Saturday, September 24, 2011

Destroy My Problem Set, Please.

Students should be able to complete this in groups without too much assistance from me. We already had a lesson on what the cube root means and simplifying cube roots to simplest form, which was also a refresher on how to simplify square roots. When it says "check on a calculator" they will have access to a CAS calculator for this lesson. I realize that if the roots don't come out rational, the calculator displays the answer with a fractional exponent. I don't know what to do about this yet. Maybe I will just put the answers on cards they can check instead of futzing with the CAS's.

The goal is for them to practice multiplying and simplifying, and investigate multiplying conjugate pairs to set us up for rationalizing denominators, both monomial and binomial.

I know the two questions at the end are kind of weird but it seems like a shame to waste the opportunity.


  • okay?
  • Crappy?
  • Suggestions for decrapifying?
  • Am I missing any relevant opportunities to make connections?
  • Or show multiple ways of seeing something?
  • How are the kids going to noob this up in ways I'm not anticipating?

Also, I am a Latex beginner so no making fun of my typesetting. It took me four hours to make this.

Monday, September 19, 2011

Everything is a Parabola

Lined up the kids at the board along a big number line. Everybody picked a number : their "x." I said, "Shawn's x is at 2. However far away you are from Shawn, move that many floor tiles into the room, perpendicular to the board." They move into a lovely approximation of the graph of f(x) = |x - 2|. "Hey what kind of a shape are you guys making?" "A PARABOLA!" Practically in unison. FACEPALM.

Saturday, September 17, 2011

Geometry Project Follow-Up

So I think I know what I'm going to do with all these Powerpoint files submitted for the Geometry project. I'll grade them as promised, but when the kids come to class Monday, I'll...

First, show them an uber-presentation of the best possible answer to each statement. This way they will get their feathers fluffed up when they recognize their work on display, but we will also hopefully resolve any lingering doubts. I did not necessarily confirm or deny all their questions while working on the project, because it was more important they keep thinking about it, and the teacher giving the answer from on high shuts down thinking. Most often this sounded something like, "Miss Nowak, will you check my ten Always/Sometimes/Nevers and tell me if they're right?" Me: "No...but I'm happy to discuss with you any specific questions you have about what words mean or what would make a statement true or false..."

Second, each group will get 4 copies of a printout of someone's presentation slide that exhibited a misconception. The task will be, what could you do to help clear up this group's misconception? What could you say/draw/show to convince them that they misunderstand, and also help them understand?" Then, we will jigsaw to mix up the groups, and everyone will share what they learned with their new group.

I always felt weird about the flow of : Work on Project, Submit Project, Teacher Grades Project and Hands it Back, and that is The End. I think there's much to be gained from re-visiting this work and catching all those lingering misunderstandings.

Friday, September 16, 2011

Geometry: Points, Lines and Planes

I was going to write up a description of this project we just wrapped up in Geometry, but luckily Allison already did it! (Maybe go read that post if the rest of this doesn't make any sense.) I liked it because it gave students an opportunity to get messy with points, lines and planes. There was a whole lot of productive struggle going on in my Geometry classes.

Here are some nice examples of student work:

Here are some examples that show misconceptions! I'm going to do something with these. Not sure what precisely yet.

Some kids really did seem to be enjoying themselves while learning, but there was also an awful lot of complaining going on. Managing all of their digital photos and getting them from their phones and cameras into their accounts was a bit of a hassle, so I can appreciate the frustration there. But it was good! There were lots of conversations, which if they were blog posts or magazine articles would have titles like:
  • Why You Are Making Us Do This
  • It Is No Fair Making Us Think
  • We Would Prefer to Just Fill Out Worksheets That Ask the Same Questions Over and Over
  • Why Miss Nowak or Anyone Would Like This Job
  • If You Give Miss Nowak Your Phone She Will Change Your Wallpaper to a Math Picture
However, there were also hopeful conversations like:
  • I Would Rather Be in This Class Where I Actually Learn Something Even Though It's Harder
  • My Friend is Jealous She Doesn't Get to Do This Project
  • Holy Crap, a Three-Legged Table Can Never Be Wobbly

Algebra 2: Graphing Absolute Value Functions

My goal with this was for students to understand "why the V shape" for absolute value functions. I think it will take three days. The TI-Nspire is used. This leads up to something very much like this, but with much more scaffolding ahead of time.

