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.

2020 Thoughts and Quotes

"How's it going?"

What a loaded question right now!  If that's how you often greet people, you might want to rethink asking a teacher that one.  We might tell you.

The reality is that, for most of us, it is not going well.  This is a hell of a lot more work than I had anticipated, and I anticipated a lot of work!  I just spent my entire Sunday working.  From 8:30 in the morning to about 5pm, I did school work.  First I had to start by figuring out how to undo the damage I did last week.  I don't want to tell that story, but let's just say it ended with tears.  Then, once I figured out what I should do for this coming week (yes, I am only planning a week at a time, but that's better than the day-at-a-time I've been doing), I had to figure out what I was going to do, how I was going to do it, then get it all organized.  So, here I am, at the end of the day and back in my pajamas, finally winding down.  Just to get up early tomorrow and put it all into action.

Okay, enough whining.  I really don't want to whine about this.  I know I'm not the only person in this boat and that others have it worse than I do, so on to the positive.

I was at a meeting last week where we had to identify something positive that has come out of the last few weeks.  That was easy.  The first Friday before students were coming in, a colleague put out a frantic email asking for help on how to set up her computer with Zoom and Smartboard and all the other stuff.  No less than five teachers jumped to help her.  This experience has bonded us together, a faculty of 170 teachers, like nothing ever could.  Colleagues from all different parts of the building, and district, have jumped to help one another with even the smallest things.  One teacher taught me how to best set up my computer so I can work with all my students.  I was able to show that teacher how to use a certain software to get the kids to collaborate more from home.

So, though it's tough, and every single day is a struggle, there are some little victories and lessons and I'm planning to focus on those.  One day at a time!

Flipping 2020

 We haven't written anything in our blog for a very long time, but I thought this would be a good time to start again.  This coming year is going to be crazy!!  I am teaching AP Calculus BC, Algebra 2, and a class we call Topics in Algebra 2 for students who aren't quite ready for the full Algebra 2 curriculum.

Our school is starting on a hybrid plan.  Students are divided into two cohorts.  Cohort 1 will be in-building on Monday and Tuesday and Cohort 2 will be in-building on Thursday and Friday.  All will be learning from home for the other three days.  So, teaching will be more of an adventure than usual!

I have always wanted to explore flipping my classroom but have never felt like the time was right or that I had all the tools I needed to do it.  Well, I'm not sure if the time is right and I'm pretty sure I don't have all the tools I need, but I'm going for it anyway!  I work with a great (much younger than me) teacher who inspires me to try new stuff and I will be sharing "our" stories as well.  Caitlyn Gironda (givemeasine.blogspot.com) gives me a lot of great ideas and has made me a better teacher as a result.  It's so nice to have someone to brainstorm with and maybe, just maybe, I'll be able to pull this off.

I have purchased and read the book Flipping with Kirch by Crystal Kirch, which has given me a foundation upon which to build my flipped classroom.  She has so much to offer but I will be doing my best to keep it as simple as possible for both me and my students.  I have developed a simple version of the WSQ ("whisk") to use on Canvas (our district's choice for a learning management system) that I will be using with my two algebra courses.

To be honest, I am way more concerned about the structure and pace of my AP class.  The schedule we have gives us less contact time than I am used to and there have been no changes made to the course.  I will still be flipping the class but using the notes and assignments I have always used in the past.  I have a feeling that this class will be going full speed ahead right from go!

I have chosen to look at the year as an opportunity to try something new, probably more than one something, and that will (hopefully) keep me sane!

Statistics Day 1

I have never felt comfortable teaching statistics in Algebra 2 (particularly the common core version) and that's because I never took a statistics class.  I know - how can I be a math major without taking stats?  I'd like to say that I got lucky, but I'm getting more and more interested in statistics that I almost (ALMOST) want to take a class.  Since I don't teach Regents-level Algebra 2 this year, I thought I'd be able to spend a little more time with my stats unit.   I'm going to use my blog to reflect on how things go and then be able to make adjustments for next year.

While some of my students will go on to take a statistics class either in high school or college, my main goal is for them to have a basic understanding of statistics and it's "ishy-ness."  We are constantly bombarded with statistics (politics, sports, health) and it can very easily be used to sway public opinion with innuendo and implications.  As an educator, I want to teach my students to love mathematics, but I also work on preparing my students for real life.  Having some knowledge of statistics can be very powerful.

Our first day was learning some basics of statistics.  I posed a question that I found at Sarah Carter's blog mathequalslove regarding the hiring practices of airline pilots and whether or not a discrimination case could be made.  The scenario describes an airline with 25 junior pilots, 15 male and 10 female.  The airline is going to promote 8 of these pilots and all are equally qualified.  To make it fair, all names are put into a lottery and the promotions go to the 8 people randomly selected.  When the results of the lottery are announced, 5 women and 3 men receive promotions.  Some of the men claim that the lottery was not carried out fairly and want to file a discrimination charge.

The question I posed to the kids was whether the men have a case?  I have not asked a question all year that got such strong, vehement responses!  It was beautiful to witness!  No one really said anything that was unexpected

The male:female ratios should be the same so they have a case.

But it's not impossible to get the results they did so they don't have a case.

Yeah, but how likely was that to happen?  It must have been rigged!

We should find the probability of this particular result.  (This was followed by lots of murmuring about how to do just that.  Since combinatorics are no longer a part of our curriculum, they had absolutely no idea how to do this.)

Then I dropped the bomb - we're not going to answer the question yet.  The plan is to learn about stats and re-visit the discrimination claim at the end of our unit.  I wanted my kiddos to start statistics by understanding that it is not about finding a specific value, which is their belief about anything with math, but more about interpretation.  This was a great intro.

