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!