Polynomials Day 2

Well... I definitely learn better ways to do things and, even more, I learn what NOT to do! First and foremost, do not overestimate what kids can do without guidance. I have had to learn to slowly build up to what I want the class to achieve and I seem to forget this lesson every time I start a new unit. Yesterday and today, students learned to factor out a greatest common factor from a polynomial. I started with a review of how to multiply a monomial and a polynomial. We review/defined the terms "product" and "factor." Then, I gave them a product and asked them to find the factors. Students did great. Our first consolidation involved me going through how to find the greatest common factor of a monomial. In my first class, I just spoke the words and I eventually learned that it wasn't enough. So, in my second class, I wrote the words. How simple of an adjustment that made a HUGE difference! (I'm definitely going to have to backtrack with my other class to fix that mistake of mine!) At that point, the students were really knocking it out of the ballpark. I use my interactive whiteboard as a common banner. I learned about the common banner from Peter L. in a workshop I attended in October. He had the common banner along the top of the whiteboards with new problems progressing from left to right. However, that does not work for this 5'1" teacher!

My next challenge is how to keep problems going onto the common banner while helping the students who are struggling the most.  I get pulled into helping and other groups are anxiously awaiting the next problems.

Our four-quadrant notes were better today.  The kids even gave me some positive feedback.  In particular, they like doing them as a group.  However, I had to promise that they could use them during their next skill-check as incentive to make sure they took them seriously.  It's all a work in progress, I suppose!

Next class we multiply polynomials!

Polynomials Day 1

 Well, today I started with my thin-slicing.  The kids did great!

I started with a simple monomial multiplication problem: 

$3x^2·8x^4$

I showed them how to expand, then rearrange the factors: $3·8·x·x·x·x·x·x$

and how it simplifies to $24x^6$

Then, we grouped and started thin-slicing! After about three or four problems, which included expressions with more than one variable and/or exponents of 1, students were asked to create a shortcut (they love a good shortcut!).

Each group understood that they were multiplying the coefficients and adding the exponents (we did some reminders that they are called "coefficients" and not "constants"). Great!!

I continued on with multiplying a monomial by a polynomial, making each example a little bit more complex than the previous. The studetns were already familiar with the distributive property but where they got themselves in trouble was what to do after distributing. About 4 of the 10 groups started combining terms and needed a little hint about how we only combine like terms. After that, it was smooth sailing!!

The most exciting (for me) part of the class was our first attempt at the four-quadrant notes. I gave them a fill-in the blank and another example. They completed the notes at their boards, then into their notebooks. I'll be curious to see how this will affect them going forward. I'm thinking of allowing them to use the notes on their quizzes for the unit, but that isn't the norm for our school and I don't want to create any issues.

All in all, a good first day. Not perfect, but good.

Monomials and Polynomials and Factoring - BTC Style

 This is the topic I've been waiting for!  Why?  Well, it feels like multiplying, dividing, and factoring of polynomials lends itself really well to thin-slicing and now I have a much better handle on how to moderate thin-slicing lessons.  I am definitely not even close to being what I would say is "good" at it, but I'm getting better.

This is the first of many posts for this unit.  I anticipate a lot of reflection on my part!!

So, the first post is about preparation.  Though I heard this during a workshop at last year's BTC conference, I really learned it through experience.  You can prepare, but it will likely be all thrown out at some point.  So, don't over prepare.  Peter says that, regarding consolidation, experience is better than anticipation.  In other words, don't assume anything but be ready to respond to what the students demonstrate in the classroom.

So, in preparation of not preparing :) I decided to start by setting up a progression for the unit.  Here is what I came up with:

Multiplying Monomials

Multiplying a Monomial by a Polynomial

Dividing a Polynomial by a Monomial

Finding the GCF of Whole Numbers

Finding the GCF of Monomials

Factoring out a GCF from a Polynomial

Multiplying Polynomials

Factoring Quadratic Trinomials (only a=1 is required for Algebra 1, but I may be able to present trinomials with a prime leading coefficient)

Factoring a Difference of Squares

I feel like this progression makes sense and I can easily connect each topic to the next.

