Celebrating Every Day: #Teach180

Before the start of this school year, I discovered the #teach180 hashtag via the post Sarah Carter made about it. I had no idea about its existence last year, and I have been so glad to be a part of it this year.

I decided to participate. This year is my 4th year teaching, and in August coming back to school it was the first time I finally felt comfortable – maybe comfortable isn’t the right word, but like I knew what was going on, what to expect, and had plans ready. So I took this chance to push myself to become better, and to start to network with other teachers across the country (and world!).

And so I was off!

I’m going to include some of my favorites of my own #teach180 posts from this year as I talk about the experience.

Every year, my kids are still talking about Slope Dude in April and May when we watched the video in August, and this tweet made Sarah’s roundup of tweets from the first few weeks of school.

At first, I sort of felt pressure to make sure I was doing really cool things. Well, no, not even really cool things. Things that looked really cool. So I could take a cool picture to post on the internet.

Some of the time, that actually resulted in me thinking of a new and helpful way to explain or organize content!

Or, sometimes it resulted in a really fun new activity!

But sometimes, it just resulted in a lot of extra stress and days spent staying late at school with me thinking, “I have such a boring day planned tomorrow I’m not going to have anything cool to post for #teach180!”

Then I realized how silly this was. And at some point, I gave myself permission to just post a picture of a worksheet if I needed to. Or some grading. Since then, there’s been nothing but LOVE for #teach180! I regularly scroll through the hashtag and find the GREATEST ideas to use in my classroom – and it’s especially cool since, in math at least, a lot of the curricula have vaguely similar sequencing/pacing, so often I open the hashtag to find an activity over something I’m getting ready to teach the very next day!

(Like this paper airplane box and whisker plot one, which got so many students not in my class to ask about the tape on the hallway floor!)

I’ve gotten a ton of questions about my Sports Statistics course (which I developed the curriculum for myself, based loosely on the first 2/3 of the AP stats curriculum), which has been really fun to share with the world.

It’s also been a cool way to share good tech tips:

Occasionally, I’ve brought my coworkers into my posts for the day, because I love using this hashtag to show off my really amazing school:

And here are just some more favorites:

It’s also been a great motivator to keep things going now that we’ve hit that end-of-the-year time when everyone is kind of like “ehhhhhh” and all the students are asking you about their grade every five seconds. It’s sort of preventing me from the urge to just coast into the finish line.

All in all, it’s a really amazing hashtag and I’m so glad I decided to participate. I am 100% going to continue it next year, and you should join! It’s also a great way to get started on twitter, because you’ll be forced to post at least one tweet per day, and you know other teacher’s will be finding your tweets through the hashtag and will want to follow you!

That May Teacher Feeling…

I feel like teachers all have a weird love/hate relationship with the month of May. (maybe for some teachers, it’s the month of June, depending on your academic calendar)

On one side, May is the time of year when so many things come together. Content starts clicking together for kids, they start seeing how it all works together and is related. They’re ready at this point to do deeper explorations and more confident in making conjectures they’re unsure of. They rely a little bit less on you and a little bit more on each other and themselves to confirm results. You all know each other really well and the classroom is just a fun environment to be in at this time of year.

On the other side, everyone is restless. The kids can feel summer coming, those days of no alarms and being outside and playing basketball in the park instead of sitting in class. The teachers can feel the sweet promise of a bit of rest and free time. And everyone’s tired. It’s been a full year of work, learning, hardships and triumphs. It’s been a lot. And it’s almost over.

I find myself always getting really emotional during May. I’m the teacher who cries at every graduation, because I can’t handle how proud I am of my students graduating. It seems this emotional state has already started, even though we have about 3 weeks to go still.

I’ve cried every day this week.

Sunday it was because I was changing my hair color and it didn’t quite turn out how I planned. (It’s fine, I was just in a Sunday night panic mode at the time).

Monday it was because a decision was made that I felt was unfair. It was made worse by the fact that when I tried to explain why I was upset to my mom, she didn’t understand what I was saying. I think that situation’s going to be okay too – I mean, I will at least be able to live with it, but I needed to give myself a little time to be upset with it.

Tuesday it was because, well, I’m trying to buy a house at the moment (which is the most adult thing I’ve ever done in my life) and I got the inspection report back and I didn’t know what any of it meant and my dad and realtor were trying to tell me opposite things of what I should request from the seller in terms of repair. That was a bad one because I HATE when I don’t understand things, it makes me feel like I have no control and like I’m going to be taken advantage of by someone who understands it better. This also ended up working out, I made a decision of what to request after lots of crying on the phone with my dad and the seller accepted it! I will be so glad when I close on this house…

Wednesday I started crying looking at Mother’s Day cards. Honestly. At that point I was like Liz, what is wrong with you.


