Tuesday, August 28, 2012

Risk your Fraction Skills

Next week, I begin my school year with 180 sixth graders (36 students per class).  My district has provided a scope and sequence that builds from where sixth graders should be starting the year skill-wise. It assumes that most students will need a brief review of adding and subtracting fractions before we launch into multiplying and dividing.  Over the years, I've learned not to count on the fact that assumed prior knowledge is in place and so I want to do a pre-assessment to double check where my students' skills lie. I really hate to throw a test at my students the first couple of weeks and - when trying to think of another way to preassess - I remembered a game I saw on Sue VanHattum's blog a while ago. Sue's game (Risk your Algebra Skills) was aimed at algebra students, but I used her idea to focus on my students' fraction skills.
Risk Your Fraction Skills
The idea of the game is that students risk up to 100 points that they will get the first question correct (they should make their bid BEFORE they work the problem). If their answer is correct, they add the points to the beginning 100 points; if they're wrong, they subtract.  They now have a new total from which to bid.  The winner of the game will have the highest point total.

The reason I like this game as a preassessment is because it gives me two important pieces of information:
  • How confident the students are in their skills of adding or subtracting fractions or mixed numbers with common and different denominators  (Where do THEY think they are?)
  • The actual skill level of the students (Where are they REALLY?) so I can differentiate my instruction and products, as needed
It also disguises the "test" in a way that will distract my test phobic students.

I'm looking forward to trying this with my kiddos .... a test disguised as a game....FUN! :) If it works well - it might be a way of disguising other sorts of assessments. If you try the "game" I'd love to hear if you think you got the same sort of information you would get from a normal test.

UPDATE: Here's an Order of Operations one ...

Risk Your Ooops

Sunday, August 26, 2012

WANTED: Dead or Alive

How many times do we - as math teachers - have students bringing their friends to the classroom between classes, after school, or at lunch just to show them a project they're working on? How many times a year do we have students ASK if they can come in at lunch to work? Maybe in art class, maybe a photography class.... but math? Hardly ever! Last year, I had students doing just that: asking to come in and work on their own time and bringing their friends in to show them their work. AWESOME! The project occurred over a three day period (we worked on it for about 30 minutes each day - rough draft and then final) and it was one of the best investments of time I made; the outcome was amazing!

Direct Variation functions seem to throw my 8th graders for a loop every year. The easy part is recognizing equations in the y=kx form or even the y/x=k form; but in the past, for some reason, every time they saw something like y=x/4 on a test and were asked to identify the type of function in was, they thought it was a direct variation - just because it was a fraction. I know they could stop a minute and reason through the equation and realize that it wasn't, but you know eighth graders! :)  I wanted to come up with something that would allow my students to explicitly think about equations that "fool" them into thinking they are something they aren't. Who else often does that? Criminals! And so...

I had my students create wanted posters for direct variation. I started out by showing them different wanted posters (they liked this!) and we talked about the different things that needed to be on a wanted poster for it to be effective. I then set them loose. And they had a blast! I've inserted some pictures below so that you can see some of the final products they came up with. There's a range of posters from my low-level kids up through the ones functioning at a much higher level. I like the fact that you can see - fairly quickly - what each student understands about the function and what misconceptions they might have. 

Each day, we worked on some fairly focused practice, both in the warm-up and in the practice activities, and then we spent the rest of the period working on our posters. It was fun to walk around, chat with my students, ask them some leading questions when they were stuck, and help streamline their ideas, when necessary. 
Several things:
  • I was surprised that many students weren't sure what "alias" meant. We spent some time talking about that, both during the powerpoint showing different wanted posters and, again, as I walked around the room while they were working.
  • We needed to spend some time talking about the terms "disguises" and "m.o".  Some of my students LOVED the phrase modus operandi and used that term on their posters. 
  • My low-level kids needed a little extra support in the writing phase, but - for the most part - they excelled in the illustration part of the project. :)

It doesn't matter how fun a project is if it doesn't produce the results you're looking for. I waited six weeks after we did the posters (so I could test for long-term learning vs short-term "remembering") and then I gave my students an assessment to see if they were still struggling with recognizing direct variations. The results were amazing - less than 6% of my students showed any indication of needing any reteaching at all! I think that makes this project worthy of another visit next year!

Tuesday, August 21, 2012

R squared equals one

Do you have a friend or colleague who always has the perfect analogy for any situation? I do. My friend, Clark, can come up with perfect analogies for any situation in a second. And they're always clever. I have another friend who comes up with analogies equally quickly; however, they never seem to make sense. To her, the analogy is perfectly clear. To the rest of us? Clear as mud!