The next task for students was to write their reactions to two different studies and the subsequent news articles written about them - no conversation, just gut reactions.  The first study was funded by a tobacco company claiming that they made "healthier" cigarettes.  The reactions were thoughtful regarding the statistics themselves, as well as the danger of smoking cigarettes.

22% less tar and 34% less nicotine means there is still tar and nicotine in the cigarettes.  You shouldn't put any of that crap into your system!

I don't think people should use statistics to make something bad look good.

The second study was regarding the IQ scores of first-born males vs. second-born males.  It was a well-presented study, giving a margin of error with a 95% confidence interval (which we haven't discussed at all).  My favorite comment:

It's only a 95% confidence level, so they really shouldn't be making such a strong statement.

We went on to discuss different types of studies, their advantages and disadvantages, as well as what types of questions should be asked when evaluating a study.  There was a lot of really good discussion and debate.   Then, they broke out into groups to discuss the class' reactions to the smoking and IQ studies.  The conversations were rich and insightful.  Even the students who are generally not the "talkers" in class were speaking up and mostly because of the issue being presented, not the mathematics behind it.

The goal for day 1 was to get students thinking about claims that have roots in science or psychology or the law and whether the information presented to them was meaningful or actionable.  The next goal is to demonstrate how statistics can support (or collapse) their arguments.

AP Calc Review

We finished our last lesson of new material for BC Calc on Monday (to a round of applause!) and started review for the AP exam in May.  Review is divided into three acts...

Act I started in February.  I gave my students the AB-level diagnostic test from Master Math Mentor (www.mastermathmentor.com).  This is a great resource that diagnoses each student's areas of strength and weakness.  I charge the students to work on the test without looking anything up.  Then I send them their results analyzed by the MMM formulas. This is great because all the work has been done and I don't have to correct anything, I just hand over the results.  I then printed the material from the MMM calculus manual and organized a file system by topic.  Students can select which topics they need and can work through them as needed.

This is directly beneath our picture of Sir Isaac Newton (drawn by a former student)...

Act II started this week and for this, I use the material and information from Lin McMullin's blog teachingcalculus.com for our full-class review.  He offers ten different problem types for the free response questions and insights and advice for each type.  I've taken that advice, printed it out for my students as a packet with the referenced FRQ's and MC questions.  Each day is dedicated to a different topic.  I started this review system last year - everyone works on the different types together as a class.  The point of this is to not only review the material, but to work on what is expected for credit on the actual test.

Act III also started this week.  We don't do a lot of multiple choice practice during the year, so it's important for them to see it in the review.  I use the D&S review books and students are assigned 15 questions each night for homework.

This may seem like a lot, and it is.  But, the results will speak for themselves.  My students will perform well on the exam (we have posters with the names of our "double-fivers" in the classroom) and they will be ready to tackle college and, they will really know their calculus!

A little introduction...

In 2003, I began teaching grade 4 in Greenwich Central School District, a small, rural district (approx 1,000 students in K-12) in Upstate New York. Upstate, as in north of Albany. I remained at the grade level for 12 years. In 2014, I had the opportunity to loop with a group of 4th graders to grade 5 and thoroughly enjoyed that unique experience. As a general education, elementary teacher, I taught 5 content areas per day, not just math. Math, as a specialty, began to work its way into my professional life around the same time as the new standards were being rolled out and my district was adopting a new math curriculum. At the same time, it began to spill into my personal life in the form of nerdy math conversations over dinner with my favorite mathematician, aka my wife, which lead me to seek out my current position in my district.

Following the year in grade 5, the AIS (Academic Intervention Services) Math position opened up and I expressed interest in filling this role. Since 2015, I have supported students in math in grades 3-5 and added grade 6 this year. I have also built an enrichment math program for students in these grades. During the 2017-18 school year, I worked as a Curriculum Fellow for Great Minds, the writers of Eureka Math. I spent half my time in my district and the other half working from home revising and updating the Eureka Math curriculum.

Before I knew it, my Amazon wish list began overflowing with professional books in the area of math. The more I read, the more my interest grew. Next, I was looking for opportunities to share what I had learned beyond the "four walls" of my district. Over the past two years, I have sought out opportunities to share my curriculum and pedagogical knowledge by presenting at local and statewide math conferences.

And now, as husband and wife, we present to the world our math blog.

Come along as we take this journey from Counting to Calculus!

Why We Have This Blog

Randy and I have been married for 13 years.  We are both triathletes and you can read about my triathlon adventures on my other blog here.   While we met through our triathlon experiences, we are also both teachers, which has given us even more to share with each other.  We want this blog to be a place where we can share with other math teachers at all levels.

Randy is an elementary teacher and has spent a big chunk of his career in a 4th grade classroom.  A few years ago, he made the switch to become his school's Academic Intervention Services (AIS) teacher for grades 3-5 mathematics.

I am a high school math teacher.  While I have taught most courses from Algebra 1 on up, I am currently teaching a class called Topics in Algebra 2, as well as AP Calculus BC.

As you can imagine, there are some nerdy conversations at our dinner table most nights!  We both work with some great teachers and have learned a lot from them.  However, we have discovered the value of sharing our thoughts on teaching math with one another.  Yes, there are many differences in what we do, but we have come to discover that there are many similarities.  We find that what one is doing with their classes can often be adapted for the opposite grade levels.

We have two goals with this blog.  The first is to individually share our experiences within our own classrooms.  The second is to share the connections we make between elementary mathematics and high school mathematics.  We plan to each write posts and occasionally we will write some together.

Thanks for reading!

~Karen