Sequences

We wrapped up second semester with a unit on sequences. I like doing sequences after studying linear and exponential functions because I feel it gives an opportunity to compare and contrast linear and exponential situation.  And when we first distinguish between arithmetic and geometric sequences, students are fairly quick to make the connection to linear and exponential functions. (yay!)

Before we even do anything officially sequence related, I usually give my students four sequences (one adding, one subtracting, one multiplying, and one dividing).  I give the first four terms and ask them to find the pattern, the starting point, and the next three terms.  This gives them a little time to transition to new material that appears easy.

This time we tried a table of contents.  (Which I got from Sarah at math=love.  You can find it here.)  I liked it a lot because I used it to makes students think (just a little) about what was on each page.

Table of Contents

Definitions of sequence, arithmetic, and geometric.  I wanted to be good about including vocabulary, but I still don't think this was enough.  I needed to also show how the words were related to each other, perhaps a graphic organizer was in order...

Definitions

I displayed a bunch of sequences using my projector and had students sort them on the right side of their notebook to practice their understanding of the definitions.  Also to show that it isn't always easy to tell...sometimes they will need to subtract and divide to check.

Sort these

Some students wanted to label this page arithmetic, geometric, and sequence...that was a little concerning and shows that I did indeed need to more emphasize the relationship between the words I was teaching.

Next we discussed writing formulas for the sequences.  Since I had asked about starting point and pattern as we went, I thought recursive was a natural follow.  Instead of using all the nasty notation (a_n, a_1, etc.), I talked with my area education association and someone recommended this now = /next = notation.  

Note:  I liked it a lot, but students wanted to just put the "now" in for the "now" in the second equation so that there would only be one equation.  I think it might have been better to use start = /next = now/.  That might help.

Notes on recursive formulas

To practice I displayed a bunch of sequences and asked students to write the recursive formula for 8 of the 12.

Practice Problems to be displayed

My notebook had the answers :)

Next we looked at writing formulas explicitly.  I tied this into the equations for linear and exponential functions.  We did a mini-breakdown of the equations in class and came up with:

• Arithmetic:  a = d*n + start
• Geometric:  a = start*r^n

Those would be very familiar to them and still used a little bit of "sequence notation".

Notes on explicit formulas

We practiced writing explicit formulas the same way we did recursive (even with the same exact sequences).

My notebook :)

My last learning target for this unit is "I can explain why a sequence is a function."  It is perhaps a not-so-great learning target because it is somewhat difficult to teach and assess in a way that requires student thinking and learning.  However, it is very closely aligned to a standard (F.IF.3...?) and I struggle with the battle between whether I should be assessing standards or learning targets.  Anyway those thoughts are best left for a different post.

Here is what we did.  It was an investigation of sorts, where students were to choose a sequence and determine if the table that represented it was a function and if the graph that represented it was a function.  However we had difficulties because my students didn't do super well with determining if something is a function.  

The goal was that students would choose a wide-variety of functions and we could whiteboard individual results, do a gallery walk, and come to the conclusion that all sequences were functions.  However it didn't go down quite like that.  I perhaps needed another day of class, but it was crunch time for semester tests, so I didn't have any wiggle room.  

I ended up having a class discussion, but not assessing that standard.  It is what it is I suppose. 

Sequence = Function (p.1)

Sequence = Function (p. 2)

Notes:

• I wish I had done more vocabulary up front (ie. term, first term, second term, common difference, common ratio). It is hard for me to remember what words are new to students.
• Also a graphic organizer to show the relationship between sequence, arithmetic, and geometric.
• As I said, with recursive formula, I would change the first equation from now = ___ to start = ___.
• I've really been thinking about how I need to spiral some review in for students.  (More on this later...probably not until this summer.)  I believe this would have helped as we looked at sequences as functions.

Please leave any thoughts or suggestions in the comments, or tweet me (@kathrynfreed).

-Kathryn

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