New Learning Targets – 2016

We just finished the 8th day of school this year, & I’ve already justified not writing a blog post (on my list for #1TMCthing) for about 3 weeks now.

This summer I rewrote the Learning Targets for Algebra I, Geometry, & Algebra II and wanted to post them in case other were thinking through the same thing. (Our school uses Standards-Based Grading and has school-wide Learning Targets for each class.)

First, here are the Tennessee State Standards (aka, the Common Core State Standards with a different title): TN Algebra I | TN Geometry | TN Algebra II

Previously, the Learning Targets we were using were just the “Clusters” (the A, B, C, etc. in the CCSS) from the above documents minus any that were colored yellow (what the state of Tennessee considered “Additional Content”). This made it so each class had important material that wasn’t being tracked or assessed except for on the end of the year state assessments. It also made it so there was a lot of redundancy in the Learning Targets where two clusters could include closely related topics, or Learning Targets that seemed to have a bunch of topics with little connection.

Here is what I came up with: ALGEBRA I LTs 16-17 | GEOMETRY LTs 16-17 | ALGEBRA II LTs 16-17

Algebra I and Algebra II are meant to mirror each other so students will go through the same basic “units” in each class. For Algebra II, I had been thinking all last year about how to better organize the course, which is why I went through the trouble of putting things into units, as well (something I did not do with either of the other courses). I don’t actually teach Algebra I anymore, so I left that up to the Algebra I teacher if she wanted to. The Learning Targets in Geometry basically act as units, themselves.

And, lastly, here are the simple syllabi I use to show what order I plan to do everything in: Geometry Syllabus BC 16-17 |  Algebra2 Syllabus BC 16-17

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Law of Large Numbers, or a Sigh of Relief

So by now, the suspense must be absolutely killing you.  My room was fine when I got back.  No one stole the dice and got caught gambling in the bathroom, and maybe half the work I left was actually completed thanks to the interventionists I had in my room.  The worksheets worked pretty well, although there was some confusion as to whether “dice” meant one or two… (at this point I cried a little bit).  So we had to end up taking a couple more days to work on probability and do the exploration the right way.  Students were wholly unimpressed that the sum of the probabilities of all of the possible outcomes was 1 for both theoretical and experimental, and I lost a dollar to a student who had a 1 in 200 probability of winning it, but we ended up with a really nice set of graphs illustrating the law of large numbers by looking at the outcomes from a partial group, 1 group and the whole class.

The green and orange were white-washed to help the students focus on the whole class data and the theoretical.

The green and orange were white-washed to help the students focus on the whole class data vs. the theoretical.

Besides that, Denver was a whole lot of fun, and I got to meet some awesome people from the #MTBoS.  It was like that movie where the kid playing the video game is actually waging an international war; All of these people in computer world turned out to be real, and really interesting!  I was going to attempt to do a few math recaps, but something I learned about myself at NCTM is that I am a very poor judge of sessions.  Still there were some good takeaways, and they may eventually find their way here.

Oh, also I report for in-service tomorrow!

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Denver (or Leaving the Nest)

It’s been an interesting Spring semester.  We ended up having another lock-down the first week of March as more insanity ensued outside the school walls. Spring break came and went bringing with it a southern snow storm (i.e., no accumulation).  We began the countdown to state testing a few weeks ago; next Monday there will be exactly 10 days left.

On April 4th, we found out there was a lot of money left from our School Improvement Grant and were encouraged to find conferences and professional development opportunities before the money disappears at the end of June.  So, of course, I applied to go to the National Council for Teachers of Mathematics Annual Conference which, at that point, had a start date less than two weeks from then.  Monday at 5pm, I found out I was approved to come.  Then today, winter storm Yogi gave me a nice relaxing day at the airport to read.