The analogy of my blog's name is perfectly clear to me. The danger with sharing the analogy with you is that you might read it and respond, "Clear as mud!"  :)
The coefficient of determination (r squared):
  •  is a measure that allows us to determine how certain one can be in making predictions from a certain model/graph.  
  • represents the percent of the data that is the closest to the line of best fit
  • is a measure that assesses how well a model explains and predicts future outcomes
  • is expressed as a value between 0 and 1. A value of one indicates a perfect fit and therefore a very reliable model for future forecasts. A value of 0 would indicate that the model fails to accurately model the dataset.
Every day we put our "model" on the line and we test it, hoping for an r squared of one.  We design (what we think is) the perfect lesson with just the right learning target and just the right set of activities to get our students there, and then we go for it. While we are teaching, we check for outliers, for students that wander near the line of best fit, for students that are right on the line, and students who don't even make our graph. A reflective teacher then tries to account for as much of the variability as we can, hoping to get close to that perfect value of 1.

That is what I try to do every day. Achieve a one.  Sometimes I get close. A few times, I've been near 0 and had to start all over the next day. Every once in a while, I "score" a one. And we all know how wonderful that feels! To me, those 1's come with the synergy created between the art of teaching and the science of teaching. They are hard-won. And they feed me (professionally) like little else does.

I hope to share my 1's with you, as well as my 0's and all that falls between. I hope to hear about yours as well. We'll rejoice in our 1's while sharing our strategies, activities and thoughts and we'll analyze the 0's so that - the next time - our outcome will be different.

Looking forward to the journey...


One of the main areas I need to assess the first week of school is how well my sixth graders understand fractions, decimals, and percent. I know my students will be all over the board in this area and so I think I'll play a game of RACKO with them. Here's how it works:

  • I'll divide the class into two teams.  
  • We'll shuffle the cards and then deal out 8 cards to each team. As I deal them, I'll place them in the order their drawn on the rack (hence the name RACKO) on the board. 
  • A student from the first team draws a card and can either discard it or replace one of the cards in his/her team's row.( The goal is to be the first team to get all their cards in order.)
  • Now, the second team has a turn. It goes back and forth until one team gets their cards in order.
While the class is playing, I'll be listening to the conversations in the teams. Who seems to have a grasp of the equivalencies? Who doesn't seem to have a clue? Who always seems to consistently offer the correct suggestion? Who remains quiet? Who can replace a card correctly by themselves? Who needs help from the team?

We'll play this a couple of times and then - another day - I'll put students into smaller teams to compete - maybe two person teams and then, eventually, one person playing another. If you plan on doing this, copy each set of cards in a different color; that way, clean up is easy!

The beauty of this game is that it can be adapted quite easily. You can:
  1. Make a set of all fraction cards if you are working on ordering fractions or on common denominators.
  2. Make a set of square root cards mixed with whole numbers or even numbers with decimals.
  3. Make a set of integer cards, mixing positive and negative numbers.
  4. Make a set of decimal cards. This will really help pull out misconceptions about place value and decimals.
The possibilities are endless. The students love this game and it's one of the best ways I've found to pull out misconceptions and examine them as a class. I love any game or activity that can do that; it is difficult to affect change at a deep level until both the students and I understand what misconceptions are held about the concept we're learning.

Friday, August 17, 2012

Foldables vs Graphic Organizers

On Tuesday, I attended the online presentation Julie did on using foldables in our interactive notebooks. I liked the variety of foldables Julie showed us and I think they would be fun for my students. I appreciated the fact that the foldables would be useful in helping my students process new content and that my students could use them to practice and to study for assessments.

Earlier, I had gone through some of my old math lab materials, looking for some things to send to someone who is teaching a Math Lab this year. I found a whole set of graphic organizers/guided notes I use with my lab students. They really like them because it makes the content fairly explicit and helps them to make connections they might not otherwise make.  I pulled out one on the angle relationships created when a transversal crosses a set of parallel lines (I've used it for years and don't know where I got it; if it's yours, let me know and I'll cite you).  It looks like this:
Transversals Guided Notes

And here's what it looked like when filled in (after some hands-on activities and practice) and glued into our notebooks.
Now, there's absolutely nothing wrong with this: it pulls all the important content onto one handy page and helps my normally scattered students focus their thinking and pull together connections they've made along the way. However, I thought it might be more useful for my students if I turned it into a foldable. So, here it is.... my first foldable! :)
I designed it so that after students take notes on it, it can be folded and opened up a variety of ways so they can us it to study for assessments. On the left, examples of each angle pair relationship is colored in. On the right, students describe the relationship as congruent or supplementary, and in the middle, students write out the relationships.

When opened like this, students see the same of the relationship and they can see if it is supplementary or congruent. Facing that page is a blank diagram so they can practice finding the angle pairs.

After they name the angle pair, they open that flap to see if they were correct. I like this; it gives them immediate feedback.
All folded up, ready to be taped into their notebooks.
While I really like using graphic organizers and guided note forms, I think this foldable is a much more valuable tool for my students. I like how interactive it is and I can think of tons of ways students could use it in class after they use it to take notes on.

Since this was my first foldable and I'm still learning (Julie-you forgot to tell us how looooooong it takes to get everything organized and lined up just right! :)),  I would love feedback. Where could I improve on the design? Would there be a better placement of information so that it would be even more useful to our students? And - for those of you who are farther along on this journey than I am - do you have any hints for streamlining the process?