Phase 1: Learn lists and spreadsheets, data and stats skills on nspire:

1. Students watch and follow along on their handhelds "Data and Statistics: Adding and Rotating a Movable Line" tutorial.
2. Close file and do not save.
3. Send students document Squares.tns containing the case vs gray squares table from the beginning of the No Sleep Til Brookline problem set. Display these directions:
  • open file
  • add a Data & Statistics page
  • create a scatter plot of case vs gray squares
  • add a movable line, and try to get it through all the points
  • remove the movable line
  • menu, analyze, plot function
  • type in the equation you know fits the line
Phase 2: Understand why absolute value graph has a V shape and what it means
1. Write a number line on the whiteboard from -8 to +8 or even longer if possible.
2. Students line up against wall (however many will fit - maybe a subset of the class). Students note their position along the number line written on the whiteboard. This is your "x."
3. Choose whoever is at 2 to be the vertex. Let's call him Jake.
4. Give Jake something to hold up like a flyswatter.
5. When I say go, you are going to move away from the board. The rule is, however many floor tiles you are away from Jake on the number line, you are going to step that many floor tiles into the room away from the board. You'll move perpendicular to the wall. Take a moment to decide how many tiles you will move....Go.
6. Students move into a V shape.
7. Display in succession on projector:
jump up and down if you are a solution to |x - 2| = 5
jump up and down if you are a solution to |x - 2| > 5
jump up and down if you are a solution to |x - 2| < 5
jump up and down if you are a solution to |x - 2| = 3
jump up and down if you are a solution to |x - 2| = -3
8. Let's call the distance you stepped into the room y. What is the equation of x vs y?
9. Reset students back to line up against the board. (Or get a new group up there.)
10. Get the flyswatter away from Jake.
11. Our new function is |x + 3| = y.
12. Who gets the flyswatter? (Let's call her Jill.)
13. Your position is still your x. Decide what your y is and move there.
14. Display a few "jump up and down" questions. 
15. Note your position! Come to the smartboard and enter your name and position in the Lists & Spreadsheet.

Phase 3: Create Absolute Value Scatter Plot on TI-Nspire
16. Send everyone the lists&spreadsheet with name, position on NL
17. column 3 call it distJill
18. In formula cell, how can we calculate everyone's distance from Jill? Try difference...note problem with negatives.
19. Show how to enter absolute value function: template or abs(position - -3)
20. Show how to sort the whole spreadsheet from closest to farthest
21. Add a DataStats page
22. Can you get the dots to arrange themselves into the V-shaped graph?
Can you add the function that goes through all the dots?
Can you add a horizontal line function that represents "4 away from Jill"?
Can you shade the region that includes people that were within 4 spaces of Jill?
Can you add a vertical line (Plot Value) that represents the average distance from Jill?  

Phase 4: Apply skills to novel problem
For the past week, I left this sitting out on a desk in my room:

Something like 100 students entered their guess. I was able to copy the column containing their guesses from Google Docs into an Nspire lists and spreadsheets page. They'll get this Nspire file and these directions, and have the period to do what they can with it.

Phase 5: Transformations on the Absolute Value Function
Students will spend time playing with this Nspire file with this investigation, to understand how changes to the parent function transform the graph. To assess, they will try to match the pictures with a function.

Thursday, September 15, 2011

Algebra 2: Solving Absolute Value Equations

You know how you can show them this way, all justified and with lots of practice untilblueintheface:

But then a couple days later half of them will do this

and the other half will do this:

So, I stopped teaching it that way. I'm starting with something much like what most of us probably do:

Allison lives at 15 Sycamore Drive, and Sarah lives 8 houses away. Where does Sarah live? 

But then, I'm sticking with that model for all kinds of problems.


Earlier in the lesson I made them write it out in words, i.e., "the distance from 200 to 3x is 896."

It was more of a pain initially, and not the most effervescent lesson I have ever delivered, but MAN, it did the trick. No more of that autopilot, forget to write two equations, forget that absolute value can't equal a negative nonsense.

This is an idea I stole wholesale from the article "A Conceptual Approach to Absolute Value Equations and Inequalities" by Mark W. Ellis and Janet L. Bryson, Mathematics Teacher April 2011, Volume 104, Issue 8, Page 592. 

Inequalities are a natural extension of this concept. Where on the number line are all the values that are more than 896 away? That are less than 896 away? 

Monday, September 12, 2011

My First Days of School

My first day schtick has changed quite a bit over the years. The first year or possibly two I drank the Wong kool-aid because I was terrified and I had no idea what else to do, and also my school gave us all the Wong book for free. That didn't work. It probably works for the Wongs but it didn't work for me. I felt like a pretend drill instructor. (Wong. Wong-wah-wah-wong-wong.)