So, I’m in Denver and missing my wife and son terribly.  But since this is a math blog, we’ll get back to that.  This is my first time leaving my students with a substitute, since when my son was born in the fall, my students were split among the other two Algebra teachers.  I struggled a lot in trying to figure out what to leave for them to do while I was gone because I’ve been a student in a room with a sub before, and great distance weakens authority greatly, right? (Thanks, high school history!)  The last thing in our curriculum before the state testing is the probability unit, so I created these two worksheets to guide them through a project on experimental vs. theoretical probability.



They’re a little leading, but I wanted to make sure they would work without me having to press them in person.  I’m kind of nervous about leaving dice with the sub.  I can imagine scores of things that could go wrong.  To make myself feel a little bit better, I left a message on the board at the front of the room that just said,

I trust you

<3 BC

We’ll see what’s waiting for me when I get back.

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Student Strategies: Exponents

Every once in a while, a student will do something unexpected.  Okay, that’s a lie.  Unexpected things happen all the time.  What I mean to say is this: every once in a while, a student will use an unexpected method that intrigues me.  As a new teacher, one of the hardest aspects of planning a lesson is “doing the math,” or anticipating how students will try to solve a certain problem.  Because of that, I thought it would be helpful to keep track of some of these methods here under the heading “student strategies.”

A few weeks ago when we were working on properties of exponents, one of my students presented me with this as his work on a quick cool down (read: exit slip) we were doing.  I’m sure you can tell from the pictures, but he was trying to simplify \sqrt{36^3} .  For some reason, he did the opposite of most of my students and decided to work it out on paper, which was good news for me!  His first steps were to take \sqrt{36} and then multiply 6\times6 (not shown), which is pretty normal.

exponents: step 1

At this point he knew he had to multiply by 6 again, but decided to go straight to repeated addition.

exponents: step 2

The willingness to do 6\times6 on paper but not 36\times6 is what intrigues me the most.

On a semi-related note, we’ve spent quite a bit of time talking about the relationships between addition, multiplication and powers (something I was glad to see @wahedahbug post on twitter today) and it’s definitely something that they get mixed up.  When a student’s knowledge of multiplication facts fail them, I see them go right back to repeated addition for help.  It’s a connection that makes sense to them.  (I even had a student not too long ago show me 3\times3 by drawing three circles with three “cookies” in each one.)  But, when they saw powers as repeated multiplication, they got so excited they now want to use exponents for everything.  We’ve been continually tempted to write x+x as x^{2}.

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Mathematical Reasoning

One of the things I try to stress to my students is that the “answer” to a problem is unimportant.  It makes no difference to anyone’s life whether the answer you got to problem 7 was “3” or “-0.8672 repeating.”  What does make a difference is that they learn how to think, reason, justify, and explain.  Today, I was not disappointed.

During my 4th block class, we had been talking about writing rational exponents as roots, and were going through simplifying rational exponent expressions.  Looking at the example 8^{1/3}, I was helping them work through how to rearrange it using inverse operations so it looked like 8=x^3 instead (I think rearranging helps the students to remember that they know the answer to this without using a calculator.  Just like they are quicker to know x=2 when looking at 2x=4 than 2={4/x}) when they suddenly got stuck. Our first step was to rewrite the rational exponent as a cube root.  Next, they decided we needed an exponent to cancel out the radical.  This is where the brakes hit.  They couldn’t decide what exponent to use.  This was almost a non-issue in the other class, but I had crickets in 4th block.  At this point I paused because a couple things happened.  First, I did not want to give the answer away for free so I had to stop to think.  Second, the principal came in to do an observation and that always makes me stop for a moment nervously, even though I love the constructive feedback I get.  When I came  back around, I asked the students to stop and take some private think time to try and figure out the correct exponent.  From there we moved to partner talk, and I fished the numbers out of the crowd. 1, 3, 9.  We went back to partner talk to discuss the numbers they hadn’t come up with themselves, and everyone decided to throw out 9.  A couple people wanted 3, but I had some adamant young men who swore the answer was 1.

