Let’s just share…Rational Expressions Question Stack

Haven’t written a post in a while – most of the time I feel like I really don’t have much to write about.  I think I’m probably too critical of my own ideas.  Everyone else seems to have such great ideas and resources that I often think… “someone else has probably posted about that already” or “mine’s probably not that great – no one would want it.”

What’s different today?  Well, I’m working on getting back on twitter in hopes of pulling myself out of a bit of a slump AND I happened to read Julie Reulbach’s (@jreulbach) post on sharing resources and “that other site” (not naming it because I don’t what to give it any extra attention!)  One of the things that drew me to Twitter and that I still love is the fact that my fellow math tweeps are so willing to share and so kind – to people they have probably never met in person.  They just SHARE… because its nice and they truly want everyone to be successful.  How amazing!  Really… just awesome!!  Wouldn’t it be great if the whole world worked that way…



So, in an effort to give back a little I decided to do a short post and share a resource.  To be perfectly honest, this is just another form of a resource that another awesome math teacher shared on her blog.  Sarah Carter (@mathequalslove) posted about a question stack that she used in reviewing rational function operations.  Here’s her post.   I totally stole her idea and created my own that included some multiplying and dividing.  Thanks Sarah!

BTW – if you haven’t checked out both of these sites – GET THERE!  NOW!  They are amazing teachers with such creative and fun ideas!

Anyway… here’s my stack.  Feel free to use it, not use it whatever!  Sharing is caring!!


Question Stack – PDF

Question Stack – Doc

BTW – My kids loved this and thought it really helped them review!  Thanks for sharing Sarah!



First Day Plans – Year 33

Its truly hard to believe that I’ve been teaching for 32 years.  First days still feel like those first days with the crazy school dream the night before and the excitement and apprehension before classes begin.  I’m sure this year will be no different.

You’d also think that, after 32 years, I’d have this first day thing down pat.  Well… I’m sure I did something pretty decent one year … but then I decided that wasn’t good enough and changed it.  That’s the great thing about teaching.  Every year is a “do-over!”  This year I’ve decided to pull together a few things from previous years.

  1. Start with the Name Tag activity.  I’ve done this one for the past several years and I really like it.  It gives me a chance to get to know my students and to give them the chance to get to know me.  I love responding to each student even though it takes some time.  Full disclosure – I may have stolen this from my twitter friend Shelli Temple (@druinok).  Here’s her post on her first day activities.   I always use the inside to ask a question each day and to respond to their answers.  I haven’t decided what I’m going to ask them to put in the outside corners quite yet.  I seem to change it up every year.
  2. Once they’ve shared some things about themselves with the members of their group, I share some things about myself.  Some years I’ve just told them some basics – how many years I’ve been teaching, married, how many kids I have etc. and then some years I’ve made them guess – quiz style.  Not sure where I’m at yet this year.
  3. Introduce productive struggle and cooperative group work.  I did this activity (Teaching kids to struggle) last year and really loved how it went.  The idea is to introduce the students to the idea that we can all learn something even if we don’t know how at first and we can all help each other learn it.
  4. At this point I’ll more formally introduce the idea of growth mindset and how that plays into how our class will run.  We’ve been using standards based grading in all our classes from Algebra 1 to Algebra 3/Trig for the last 4 years or so.  I have a mix of 9th – 12th graders in my Algebra 2 classes.  Some will have experienced it and some not so its important to me to be sure everyone is clear on why we do it and how we do it.  I use a mix of videos and a prezi to help out here.

You’ll notice that there is really no place here where I say – Go over the syllabus or class rules.  Nope – don’t do it.  I cover procedures and expectations as we need them and the students first task to do on their own is to review the syllabus.

So that’s it… Day 1.  I have found that I really can’t do much more.

Post script – Teachers start back tomorrow (8/15) and our kids come back a week from Wednesday.  While I’ve been thinking about these plans and sketching them out in my head and as a draft here for several weeks, recent events have caused me to pause a bit.  You see, I live in Charlottesville, VA.  I teach an amazing, extremely diverse group of students at the local high school.  I’m honored to work with colleagues that truly love and care about our students and their families and each other.  That’s what we do – WE LOVE, we don’t hate.  My students need me and I need them.  We will rise!