It’s just May. There’s a lot of stress and a lot of emotions whirling around everyone, and I’m frustrated because my attendance is dropping by the day and students are demanding to know their grade and how they can raise it every five seconds.


This post is just to let you all know that if you’re feeling this way, if you started crying in the card aisle at Dollar Tree as well, that it’s okay. It’s May, and you’re a teacher, and you’ve accomplished SO MUCH GOOD this year. It’s okay to be a bit weak and emotionally silly this month.

Let me know if you’ve cried at any really stupid things lately…

(I haven’t cried yet today, so we’ll see if that holds up)

Sprint to the finish, friends! Enjoy the great things about the end of the year, don’t wish them away too soon!


Update: Thursday, I cried at the choir concert, when one of our students sang Leonard Cohen’s Hallelujah and the whole choir spontaneously joined in on the last chorus. I LOVE MY LITTLE COMMUNITY OF KIDDOS SO MUCH.

Know Your… (#DCSDblogs Week 4)

It’s the last week of the #DCSDblogs challenge for our district! I’m really proud of myself for coordinating this challenge and it’s been so fun to read and write blog posts with other awesome teachers in the district!

This week’s prompt is Tips and Tricks, which asks us to write a post with advice for a brand new teacher.

I think this is pretty hard to tell from the outside unless you knew me well that year, but I spent a lot of my first year teaching being extremely overwhelmed, stressed, lost, and anxious. The hope is that my advice could help some other brand new teacher avoid feeling like that!

It all boils down to:

#DCSDblogs week 4-April 23 - april 29



Know Yourself:

I knew going in that one of my biggest struggles in teaching and in being part of a school staff was going to be the fact that I am a huge introvert. I often prefer being alone to being with people and I’m not going to be the one to begin a social interaction. So the first thing I knew was that it was going to be difficult for me to get to know my coworkers and feel like a part of the group. So my mantra for year one was say yes to everything. Every time a staff email went out to invite people for drinks after school, I went (unless I already had plans). When people asked me to eat lunch with them, I did. If they requested volunteers for a district math curriculum writing, I went. I took every opportunity I could, even when I just wanted to go home and be alone.

And I made friends. Not as many my first year, but as my second year started, I realized that because I had said yes to all those things year one, people started inviting me to other things. Birthday dinners, barbecue at their house, to go to the play at another high school with them.

That worked for me because I’m introverted and knew I needed to force myself to get to know everyone, but it could be something else for you. Figure out what your barrier becoming a part of your school staff group is going to be and find a way to combat it.


The worst piece of advice I’ve ever heard for teachers, and I keep hearing it, and I’m sure you’ve heard it, is “don’t smile until November”. What the heck?! People kept telling me that, and I knew it wasn’t me. I’m never going to be a strict disciplinarian, it’s just not my personality. (I’m not even going to get into the fact that I think this is horrible advice even if you are a stricter teacher – especially if you’re working with disadvantaged students. They need smiles.) I knew that advice didn’t fit me, so I ignored it.

You need to know yourself to be an effective teacher. You can’t pretend to be someone you’re not in front of a classroom of students. They will know, and they will not respect you.


Know Your People:

Teaching is an incredibly unique profession. It’s a job that everyone thinks they understand, because everyone went to school and had teachers, but that’s not the same thing at all. We carry all the burdens and celebrations of our students with us, and no matter how hard we try not to, teachers often take the failures of individual students to heart.

It’s been proven that psychologically, teachers suffer many of the same effects of PTSD just by being exposed to all the struggles of all of their students. That was a fact that totally floored me when I first learned it.

I’ve discovered that what you need is your people. You need to find people on your staff, just two or three, that you totally and completely trust. And sometimes, when you’re having a really rough patch, you need to go find one or two of your people and just vent it all out. You need to know that these people won’t judge you for your frustrations, you need to trust them to not gossip about what you’ve told them.

These vent sessions should be contained – once all parties are done venting, you move on and leave it all there. It can’t turn into a negativity spiral, which is sometimes hard to do. But what I’ve found is that you need people like this THAT ARE TEACHERS, THAT WORK AT YOUR SCHOOL. You can’t take all of these burdens home and place them on your family and friends. Plus, they won’t completely understand anyways, because they aren’t within the climate of your building. You need someone who will understand what you’re frustrated about, and why it’s frustrating.

Then these people are also the people you can go to when you want to celebrate something – get really excited about a student who answered a question about something that happened three units ago, or show them this really cool lesson you’ve developed.