Then for a while I got kids to fill out a sheet about themselves that I stole from Dan Meyer. I had hopes these pieces of paper would capture the essence of each child, and had every intention of perusing them leisurely with loving tenderness. In reality, I just scanned the "Anything going on you want me to know about?" section so I'd have a heads up about all the imminent divorces and cancers and then I threw them in a drawer. Since that only took about ten minutes of day 1, I would just start in on a lesson for the rest of the period. Giddy up!

This year is the first one I feel I started out semi-competently. The cluebird is circling and coming in for a landing. My only measure for this is the proportion of students that will make eye contact when they talk to me.  I considered what I actually wanted to achieve from the interaction (imagine that): I want them to know they can be successful. I do want to start getting to know them, but just as importantly, I want them to know that I want to know them. I want to know who has the tech in their pocket that we can exploit for the learning. I want them to think about what they want to improve about themselves as students, and I want it to dawn on them that it's in their power to change whatever that is. I want them to feel comfortable in my room, to not feel trapped and helpless, at least know some of their classmates' names, know that I expect them to work hard but I'm pretty darn reasonable, and to tell me where funny things are on the Internet. Okay I'm going to stop with this paragraph now because it's getting out of control and was obviously was too ambitious for 43 minutes with 121 strange kids. Ahem.

Phase 1: Snowball Icebreaker
(I saw this on somebody's blog but I can't find where. Speak up and I'll happily give you credit.) Everyone gets 1/2 sheet of paper and writes three distinctive (you will have to give examples of distinctive vs non-distinctive) characteristics about themselves, but not their name. Everyone crumples up the paper, and we toss them around the room (suggested verbiage: "My biggest rule is that you show respect for each other and for me. If you can do that we will get along fine. So there will please be no whipping your snowball at anyone's face. Now, all together, let's pick a direction and gently fling our snowballs.") Everyone picks up a snowball and uncrumples it. Their job is to find its owner and write his/her name on it, and be prepared to introduce this person to the class. Their job is also to facilitate being found. As soon as they do both things, they sit down in the closest empty seat. (Suggested verbiage/modeling: "This is what I don't want to see: (hold up sheet in someone's face) 'IS THIS YER SHEET?!' (pause for giggling.) This is what I do want to see: 'Hi! My name is Kate. What's yours? Audrey? Nice to meet you! Tell me, Audrey, are you allergic to wheat? No? That's too bad. Although fortunate for you, I suppose. Ok, is there anything you'd like to ask me?")

Once the dust settles, start asking kids to introduce each other. This was great fun, because I kept asking for more details that would let them show off a little and/or amuse us all.
"This is James. He works at Wegmans and plays guitar."
Me: "Cool! I wish I could play guitar. What is your best song?"

"This is Angelina. Her brother is a pilot who lives in London."
Me: "How old is your brother?"
Angelina: "30"
Me: "Is he single?"

So that all took 15-20 minutes. I liked it because it fit one of my goals for this year: everything we are doing is for a reason, and we will follow up on things and not drop them without processing them and assessing you and making sure we all got the point (more on my lofty '11-12 goals in a later post.) It also hit several of my goals for day one: some kids learned some other kids' names, it was low-stress and all the chatter made the room feel inviting, I got to learn a little about them and they saw me being interested.

Phase 2: Distribute Books and Collect Data
Then I passed out my heavily modified Who I Am sheets, and the kids worked on them while I passed out textbooks and made smalltalk. This exercise felt different than previous years, because they weren't filling them out all scared in stony silence, but there was productive chatter and informal sharing and it just felt nicer.

Since I asked them questions I actually cared about, it was no drudgery to take all their filled in sheets and read them thoroughly and enter them in a spreadsheet. I sort of had the idea that analysis would yield some interesting things but I don't know what I was thinking. There are six days a year where three of my students have a birthday at the same time...that's kind of neat. I got a roughly even mix of math-likers and -haters:

And we have mixed opinions on whether Mathematics is invented or discovered:

And I suspect that some of the kids who claim they like math, don't know what it is, and like it for all the wrong reasons:

But we will see what we can do to change that.

Phase 3: Some Blah Blah
I spent the last 5-10 minutes telling them about supplies they need and what to do if they have to use the bathroom, that sort of thing. We didn't do any math. I don't feel bad about it. I feel better about the way this year started than I ever did. I'm excited about how we're learning, too. More soon.