First day plans – LOVE my KIDS!  ALL OF THEM!  Every. Single. Day.

New Year – Same Goals

So, as I mentioned in my last post – this time last year – I’ve been at this a long time.  This is year 33.  I’ve consistently said that, when I stopped having fun, I’d leave.  Well, last year I didn’t have my normal crazy amount of fun. There were moments of joy.  I seriously love teaching but I was like this the majority of the year.   Running like crazy and completely overwhelmed!

Now, I really can’t blame anyone but myself.  You see, I have a problem – several actually.


  1.  I love to learn – so I tend to say YES to anything that remotely interests me.  Seriously… I’m crazy like that.
  2. I like to help – so anytime there seems to be something that needs to be done I decide to do it.
  3. I’m a perfectionist – so I can’t do anything halfway.  Which is also why I can only seem to write one blog post a year.

Now the question is, if I said I would leave when I wasn’t having fun anymore, why am I coming back?  Well  I just can’t leave on a bad note.  I need to retire on my own terms and when I feel like its time.  Its not yet.

So, my goals for this year really aren’t much different than the were last year.  Hopefully I’ll be more successful.

  1. Do MUCH less – now there is a certain amount that has to be done but not every lesson has to be a day in Disney World.  Not everything has to be perfect and I don’t need to revise something just for the sake of revising it.  
  2. When I have a choice, do ONLY what fills my SOUL – I’m cutting out those things that I felt burdened by last year.  
  3. And give myself the GRACE to be ok with saying No.  I can’t say it better than @mathequalslove said it in her post (please read it …it really says it all…My ONE GOAL for the 2017-2018 School Year.)

I’m hoping those goals will lead me back to what I love – collaborating with my colleagues and loving and teaching my kids.  I’m counting on all my friends to hold me accountable!  Help me out – PLEASE!!

31 and counting…

Back to school backgroundSo today I officially started my 32nd year as a teacher.  I have said that, as long as I’m having fun and not exhibiting signs of losing my mind (no comments please), I’ll keep at it.  I’m hoping that someone lets me know before I’ve officially stayed too long at the party.

Even after 31 years, I still don’t sleep the night before the kids come and I still feel like I’m attending a great reunion of friends at the beginning of the pre-school work week.  I would definitely miss all the relationships I have with both kids and adults if I left.  I also really enjoy learning new things and am constantly searching out new opportunities to learn.  I’m sure I could search out those opportunities if I didn’t teach but they are always presenting themselves in my life as a teacher.  My fellow teachers know exactly what I mean here!

So, you might ask, what’s the problem… sounds great!  And it is great!  But, like almost anything, there is downside for me.  They say that the first step in solving a problem is admitting that you have one……..Here goes……006CFD531000044C-3512147-image-a-1_1459151551854

I say YES a lot and then find that I’m overwhelmed.  And, yes (there I go again), I’ve had 31 years to figure out how to NOT do that – so far I’m failing that learning target.

Its just not in my blood.  I’m a helper.  Its just who I am.  So the purpose of this blog post today….

  1.  To get these thoughts down – Shelli Temple, here you go.   Thanks for the push!


      2.  To ask for HELP!!  So, other than learning to say NO more, what kinds of things do you do to create better balance in your life?

What goals do you set for yourself that help you maintain your sanity?

What habits do you have that you find help you relax and not stress out?

Where do you draw the line and say …nope, I’m taking a break?

I’d love to hear your thoughts.  I absolutely LOVE my job but I often say I wish it was less hectic.  Turning to my online PLC to help me find ways to make that happen.

Happy New School Year to all!

Sum it Up! – Complex Numbers

I have been very lazy about blogging this year – even though I promised myself I would be better.  I seem to never feel that what I’m doing or the experience I’m having is really that interesting or different.  Lately though, I have run across a couple of activities that I found particularly successful for my kids and that my kids seem to enjoy so I thought I’d share one more.  Just to be clear – none of these are really my creation – I’m not that creative – but are “borrowed” from others.   Love how the MTBoS shares!!  This one is an adaptation of Cheesemonkey’s (@cheesemonkeysf) placemat activity described HERE in her blog – Cheesemonkey wonders.

