a day, and doing my very best to make it worthwhile.  But I've been mostly clueless.  I tried to do a lot of pre-intervention with them... (see Micheal Pershan's argument for this type of intervention here) which looked like covering the coordinate plane and plotting points before we graphed linear functions in Algebra, for example.  But I still felt like my students weren't able to use much of what we were doing in a way that truly benefited them in the regular classroom.  Not that I actually had any real way of measuring it...

Then the AEA shared with us about a presentation they heard from a school that has a similar model in 8th grade.  They decided that vocabulary was really important, because IF THEY DON'T KNOW THE VOCABULARY, THEY CANNOT ACCESS THE LEARNING IN THE GENERAL CLASSROOM.  This was an argument I had never heard before.  Now I always knew that vocabulary was important.  And I teach it...sort of, but I've never really emphasized it.  This has changed that.  Along with other things that came from this summary of the presentation, I have changed some things, and I think it is for the better.

I now have week-long units.  These units focus on a particular skill and the vocabulary associated with it.  For example we studied exponents prior to working with exponential functions.  We had vocabulary:  exponent, power, base, exponential, reciprocal, expanded form.  We studied the vocabulary each day in different ways:  matching; create your own example; which could be used for x, which would be used for y; etc.  We also had scaffolded lessons on simplifying exponents.  We started with whole number exponents with only positive numbers.  Then we discussed things such as -2^4 vs. (-2)^4.  We simplified expressions using the order of operations (with exponents).  And finally we saw negative exponents.

I was actually able to see my students apply what we were learning in class (no actual way of measuring other than my observations).  I depended on them to lead the other students when we saw negative exponents.  This felt amazing.  I am excited to continue using this model, although I am very frustrated that it took until November for me to find something this useful for interventions.  Hopefully as I continue to apply it I will see more improvement from these students.

But I am also frustrated by the fact that it is still group all of my intervention students together.  What if student a needs this and student b needs that?  How do I make that work?  How do I know what they need?

Just for an FYI here is an outline of the exponent unit:

-Kathryn