Here's the front....
Transversals 2
Here's the back...
Transversals 1

Sunday, August 5, 2012


I’ve taught for quite a few years and I’ve learned that no matter how wonderful an activity is, there is much a teacher needs to bring to it that, in my opinion, goes unsaid in most lesson plans and ideas. A few years ago, I mentored a group of teachers who were interested in learning more about brain-compatible instruction and about ways of increasing relevance for their students.  A couple of teachers would try the things we worked on in our weekly meetings and would come back to report less than stellar results when others were reporting the opposite. Finally, I went in to observe some teachers who were reporting success and some who weren’t. I quickly realized the problem.  Although everyone was implementing the activities correctly, some teachers went no farther than implementing the activity itself. 

 I asked one particular teacher what she had learned about her students and about what they knew/didn’t know from observing their involvement in the activity. She wasn’t sure what I meant and so I explained to her what I was thinking and observing during the lesson (who was on or off task, who got the “point” of the activity and who needed me to ask a leading question or two, who wanted me to tell them the answer, who didn’t do anything at all, who seemed to have a misconception that needed addressing ….) She said to me, “ I was just trying to implement the activity correctly; I didn’t know I should be watching for those things!" 

Her comment caused me to stop and reflect on my own teaching practices and how to make what seems apparent to me apparent to everyone else, whether it be a teacher I am mentoring or a class full of students I’m teaching. As I share some of the activities and lessons I’m using, I’ll try to remember to share some of the “hidden parts” of the lesson. For most of you, it will be things you are already thinking or doing; for one or two of us, it might make the reason behind the lesson a bit more apparent. I know it will help me as well as it will remind me of what my overt and covert intentions are and will help me focus more intently on those goals.

All of us best process information in one (or more) of several ways; one category of those are our visual learners. I enjoy using this activity to see who in my classes tend to be good at picking up information visually.  I’ve used it on and off for years and have no idea where I got it. I do know that it was intended to be more of a brain challenge, but I use it in the way I’ve described above.  I use the Sherlock activity to begin a discussion about how we all might learn a little differently and to talk about our learning styles and preferences.
Sherlock 1

I hand out the first picture of Sherlock Holmes’ room and ask students to observe it for a bit of time (I’ve tried 3 minutes and 5 minutes… adjust the time to fit your particular students). At the end of the time, I collect the pictures and I give them the blank one with the questions. I then give them a period of time to draw in the objects as they remember them. At the end of the time, I project the original diagram on the Smart Board and have students circle all the ones they got correct. We then begin a discussion about ways we like to learn and that even though someone may have got very few correct, if I gave them a similar activity where I described where everything was, they might get them all correct – we all simply learn best in different ways. And that’s okay.  I want to set a tone of openness and acceptance as we all discuss how many we got correct and how we feel we learn best.  Finally, I ask the students to reflect on their own preferred ways to learn and I have them write me a short paragraph telling me how they feel they learn best and telling me how I can best help them this year. This will be their exit slip.Sherlock 2

Now, here’s what I do while my kids are busy… I’m watching and learning. I’m learning sooooo much about my students, about my class make-up, about how each particular class relates to their fellow students … Here’s what I’m watching for…..

1.        Are all the students involved? Who is looking around the class or out the window (or whatever) instead of participating in the activity?
2.       Who is very focused and studying the diagram intently?
3.       Who simply HAS to talk with his or her neighbor as they process what to do or process how many he or she got correct?
4.       Who has to check in with me to see if they’re doing it correctly?
5.       Who seems to want to sit back and observe what everyone else is doing before they jump in?
6.       Who calls out comments as they’re working?
7.       Which students simply can’t concentrate for the short time period you give them to study the diagram? What are they doing? Getting up? Beginning conversations? Fiddling with their “stuff”?
8.       As I walk by to see how the students are doing, who wants to talk with me or show me something? Who wants to simply work without interruption?
9.       As we discuss our findings, who wants to share? Who seems to want to add their comments more than expected?
10.   Which students try to steer the discussion off topic?
11.   Which students don’t share at all?

Now, check the writing the students do for the exit slip.
1.       Which students write well? Poorly? Who struggles to communicate their thoughts?
2.       Which students invested themselves in the writing task? Who sort of blew it off?
3.       Were students able to stay on topic or did they meander all over the place?
4.       Who didn’t write anything at all?

By the end of class, I have a pretty good feel for the class as a whole and for my individual students. I’ve already discovered some potential challenges I might have with each class and I’ve begun thinking ways I might structure each class to help with those challenges. Tomorrow, when we talk about class norms, routines, and expected behaviors, I already know where I might need to focus a little differently with each class. I know which of my classes are more social, which might need more prodding during discussions and which classes have students who need a little more help (and I already have an idea of who it might be best to pair them with). All in all, a very fun and productive first day activity!