What I did….

I changed the problem cards some because I wanted to include some more basic adding and subtracting ones and some division ones.    I did start using the placemats in my first class but quickly realized that they really weren’t using them so I punted on that with the remaining classes.   My kids are in groups of 4 so it worked perfectly to have them in their groups and just do their work on the desks (my desks are perfect white boards.)    Just like Cheesemonkey, I had the kids add up their answers and check their work with me before they could move on.

Here are the problem cards and the placemat template.

How it went…

It was a great activity – the kids struggled just enough and quickly figured out that they had to help each other to find their errors.  Once group decided that they would rotate their cards when they didn’t get it right the first time.  Working each others problems helped them find errors.  Another group made it a norm that you had to work out the whole problem on the desk so everyone could see and then they’d help find errors.  The discussion was FANTASTIC!  Just like a previous activity, they begged to finish it the next day and commented on how much they learned.   I will definitely look for ways to use this again!

Unexpected Win – Rational Expressions

I teach two levels of Algebra 2 – academic and honors.  The academic kids are just a little behind where my honors kids are currently – they just need a bit more time to absorb, practice and really pull all the concepts together.  So, as I was starting the rational expressions unit with my academic class, I spent some time searching for some activities to help scaffold the concepts a bit and to also try to find something that the kids might enjoy – rather than just a worksheet.  Well, I really didn’t feel overly successful in my search.  I had done speed dating with my other class but my academic class wasn’t quite ready for that.  Then I hit pay dirt.  I ran across this worksheet from a school division in Texas (thanks Birdville schools.)  I’d love to give the teacher credit but I haven’t figured out who wrote it yet.  I liked the scaffolding that this provided for my kids and it was somewhat “game” like.

I tweaked it a little to include some basic simplifying ones.  Here it is…. the tweaked version.

I ran the original  on colored card stock thinking I’d cut it up and have them sort.    I didn’t have a chance to cut them up before class so I just handed them out to the kids and said… “hey, cut them up, sort them and write your answers on your answer sheet..”     My kids went… “Why?   Why can’t we just do them on our sheet and match the letters.”    DUH!  (I think they secretly hate sorts but just cooperate to be nice.)

So they went at it… and I mean they went crazy working those problems.  They wrote their work on their answer sheets and then looked for the answer on the original – crossing it out when they found it.  (This led to me putting the originals in sheet protectors so they could mark them with a dry erase marker.)  I mean they WORKED, worked and worked some more.  They discussed together, they compared answers, they found each others mistakes.  It was AMAZING!!  When they didn’t finish, they begged to finish the next day.  I was blown away – what seemed like a worksheet to me was a puzzle to them.  WIN, WIN, WIN!   Why?

So, of course, I had to ask why?  Their responses –

“The answers were there so I knew I could get it eventually.”

” Having the answers and steps took some pressure off.”

” I could figure out my mistakes when I could see the steps in between.”

“Working the problems with the steps and the help of my friends made it easy!”

“Having the steps shown helped me understand.”

“I’m still struggling with factoring, so having them factored helped me get to the answer…. and my factoring got better too.”

Will I use it again?   Already did – created one for addition and subtraction too!   Still not completely sure why it was such a win… but I’ll take it!

Here are the links to the word version of both.

Simplifying, Multiplying and Dividing Rational Expressions

Adding and Subtracting Rational Expressions

Can an activity be a success and failure at the same time?

The short answer – YES!   Now for the longer answer….

Anyone that’s taught Algebra 2 knows that the rational functions unit can be a challenge for students – “WHAT Fractions with letters in them!  You’ve got to be kidding me, Ms. Burke!  Do you really hate us this much?”   So we’re in the middle of that unit…. and kids are, of course, avoiding the “home math enjoyment” I’ve been sharing with them.  They were in desperate need of some practice that didn’t involve me showing them or explaining.  THEY needed to practice and explain.  So enter @k8nowak and her speed dating activity.

