Monday, January 9, 2017

Sum and Difference of Cubes

We are about four days into our polynomials unit (Algebra 2 Honors) and want students to see that being given a polynomial in standard form is simply not as useful as having it in factored form. So, we're going to spend a day practicing factoring.

That said, they've never seen how to factor the sum and difference of cubes. And we didn't want to just TELL them. So, instead, we thought of a way to have them figure it out!

Follow the link here for the whole handout, titled "4. Factoring"

Overall, I think it's helping students to make connections! Thought I would share the resource!


Warm-up
1.     Given 4 is an x-intercept of f(x), what must be a factor of f(x)?



2.     A portion of the graph of a polynomial is below, what (if anything) do you know about…


a) the degree of the polynomial?



b) its end behavior?



c) its roots? Their multiplicity?





3.     Identify the end behavior and x-intercepts of

a.  h(x)=x(x+3)^3(2-x)                                             b)g(x)=-x^5 +4x^4-4x^3        






c)  f(x)=x^5-3x^4-x^3+3x^2







Almost all the polynomials we have explored so far have been in factored form, which is convenient as we can easily find the intercepts.  Sadly, polynomials are often in standard form, which is far less convenient.  We already know to take a quadratic (a second degree polynomials) from standard from to factored form (by factoring).  This worksheet will serve as a review of those methods as we all exploring a few other ways of factoring as well. 

Sum/ Difference of Cubes

1.     On Desmos, or your graphing calculator, graph y = x^3 + 8 
           a.  What is the x-intercept? What is its multiplicity? 



            b. Given your answer above, what must be a factor of x^3 + 8? 




This means that (x^3 + 8)=(answer from above)(something).  But how do we find that something?

            c.  What degree does something have to be? 



Therefore (x^3 + 8)=(answer from above)(ax^2 + bx +c). 

            d.  Using what you know about distribution, what does a have to be?  c? 




            e.  Find b. 






            f.  So factored, x^3 + 8 = (             ) (                               ). 

2.     Try to same process to factor, x^3 - 27 








 (this is on the next page)
What you hopefully found is that
(insert answer to first)  and (answer to second)        (check your answers)

These are categorized as the sum and difference of cubes.  And are factorable as

            Give formulas here!  



Sunday, January 8, 2017

These are two of my favorite things!

This week, my phone has changed the way we do things in class (and the suggestion by @cmmteach of the #MTBoS). Often, I find that I want to students to share their work in class, but it always takes so long for them to write it on the board (especially in Geometry, when we have sometimes complicated figures that go with our work). SO, I have begun snapping a picture of their work, then sending it to Google Drive (must have the app on your phone) and almost immediately opening it on my computer! The student will go up and walk us through the work. Something that might be interesting as well is to take a lot of photos during their work and pause when I come to a really interesting solution. Then we could display that one and see if other students are able to explain it.




 I understand here that I could air drop, but it doesn't
 work when I'm on the school's network!





It's a small thing, and one a lot of folks already use...but I have been loving it!

The second thing is less related to work, but has been super fun and works with my Type A-ness. That thing is Bullet Journaling. I live for being able to check items off a list. And my weekly layout is essentially a combination of my to-do lists for each day. On top of that, it also allows me to track certain things, like whether or not I made my bed or drank enough water. This might sound terrible to some, but I have been digging it! I get most of ideas for layouts from either Pinterest or Instagram!




Those are the two favorites for me in this moment! I'm looking forward to reading about everyone else's favorites!

Monday, January 2, 2017

You say Goodbye, I say Hello!

I was telling my wife as we were setting goals for 2017, that it almost doesn't feel like a new year to me. August feels more like a new year as a teacher. That said, we're getting ready to start a new semester tomorrow and it will be a great time to try some new things and make some new goals!

2016 has been an interesting year! I had spine surgery in May that took me out of the classroom for the rest of the school year. Because of that surgery, I feel like a new person! I didn't know what it was like to NOT have chronic pain!



My wife and I bought a house this summer. We did a lot of work on it ourselves, turning our garage into a home gym, painting 5 rooms, removing a popcorn ceiling (okay, so I personally did NO work on that one...), and we have some upcoming projects too!



I started my ninth year of teaching in August! Along with the other Algebra 2 Honors teacher at my school, we ditched the textbook (as a main resource anyway) and chose to do our own thing. It has been very time consuming, but worth it! That said, with two other preps, those took the backseat. My goal is to put more effort into those other two classes this semester. I went into work over break to work on all my classes, and I feel like I'm in a great spot to start the new semester! (I'm happy to share this with you if you'd like it - send along your email address in a DM on twitter and I'll give you reading permissions on our Algebra 2 folder in Google drive!) (Our naming conventions changed during our last unit).

Goals for 2017:

I want to blog more consistently. I set my goal at 1-2 times per month for now, as I think it's completely doable. If I do more than that, so be it!

I want to read at least two professional books - starting with Tracy Zager's new book! I sent an email to my department chair asking if I could buy it with department money....we'll see what happens!

I want to re-engage in the #MTBoS on twitter. After the election, I crawled into a hole and didn't want to face the world....but I need my math folks. You all make my life better!

Apart from my professional goals, I set a goal of working out at least 275 hours in 2017 (my #fitbos goal) and I want to workout an average of 5x per week - for a total of 260 workouts (including hiking). I want to learn calligraphy this year, work on my friendships - both new and old, and make sure my wife and I have at least one date night per month!

Happy 2017 all! I'm looking forward to being/staying more connected with all of you!!