The next step was to create thin-slicing questions for each of these topics.  To thin-slice, each subsequent question should be similar in difficulty to the previous question, or add a new element.  I admit that I start with Chat GPT here.  I ask the program to create a set of 10 problems for a given topic that progresses with those parameters.  Once the set is created, I may modify it or just print it out and modify on my own.  It's important to remember that Chat GPT is not a mind reader and that reviewing the material is necessary!  That said, I have found it to be a great starting point.

Of course, I have modified my plan quite a bit before the first day.  Sometimes, we can overthink and just need to jump in!

I Finally Did It!!!

 I finally... FINALLY left a class feeling like it was great! 

As I have been slowly building my thinking classroom in Algebra 1, I have learned so many things that haven't worked.  What I've learned the most is really two things.  

First, have a well thought out plan.  One that includes how to launch a task and how to monitor the task. 

Second, be ready to throw that plan out the window.  I've done this a lot, usually because I've realize that my well thought out plan was not as well thought out as I envisioned.

I teach two sections of Algebra 1.  I always feel bad for the first group because of how much I modify for the second group.  Yesterday and today, I introduced my students to arithmetic sequences.  We started by making tables for the following sequence of pictures...

I learned from class #1 that I needed to clarify that there were THREE pictures here.  Who would've thought that this would be considered only ONE picture? 

So, students were asked to make a table to show the number of stars for each picture (picture #1, #2, etc.)  We went up to picture #3 and I added the number 57 to the "picture #" column.  Well, that was fun.  Most groups were able to answer the question using the patterns they saw in the first few rows but I realized that they were using the number 57 to get their answer:  (57*2)+1.  So, I then asked them to determine the number of stars in picture 57 based on the number of stars in picture #1.  Students were originally coming up with 3+2(57).  After some discussion and questioning from me, we were all able to figure out that it was 3+2(56).  Yeah!!

Then came the algebra.  I added "n" to the picture # column.  Audible gulps and sighs.  But, off they went and off they succeeded!

We changed up the picture and I asked them to repeat the process and they were quickly able to come up with the appropriate rule.

Consolidation came next. 

Fill in the blank:  ______ + ______ (n-1)

Students came up with "first number" and "change" which I defined as a "common difference."  So, off they went to answer thin-sliced problems to write the rule for the arithmetic sequence.  At this point, they were simply writing the expression ___ + ____ (n-1).  There was no equal sign.  There were no subscripts.  I eventually flipped it around and gave them the expression and they had to produce the sequence of numbers.  Success!!  Not only were they nailing it, they were claiming it was easy!

I did not use any vocabulary other than "common difference" and I didn't even emphasize it.  Next class, the students will pick this up again and we will expand to add geometric sequences.

I'm keeping my fingers crossed for two solid (or semi-solid) BTC lessons in a row.  That would be a record!  

So Much to Learn!!!

 In July, I attended the Building Thinking Classrooms Conference in Phoenix, Arizona.  It was absolutely amazing, yet very overwhelming.  There is just so much to learn!  When I left the conference, I was feeling empowered about building my thinking classroom for Algebra 1 this year, instead of "dabbling" as I had done in the past.  I am now one month into the school year, and I have so much to learn!!

As luck would have it, I had another opportunity to attend a workshop with Peter Liljedahl at a local high school.  This opportunity came at the perfect time for me.  Though much of it repeated a lot of what I have seen and heard before, I now have some experience.  I would say that my practice has been, at best, hit or miss.  There are just so many nuances that I have been floundering over.  Attending this workshop has helped me to identify what I'm missing and what I can do to improve.

When I returned to school today, we were learning about slope.  I started with a task that I modified from a  Math Medic activity, which can be found HERE.  This allowed us to explore rates of change and we connected it to the slope of a line.  Our second task, with a new group (something I picked up in Friday's workshop), was focused on using the slope formula that students had learned last year.  They worked through the thin-sliced problems and it was great!  I was able to walk around and give help where needed.  I consolidated by doing the "sort these three problems" conversation, which was pretty quick, then we went off to take notes.  I am NOT doing the four quadrant notes at this point.  One of the best parts about this particular modality of instruction is that it is not necessary to go all in with all fourteen practices.  I am picking and choosing which ones work for me and my students and doing the four-quadrant notes is not in that list quite yet, never mind the fourth toolkit!