Now full disclosure here – even though I had the great privilege of working with Kate last year at CHS, I never tried this activity.  For those that might not have heard about the activity it goes something like this…..  Each student is given a problem on a small card to work.  They work their problem and check their answer on the back.  If they need help, they ask the teacher for help.  Once they are done, the students are split in half and seated across from each other.  The students exchange problems and work each others problems.  If they run into a problem, they consult the “expert” sitting across from them for help.  Once they have each completed the problems they exchange back, stand and all shift one student over – change partners.  Exchange problems and it keeps rolling.  Here’s Kate’s much more detailed explanation.

I just didn’t go there….It seemed like a lot of work – setting up the room etc.  This year, it dawned on me that I could use the counters in my hallway as our desks (saving the rearranging of my room ) so I decided to give it a whirl and I was desperate for something other than a worksheet.  I was smart enough to try it first with my easiest class.  You know what I mean – the class that just goes with whatever and won’t fly off the handle.  Thank heavens!

2016-03-02 15.45.05

Lessons learned – 

  • Don’t turn 20 teenagers loose in the hall count on them dividing themselves up evenly.
  • Don’t wait until you go into the hallway to have them work their first problem.
  • Remember to encourage the students to become “experts” on their problems – think about what their classmates may ask.
  • Be persistent about NOT helping once they start working in partners.
  • Don’t believe that your students have any idea how speed dating works (thank goodness!)  You will need to help them shift the correct direction the first time.
  • Gauge the difficulty of your questions carefully.  You probably want to try to find problems that are of somewhat equal difficulty so students will finish about the same time.  I just grabbed the problems I set aside from last year and some were definitely harder than others.
  • Don’t necessarily think you’re going to get through a lot of practice.  Especially the first time or if your problems require more steps.
  • Create an answer sheet for them to record their work – hold them accountable (I actually did do this!)

So… you can tell what didn’t go well. (Gosh… that list looks long!)  Some of those were …DUH…of course that went badly.  What was I thinking?   What did go well?  Well, we had some productive struggle!  The students worked hard to get those answers on the back of the card.  They weren’t complete experts at their problems so they had to work hard to get through.  I was pretty persistent though – I made them help each other and only gave limited suggestions.

The verdict – it was successful!  An even more successful with the other two classes where I fixed a few of these things!  The kids felt accomplished and got that practice I wanted them to get.  And even more successful, because as I have told my kids, we all learned something, even me!  We all grew some synapses in our brains because we all made mistakes!

Thanks Kate for sharing – we’ll speed date again soon!!


Domain and Range – $25,000 Pyramid Style

Anyone remember the $25,000 pyramid?  That show where one person faces the board of words or phrases and has to describe them well enough for the other person (facing away) to say the word.  This lesson is similar to that idea – only with graphs of functions! And yes, it was just as fun!!

This is a lesson that one of my PLC members, Hank Sohn, developed and shared with us.  He doesn’t blog so I asked for permission to write about the lesson for him.  He wanted to be sure I mentioned that he completely “borrowed” this idea from ones online.  As we all know, sharing is caring!  So this is not necessarily unique, just an adaptation of others excellent work and ideas.

Domain and Range – UGH!  Somehow doesn’t seem that difficult to all of us but is super hard for Algebra 2 students to grasp.  They just struggle really interpreting equations and graphs to make sense of it all.  In my many years I’ve tried teaching it several different ways, all with varying levels of success.  Basic notes telling students about domain and range, Sketchpad examples of “squishing” a graph to the x and to the y… you name it, I’ve tried it.  Because of that limited success I’m always open to something new.  At this point in Algebra 2, we’ve taught composition, inverses, and function families.  We’ve mentioned/reviewed the vocabulary of domain and range but not specifically discussed them.

This lesson starts with students in pairs – back to back.  One student has a blank graph and the other student has the graph of a function or relation.  The student with the graph begins to describe the graph to the other student while that student draws the graph.  They have a few minutes to try drawing the graph before they can check to see how they’ve done.  They trade places – one drawing before describes now and visa versa – and try another. Originally we’d planned on giving some points and making it more of a competition but they didn’t really seem to need this. They had a great time trying to get it right and were super excited when their graphs turned out looking the same.  We did this through three or four graphs before we moved on.  The graphs we used are below.