Find your people. Share the ride with them.


Know Your Students:

Getting to know your students as people is so, so important. They aren’t just vessels to dump content into. They. Are. Humans. Should I throw my favorite quote in here again? I think I should:

Every human life is worth the same, and worth saving.

– Harry Potter and the Deathly Hallows

If you can think of your students as people, it becomes a lot easier to find worth in all of them, even the ones who aren’t doing as well in your class. You should definitely do surveys at the start of the year, or find some activity that fits you that lets you get to know them as people.

This is my favorite section of my intro surveys:Capture

Many of my students struggle to fill in 10 things that make them happy, which is telling in itself, but it definitely shows their priorities immediately. The personality traits one I use to roughly sort them into a Hogwarts House (did you see that one coming????). I do that because as someone who is very invested in the series, it works almost like a Myers-Briggs type personality test would and tells me a lot about how students will work best and what methods will probably help them understand content the best. If I know they’re a Gryffindor, I know there’s probably a lot of pride involved, and that they most likely will struggle to ask for help when they need it, for example.


I also mean know your students in another way. The thing I most wish I had learned about in my teacher ed program and didn’t at all was mental health and childhood trauma. I do work in a specialized environment where all of my students are at risk students and so most if not all of them deal with either mental health or childhood trauma, but I think it is so, so important to know about even if you don’t work with a specialized population with a high incidence of that type of student.

If you’re a new teacher (or a not so new teacher) who feels like you haven’t been trained on mental health or childhood trauma, I strongly suggest you seek out your school counselor and ask them for resources. In the meantime, two of the things I’ve found most helpful are below.

The Adverse Childhood Experiences (ACEs) study, originally done in the 1990s in California, explores the link between different types of childhood trauma and the effects on later life. It’s shocking and upsetting how prevalent these are and what they can do to people’s brains. I’ve now taken a class on it, watched two documentaries, and read a lot about it and it’s quite literally changed my life. It’s all about the fact that this isn’t a choice; trauma truly affects your brain’s development and changes the ways you encounter the world.

Mindfulness exercises can really help students who struggle with anxiety, but also any student, as they struggle to focus and keep themselves calm during stressful class situations. Our mental health therapist did a professional development with our staff on mindfulness, and I’ve linked to some of the resources she gave us, including a mindfulness toolkit.

Note to Self (#DCSDblogs Week 3)

The theme for week 3 of the #DCSDblogs challenge is Oops! The goal is to talk about a mistake you made in your classroom recently and how you addressed it.

This year is the first year I’ve taught Algebra 2. Starting last summer and throughout the year, I’ve made sure to start my planning for this class a bit earlier than normal so I can process the content I need to teach and get my mind around the best way to present it. This is also the first class I’ve used Interactive Notebooks in, which has actually overall helped me with finding the core ideas of the content and finding the pieces that are going to resonate most with students.

The unit we just finished covered rational functions. I took a bit to re-acquaint myself with the process of finding asymptotes, adding/subtracting/multiplying/dividing and solving these functions. However, this unit came right in the spring break / Iowa Assessments time of year, and so my planning all got a bit wonky.

I definitely didn’t leave myself enough time to do the planning of these lessons justice, and it showed. Here are my notes to myself for teaching this unit in the future:

We started with sketching graphs of rational functions. The very first thing I realized is that my students, while proficient at factoring quadratics, have not gotten very efficient at it. This meant that every single problem seemed more complex, because pretty much regardless of what you’re doing with a rational function, the first thing you need to do is factor the numerator and denominator.

*note to self: more practice factoring quadratics to enhance efficiency

I also realized that in my process of sketching a graph – finding x and y intercepts, vertical and horizontal asymptotes, holes, etc., I had them finding the intercepts first. This ended up not making sense, because if there’s a hole at one of the intercepts, that point isn’t actually an intercept, so the holes need to be the first thing you find. This one was a fairly easy fix because I just had them write on the inside of the foldable “move step 2 to after step 4” and explained why we needed to do it in a different order. Everyone was fine, and we moved on.

*note to self: use a few examples to make sure the order of your process makes sense

Then, we hit the exit slip problem I had included on their foldable. They were feeling okay about finding the characteristics of the graphs, not so great about actually sketching the final curves amongst the asymptotes, intercepts, and holes, but I figured we could give the exit slip a shot and then come back and discuss it the next day. Rookie mistake: I had taken the functions I used for the foldable from one of their textbook’s worksheets for the section, and I hadn’t graphed the exit slip one myself because I wanted to leave it blank in my teacher INB since the students were supposed to complete this one on their own.