Thank you to Bethlehem High School and to Peter Liljedahl for another great workshop!  I'm sure we will meet again!

Algebra 1 students and rate of change

Me and Peter L.

Introducing My Thinking Classroom

 I just started my 28th year teaching high school math and I had the best first day(s) ever!  I teach in a block schedule, so last week I had two classes on Thursday and three on Friday.  I am teaching AP Calculus BC and Algebra 1.

In my calculus class, students completed two tasks at the boards.  I modified two lessons from MathMedic. The first was to find an estimate for an instantaneous rate of change based on a free fall by Austrian Felix Baumgartner, starting with a video of his accomplishment.  Instead of completing the entire MathMedic lesson, I had the students use the graph from the lesson to come up with an estimate of his velocity 30 seconds into his freefall.  We discussed how to make the best possible estimate since a rate of change is (generally) over time.

In the second lesson, I used a graph that showed the velocity of a motorcycle over time and asked the students how far the driver (me) travelled.  They were able to determine that they had to find the area under the curve and were able to estimate using geometric shapes.

The two activities were designed to introduce them to differential and integral calculus and we had a discussion of what calculus is all about.

In Algebra, students started with The Tax Collector.  Then we did some housekeeping items and moved into The Answers Are.  Both activities, found in the book, were great!  Students were engaged, getting ideas from other groups and having some great conversations.  We debriefed after both activities.  I asked the students three questions.  First, why did we use dry erase boards and not paper?  Paper just feels too permanent so we all feel better when it's easy to erase things we don't want to keep.  Second, why were the groups created randomly?  Some good conversation here about getting to know others as well as my explanation that it goes beyond the social.  Sometimes our classmates simply understand something better or can explain it better than Mrs. Swift.  The third question was about standing up.  One young lady shouted out "blood flow!"  Awesome!  Yes, standing allows us to move a bit and gets our blood flowing.  That, in turn, means our brains are more engaged and we think better.

All in all, a great first day!

Building Our Thinking Classrooms

 It's been a VERY long time since we blogged.  My thoughts are that we got so turned around during the pandemic that it took a couple more years to feel like we are back on our feet.  This feels like a good time to get re-started with blogging about our classrooms.

Speaking of our classrooms, both Randy and I have really gotten excited about Peter Liljedahl's (@pgliljedal) book Building Thinking Classrooms and most of our blogging will be about our experiences creating and growing our own Thinking Classrooms.

Randy is still doing academic intervention for grades 3-6 at Greenwich Elementary School in Greenwich, NY.  In addition to this, he is co-teaching mathematics in a 4th grade classroom, where he and his partner teacher are beginning their second year with a thinking classroom.

I am still teaching high school mathematics at Saratoga Springs HS in Saratoga Springs NY.  This coming year I will be teaching AP Calculus BC and Algebra 1.  It is in my Algebra 1 classes that I am planning to implement the first two (two-and-a-half?) toolkits of a thinking classroom.

We were fortunate to have had the opportunity to attend workshops this past May in Connecticut with Peter Liljedahl as the facilitator.  The workshop I attended was focused on how to close tasks, which I found extremely helpful as I was getting better at giving tasks (especially thin-sliced tasks) but did not feel comfortable with this next step.  Randy (and his co-teacher) attended the workshop about assessment, and they found it very helpful.  

At the end of June/beginning of July, we traveled to Phoenix, Arizona to attend the 2nd Annual Building Thinking Classrooms Conference.  This was AMAZING for us!  We were able to speak with others who are just as excited as we are about this framework and got so many ideas that we are eager to put into action in our respective classrooms.

This upcoming year, in our BTC journey, Randy will be focusing on assessment and how to assess within the BTC framework and compare/contrast it to the more traditional forms of assessment and grading.  He is using the grading rubric provided by Tim Brzezinski (@TimBrzezinski) and Melisa McCain (@mccainm).  I am going to be focusing on tasks (creating/finding, launching, closing) and note-making.

So, this is what the 2024-2025 school year will look like in the Swift household.