Once we finished playing we debriefed by asking the students what words they used to describe their graphs.  We got the ones we wanted – highest, lowest, point, furthest right, left, hill, valley, flat, root, intercept etc.  We also got some unique ones – whip (perhaps from the whip-nana dance craze!), wave, swoop.  The class had a discussion about which words seemed most useful.  We used those to bring them around to domain and range.  That discussion moved over to a more formalized set of notes covering the basic examples and a sort below.

These notes adapted from these.

Would love to hear how it goes for others.  So far our kids seem to be doing better than I’ve experienced in the past – A WIN!


PS – One thing I’d change or add – some discontinuous graphs – rational functions with asymptotes – for them to draw.  I’d also include those in the notes.

What do I make?

When I tell someone I’m a high school math teacher, usually one of two things happens.  They tell me how much they hated math and how bad they were at it or they make a comment about how they could “never do THAT JOB” and usually includes a look of pity..   I’m not especially surprised at the  “I hate math” or “I was never good at math” statements.  Somehow it seems to be okay to say that you’re bad at math but we’d never want to say that we can’t read.  Its the other statement and particularly the look that bothers me. Pity…. really… you feel sorry for me. Well let me clear a few things up.

First,  maybe you couldn’t be a teacher- I don’t know you.  It does require the patience of Job, the energy of that
battery powered rabbit, the ability to make a million decisions every hour, passion, curiosity and a commitment to
OTHER people’s children that is a rare thing.  Secondly, because of all of those things, its not a JOB, its a calling.
And yes, its hard!  But for all that difficulty, comes lots of benefit.  So, what do I make?

Hum…..Many of you may have seen this video – What do Teacher’s make? – its great!  I highly recommend it. Yep, I feel like I make a difference and every now and then I get reminded about that difference – that ripple in the water –  and that reminder is so much better than any big check anyone could ever send me!2013-12-11 18.31.27

The last few weeks, I’ve been reminded and I honestly wanted to use this blog so I would remember when maybe I’m not having as great a day.  First, a current student emailed and asked me if I knew any CHS alumni that had “cool” jobs (they wanted to interview them for the school paper.)  I didn’t remind her that I had a pretty “cool” job but did send her a rather impressive list.   Here’s just a few of the “cool” jobs my “kids” are doing – working with teachers through The Math Forum, associate director of the Center for Peace and Global Citizenship, members of the band Parachute, Stanford researcher developing clean water sources in poor countries, an engineer for Exxon Mobile, a police officer reconstructing accidents (using lots of Geometry), developer at Google, interactive graphics editor TIME magazine and of course at least four are teachers with one more in the wings.  There are many, many more – plumbers, carpenters, administrative assistants, tech folks, moms and dads – all making their own impact.

Then last night I got to spend the evening with a bunch of my kids – the class I sponsored.2015-10-23 19.31.11 They are, of course, not kids anymore but in my eyes they’ll always be my kids.  I couldn’t be more proud of them as adults, as “my kids” and now as my friends.  The hugs, the old stories, the connections – it was an amazing evening.  I look forward to keeping those connections for many
more years to come.

2015-10-23 20.11.10So, I’m a teacher.  I’m not bringing home a huge pay check.  I probably won’t own a beach house.  But I have made my ripple in the water – hopefully for the good.  If these kids are any indication, I’m the winner.  Pity me – everyone should envy me!  I’m looking forward to the day when someone says – WOW, teaching, that must be an amazing way to spend your day!   It is!  Thirty-one years later, I’m still having fun and I’ll keep at it as long as I am.  It is truly a gift!




Algebra 2 SBG Learning Targets – Virginia Edition!

Ok… I’ve fallen off the “one blog post a week” bandwagon but hope to jump back on this weekend!

In the meantime, I’ve had several folks ask for these so I thought  would go ahead and post them.  These were definitely a collaborative effort between several teachers at my school over several years (including Kate Nowak – @k8nowak – last year!  We miss you Kate!)

I am in Virginia which is NOT a common core state so these may not be appropriate for someone in a common core state!

They seem to be constantly a work in progress so please feel free to ask questions, make comments etc.

Sorry for the tiny picture here.  I’ve tried to make the learning targets big enough to read but feel free to download to see a bigger version along with the success criteria or use the link below to get a copy for yourself.

Here is a link to a document that will allow you to make a copy for your own use!