Sunday, January 19, 2014


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...
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)
  • 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).


Sunday, January 12, 2014

Solving One-Variable Linear Equations

Confession:  I taught an entire semester of Algebra 1 without spending time solving linear equations. 

Why?  Because I knew my students had seen a lot of it in 8th grade, and I also knew that they were all at different places and ready for different challenges.  I also knew that they had forgotten some of what they had learned and a little reminder might take them a lot further.  All of this is hard to address in a class of 20+ students.

However, I knew it was something I needed to address at the start of this semester so that we could solve systems and eventually some quadratics.  I gave a pretest, but it was evident that not a lot was immediately recall-able for them.  So I spent a few days focusing on solving 2-step equations.  Here are a few reasons why I chose to start with 2-step equations:
  • I can say they can all be solved in 2-steps, which helps the students process what they need to do
  • I can address issues like "Ah, there's a fraction!" (but it really just means division) and I can even throw in some parenthesis
  • I can challenge all students with things like -t + 10.2 = -23.1 (the negative variable is really tricky the first few times they see it)
  • It is not out of the reach of most students (I do have a few students who still struggle to solve one-step...but those are students who did not take 8th grade math in my district last year.)
  • It is enough to bring back a lot of what they learned about solving equations
I have one Algebra class that is a little bit quicker than the others, so I got to move away from 2-step equations with them on Friday.  I couldn't decide exactly how to do it, because there are SO MANY ways to solve different equations.  I really just want to ensure they are aware of the various options they have and give them some practice at choosing what to use in what scenarios.  I don't want to say:  "always distribute" or "put the variable terms on the left and the constants on the right" or anything that shows that there is only one way to solve.  I want problem solvers, not procedure followers.

Anyway, here is what I ended up doing.  I just gave them 4 "challenge" equations to solve.  And for the most part, they worked HARD for 30 minutes to work out solutions.  Here are the equations:
  • 2(-1x + 6) = 22
  • 2x + 2 = 32 + 5x
  • 3 - 2x + 6x = 15
  • 5x - 7 = 2(x + 1)
I put the four equations on the board and said "I challenge you to use the resources available in this room to find the solutions (and perhaps a solution method) to these equations today."  We discussed what resources were available and then they got to work.  The review we had done with 2-steps was enough for several students to remember even how to move variable terms to the other side.

There were a few students who "tried" and then gave up and didn't accomplish much at all, but I would say at least 85% of my students worked hard during the time I gave them.  I was SO PROUD of them.  I just wanted to brag a little about my students, because it was everything I hoped it would be.  

We will continue to work with advanced linear equations on Monday and Tuesday to get them more fluent at being aware of the strategies that are available and choosing an appropriate one.  Perhaps with some whiteboarding...


Thursday, January 9, 2014


I need to get back onto the blogging wagon, and I have been trying unsuccessfully.  So instead of blogging about something thoughtful, I'm just going to share something I learned about at our PD technology day on Monday:

On this site you can create "sets" of flashcards and study them in many different ways.  Since I've been working at pushing vocabulary in my intervention class, I decided that this would be something to try with them.  Some of the reasons why I decided it was worth trying:  it has a math option for "language" and you can add pictures to the flashcards.  A set can be public, so a set I create my students can use to study.  Also there are apps available for both android and apple devices.

So I created a set for students for this week's vocabulary and had students create an account and join the class to practice for a while today.  We are a google apps school, so I had students use their google account to create a quizlet account.  Then I had them join my class so they could easily access my set, but you can share it by link like this:  Variables and Expressions Set.

What I like is that the students can do several things with the flashcards:

  • use them as flashcards...front - back - front - back - etc
  • "learn" gives definition; student types in word
  • "speller" says word and gives definition; student types in word
  • "test"...set up a test with x fill-in-the-blank questions, y matching, z multiple choice, and w true-false
  • "scatter"...a matching game where students drag word and definition together; if it matches they disappear
  • "race"...definitions move across the screen and student has to type in the word before it gets across; speed GRADUALLY increases
I think both of the last two will keep a rank of the students and that helps encourage some healthy competition.  I played first so they would try to beat me :)  (However when using the app it doesn't record their scores in comparison to the rest of the class.)

I can think of a lot of things this would be useful for, but you can figure it out too.  So far I'm using the free version, but a better version is available for $15/year or even better for $25/year.