From here I cannot explain what happened to my students.  It’s like they ran into the phone booth and came out as mathematicians.  We moved into public discussion and both sides presented an argument.  Both sides were doing an amazing job using definitions, examples and prior knowledge to justify their reasoning and prove their stance.  The “1” team was actually doing a little better at it, which I thought was hilarious because they were wrong, of course, but I did not comment on the correctness of either side.  I eventually sat down with the main “3” advocate and encouraged him since he was having trouble putting his thoughts across (and I needed him to not give up, he had the right answer!).  He proceeded to explain to me in detail why the answer was 3 citing examples from what we did yesterday, so since it was obvious he already knew what he was doing, I helped him figure out how to show his reasoning to the others.  Then I invited both sides to give what would be their final arguments.  It turned into their final arguments, not because the “1” group was convinced, but because they finally were able to explain convincingly what I had suspected earlier… it was time to intervene.  The one group was arguing that the answer was 1 because we had just talked about how with x^{a/b} “a” becomes the exponent of the radicand.  They had misunderstood what we were looking for.  So was the class a waste of time, arguing an answer to the wrong question? HECK NO! The fervor with which the “1” group argued their point only caused the “3” group to work even harder to justify, and BOTH groups ended up successfully justifying a completely correct statement in front of the entire class using mathematical language and examples.  It was around this point that the principal left while giving me a thumbs-up (good sign!) and we debriefed what had happened.  For the rest of the class the students were “on.”  Math had become a class where they don’t just sit and listen but they are encouraged to get up and reason.  There was a noticeable difference in the way the students listened to what I had to say before and after our discussion.  Somewhere in the middle of the chaos of reasoning, a student said “I feel like we’re in court,” referring to the way the two sides where going back and forth building and presenting arguments, and I thought, “YES! Yes, you are! That is what this classroom was made for!”

I think there must be gas leak in my room or something.  This whole week my students have been on.the.ball.

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Two Good Days

The last two days in class have been amazing! I have been hitting my students with BIG expectations for the new semester and they are knocking it out of the park.

Yesterday, I was planning a review day to make sure everyone was ready to move on to rational exponents today, but I knew the small groups I had planned based on their exit task from Friday were not going to be enough for an engaging class.  As I sat thinking about how I really should have spent more time coming up with ways to engage my students, I thought, “I’ll have them do presentations on what they review in their groups.”  I’ve wanted to get these students presenting something since the beginning of they year, but we had yet to do it.  So I typed up a quick scoring guide and decided I would have them grade themselves and average their score with mine (document to follow).

This semester I have TWO interventionists with me all day, every day (more on that later), so we each took a group and were able to give focused, one-on-one help to every student, which in itself had me feeling good.  Then, the presentations started and the feeling in the classroom was so awesome. Not only did most of the students do a great job given that it was everyone’s first presentations, but it was so easy to tell who knew what they were doing and who needed more help, I felt like I was cheating.

Today, we started rational exponents. (I know. I could just end there. Amazing, right?) I, trying to avoid a lecture (theme), decided, thanks to a post by Ian Byrd, to take the inductive approach.  So, I put up 10 or so solved examples of simple rational exponent problems (numerator of 1) that they would know the inverse facts for (e.g., 4^(1/2)= 2, 64^(1/3) = 4).  They started by thinking individually about what the fractional exponent was doing and worked their way through dyads and table discussions interspersed with a few whole class clue-sharing sessions as students discovered different things.  We took the whole class period, and it was awesome.  I was careful to only reveal what was necessary at the time and when they got stuck, we went right back to individual thinking time to let the students come up with new ideas, and climbed up the discussion ladder again.  By the end of the block, the tension in the room was becoming so palpable as they started getting closer to realizing that x^(1/2) = sqrt (x) that I think I almost leapt into the air when they got it.  My 3rd block class did especially well, so I e-mailed the rest of the 9th grade teachers to tell them to congratulate the students on their status as amazing mathematicians when they see them tomorrow. I was so proud of their inductive reasoning and perseverance I literally came home and told my wife, “I think I’m becoming a good teacher!”

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