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!

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!!

Thursday, September 22, 2016

Sharing My Story

Recently (read as at least a week ago), I read Sam Shah's blog about his Notes on the Beginning of the School Year.  I loved the idea of getting to know my advisees better by first sharing some information about myself and then making individual appointments with each student. I framed it as telling them "My Story." least parts of my story, because...I can't tell them my story in 8 minutes.

I talked about growing up in Michigan, my parents getting divorced when I was in second grade, living with my grandparents through high school, and working really hard to make sure I went to college. I told them about teaching at my first school and how I realized my last year there that I really wanted to leave Michigan. I told them the story of how I came to end up at my current school. And how when I left everyone I ever knew and all the preconceived notions of who I was supposed to be, that I was able to be my most authentic self. I was able to come out as gay when I moved to Colorado. I found my wife, I embraced who I am.

I would like to think it was a pretty powerful 8 minutes. I have set up meetings with them over the next two weeks and I'm really looking forward to hearing their stories!

Just because. 

Wednesday, September 21, 2016

Developing Understanding with Desmos

It's hard to believe it's already the fifth week of school! As one teacher put it at lunch the other day, "the shininess has worn off."

Although, I don't feel like that's true for me! I honestly feel like I'm having an amazing year. It's not perfect (could someone PLEASE help me spice up Pre-Calculus? The kids wouldn't even buy into the "What is a sandwich?" activity! It's dreadful! How do I get them to loosen up and interact?! Anyway, that's another blog post!)

I am teaching Algebra 2 Honors for the first time this year. Though it is incredibly time consuming to have a new prep, I'm trying to be thoughtful with how I do things instead of flying through it to save time. I've been spending many of my 4am-5am's (while I wake up slowly and drink coffee) exploring Desmos activities.

In our first unit, I asked students a few time to draw a plausible graph for a situation. I was trying to get at how they do with taking a word situation and/or picture and being able to convert it into a plausible graph. It's not something I directly taught in that unit, but I wanted to see how they would do. It was a mixed bag! So, enter the desmos activity "Function Carnival." Students drew possible graphs for situations then they were able to "play" them and then adjust if their graph was wrong.

As I walked around, I noticed students drawing things that weren't functions. When they would "play" the graph, all of a sudden there would be two cars on the screen. Without me having to tell students, they were able to figure it out (sometimes their neighbor helped them too!)

Fast forward to this week, when introducing functions for the first time. We did "Function Carnival part Deux." It introduced the notation of functions and the general concept of function/non-function. I loved having a situation/context that I could refer back to - and will continue to refer back to!

In class, after the Desmos activity, we did an "Intro to Functions Handout." can you go wrong when it's "edited with love?"

Friday, September 16, 2016


For the past two years, I have been lucky enough to have only 2 preps (Geometry Honors and Pre-Calculus). This year, I have a new prep (Algebra 2 Honors) on top of the other two. Now, it's not the first time I've had 3 preps. But it's the first time I've had three preps while married, and while trying to make sure i have a great work-life balance. When I first moved to Denver and was single, it felt easier to spend all my waking hours preparing things for my three classes. Now, I feel like I'm taking away valuable time from my family (and I don't even have children...I don't know how you folks do it!)

This is the second year that I'm teaching a PrBL Geometry class, and I still do each problem before students do them. I could probably "get away" with not doing that, but they go a lot faster this year and I think I'm reminded of past approaches as I walk through a problem. I don't want to give that up. In Algebra 2 Honors, it's a new prep. I can't get away from creating things! I have a lot of material to work with, since a teacher who taught the class for many years shared all her stuff with me (and she is totally the type of teacher to take and make things better year after I didn't simply inherit materials that have been static for years! Woohoo!) I still have to work a lot to make those materials "my own." Thankfully, another teacher is teaching it too - so we can split up some of the work.

Then, there's Pre-Calc. The class I've been teaching every year since I started. And I feel like this one (in terms of prep) is always last on my list because I have so many of my old resources to use. And I feel like it's getting an unfair backseat. I recognized this week how much different this class FEELS than my others. Is it a culture thing? Do I treat them differently because they're older? The class is SO quiet. It's sometimes hard to get them to interact. I have to figure out a way to change this....

I have a test planned for Monday (I already don't like giving tests Monday) and I'm thinking of postponing it. We could spend a day finishing our "Me by the Numbers" presentations, doing a class-building activity (thinking about one I recently read about having students argue their favorite numbers using research), and a Desmos activity to practice fitting quadratics (since that's part of what we're studying this unit). Okay. Writing that paragraph made me realize that I really need to do that. AND IT SHALL BE DONE.

And moving forward, I need to put a little more effort into this class. No more backseat for Pre-Calc. Three can sit in the front seat....right?

Happy Friday ya'll!

Thursday, September 8, 2016

Selected journal responses...

I had my freshmen write a journal response last week and here are a few highlighted pieces:

"So far, I've been absolutely rekd by a couple math problems, but I know that I can ask my dad for help. It has SUCKED to have so few classes with my friends and (insert name of person here). But I've channeled my inner-early-career Beyonce ('I'm a survivor, I'm not gon' give up.') I've learned that I can do more math than I thought, and that I can think on my own more than I thought in math terms"

Early-career Bey

"The challenges I faced [this week] were correlated to my successes. I had to push myself to try and find clues that would help. This mostly took time, and was a good thinking struggle."

"I just love the new style, plain and simple." (talking about PBL)

"It was very shocking to find out that there is no such thing as a natural math student. Since Kindergarten even, it seemed like some students were able to grasp concepts easier, however I was surprised to find that we just need more information to learn more."

"In the past, I've struggled with participation and now, I feel super comfortable with my class and sharing my ideas."

Do yourself a favor and grade that stack of journals at school if you can!

Grading journals is incredibly time consuming, but I think it's one of many things I have done in the first few weeks to build a classroom community that I'm convinced will help us learn math together. I got to laugh a bit, and I learned a ton about my students! I hope to blog tomorrow or sometime this weekend about the other things that we have spent a good chunk of class time doing (apart from the math) that have been worth every minute!