Turns out, this particular rational function has no asymptotes, which we had not seen any examples of and so every student completely panicked. They correctly found that there were no vertical and no horizontal asymptotes, but then they all just stopped working because they were convinced that wasn’t possible for a rational function and they had done something wrong.

*note to self: check the exit slip problem. Also, don’t assign a unique case example for an exit slip!


Next, we covered simplifying, multiplying, and dividing rationals, along with complex fractions. This section actually went really well, and my students felt really good about themselves after having a freakout when they saw the complex fractions and then realizing that they had all the skills to deal with them already! The only thing I want to change here is…again…the order of the steps. It makes more sense to rearrange the problem into a multiplication problem before factoring. My students were the ones who figured this out, because they’re awesome.

*note to self: seriously, check to make sure the order of your process make sense.


Adding and subtracting rational expressions is probably the most complex process in our district’s Algebra 2 curriculum. Either that or factoring polynomials above degree 2. Regardless, I did not do a good job of presenting this, or practicing it, or anything. I kind of botched this one big time.

First, the foldable didn’t leave enough room for anything to happen.

*note to self: give students enough room to do math on the paper!

Then there’s the fact that I just…didn’t explain this well. There’s really no way around it. I did not teach this well. My students didn’t know when they were finished with a problem, what to do next, they kept getting lost in calculations.

*note to self: spend some time doing more problems with adding and subtracting rationals yourself, so you can break down the structure better

*note to self: search the #mtbos and other online resources to see how other people break this skill down

*note to self: really, just scrap this section and start over from scratch for next year

I can end this post on a good note, though, because I made sure to set aside extra time to plan for the last skill in this unit, solving rational equations, and I think that turned out pretty well. My students loved making the pockets for their INBs and getting to stick the practice problems in them, which we also did for simplifying rational expressions, and it was a good way to fit more practice problems into their INBs without taking up more pages.

They also showed me that they really had mastered solving quadratics earlier this year, because that’s what you end up having to solve when you’re solving a rational equation. I was really proud to see them pulling out the Quadratic Formula or factoring again and just going at it!

*note to self: good job on this one 🙂


I learned from this section that I need to be more intentional about planning, especially with content I haven’t worked with myself in a while. I have stellar students in my Algebra 2 class, so we were able to overcome my shortcomings in planning without too much trauma, but they did get lower quiz scores over this content than I’m used to from them.

I’m hoping to have a bit of time left at the end of the year to come back to this content before their final, but I don’t think it would be productive to keep pushing forward with it right now. They need a break from it after the train wreck I put them through.

Please let me know if you have any great lessons over rational expressions and functions – I would love the help in improving this unit for next year!

*note to self: word processing systems don’t think asymptote is a word and it’s incredibly frustrating.

You Won’t Do This Alone (#DCSDblogs Week 2)

Last night, I went to see one of my favorite bands, With Confidence (and also Don Broco and State Champs) in concert.

One of my favorite songs by them is called Voldemort – this is also one of the ways they originally caught my attention, because obviously I’m going to be intrigued by any band who titles their songs after a Harry Potter theme!


As I thought about what I would write for this week’s #DCSDblogs post (The theme this week is Teachers Learning from Teachers), I started to realize that this song encompasses a lot of the things I wanted to talk about.

I remember the first night that she said
“Oh maybe I can do this on my own”

I am an incredibly stubborn and independent person, which is sometimes a flaw and sometimes an asset. When I set out on my first year of teaching, I was convinced that I could do everything myself. If I didn’t know how to do it, I could figure it out. The song is from the perspective of a friend who insists on being there for the girl described, even when she says she can do it on her own.

And I will try to hold you up
Through those times when you are gone
Despite the weather, it gets better
You won’t do this alone

In case any of you out there didn’t know yet…teaching is hard. My first year, I was overwhelmed and barely keeping afloat at times, and yet I continuously refused to ask anyone for help. Most of it was a little voice in my head that went, “you don’t have a specific question to ask, so you’re fine. You’ll figure it out.”

Heather, one of the other amazing math teachers in our building, worked herself into the cracks in my stubbornness over the course of the year. At the start of the year, when I refused all of her offers of help, she left me alone for awhile. As the year wore on, she would drop in occasionally and ask how things were going. Her questions got more and more specific – “do you need help with anything?” started to become “Which class is your most difficult this quarter? Is there anything you wish you could do to work with that?” and she made it harder and harder for me to just brush her off and pretend like I had everything under control.

I won’t even pretend this is a finished process today, because I still tend to think I can do things for myself, but Heather has helped to convince me that it’s not weakness to reach out for help in your classroom. We’ve built a relationship of bouncing ideas off of each other that now often will start a conversation off with “Okay, so I’m going to tell you this idea and I want your honest opinion even if it’s bad.”

I remember the first night that she went
To find her little place inside this world

The other piece of advice I’ve gotten that has hugely impacted my teaching was from one of my cooperating teachers in student teaching. On my first day in his classroom, Brian told me “I never take any work home. Home is my family place. If I bring things home, it starts to bleed into my time with them and hurts my relationship with them.”

Over the time I spent with him, he expanded on this idea to say that it doesn’t work for every teacher to take nothing home – he preferred to stay a little bit later at school in order to keep his home a work-free zone, but that the main point was to build boundaries for yourself.

It’s far too easy as a teacher to occupy yourself with the goings on of your classroom and your students every waking moment (and as I’m sure you can all relate, sometimes they spill into the sleeping moments too!). Obviously, this can be harmful to our existence as humans outside the classroom – our relationships with friends and family, our outside interests, etc.

In my first few years of teaching, I firmly adhered to Brian’s model of taking no work home – I didn’t mind staying at school later if I knew that when I got home, I wouldn’t have any work responsibilities at all. It’s only in the last year that I’ve been able to reflect on and reshape the model a little bit to better fit my own mindset and lifestyle – I still usually stay at school to do most of my lesson planning and grading, but now I’ll throw in a spare hour here and there at home to blog, or read blogs, or get the ideas down for a new activity before I forget them. The point is that you need your “little place inside this world”, like the song says, that separates your work from your personal life. We’re in a weird profession where the work follows you everywhere, and for our mental health it’s important to create those boundaries – whether they are physical, mental, or both.

As of right now, my boundaries are pretty much that I complete all must do work at school, and then extra things like blogging or a new idea or possibly getting ahead of the game if I have spare time and really feel like it can be done at home. That’s what works for me!


There’s a lot of things I’ve learned from other teachers so far, but those are the main few.

  • you need help, and other teachers can give it to you (admitting it is not weakness)
  • create some boundaries between work and your personal life
  • having a teacher you can trust to bounce ideas off of that will respond with honesty and without judgement is golden

And I know that you’re holding out for better weather
And I can’t promise you that I’ll be round forever
If there’s one thing I know it’s that we’re good together

We’re good together. The people around you have a lot to offer you, I promise 🙂

Book Recommendations (Vol. 01)

Reading is my favorite hobby.

At the end of 2016, I posted a wrap up of the year and when I wrote it, I wanted to include a top 5 books list. Two things stopped me – first, it didn’t seem like it fit with the theme of “professional top fives”, I just really wanted to make book recommendations to people. Second, I had read too many books to choose five from the whole year.

So this year, I’m going to try making these book recommendations posts every 3 months (a quarter of a year). They say “everyone is a teacher of literacy”, so here’s math teacher me, trying to convince you to read more for fun. Amazingly, the first three months of 2017 have already passed?!

As a disclaimer, the summaries are probably going to be pretty vague because I don’t like to go into books knowing too much about them, but I’ll give you the gist!


I’ve already read 15 books this year – here are my 5 favorites:


The Problem with Forever – Jennifer L. Armentrout

Mallory is starting her senior year, returning to public school for the first time in several years after being homeschooled following a trauma she endured while living with a previous foster family.  She encounters a face from that time that she never expected to see again, who was a light in the darkness of that home.

This book hit me really hard emotionally and I ended up crying through much of the last third or so. It is a very realistic portrayal of many of the hardships that our underprivileged, at risk, and traumatized students face on a daily basis. It is very painful to read at times, but it is also full of hope and the path of rebuilding trust and connection after it has been lost.

We Are Okay – Nina LaCour

Marin left her entire old life behind when she left California for college in New York. It’s now winter break and she finds herself alone in the dorms, the only one who didn’t go home or on a vacation. Her best friend from California, Mabel, shows up to visit and forces her to confront everything she left.

This book is such a short read, and is a really raw examination of everything that comes with grief and change. It’s another one that’s about broken connections and how to rebuild them, and how to re-imagine your life when you discover that things aren’t exactly as you thought they were. It’s written so that you feel all of the emotions along with Marin, and go on the journey with her of confronting what happened.

The Hate U Give – Angie Thomas

This book was inspired by the #blacklivesmatter movement and starts out with high schooler Starr witnessing one of her good friends be shot and killed by a police officer. The rest of the story follows the unrest in Starr’s neighborhood, her struggles to reconcile the world of her mostly white and privileged school with her friends and family in her mostly black and lower class neighborhood.

This one is really powerful – I read the whole thing in pretty much one sitting. “What’s the point of having a voice if you’re gonna be silent in those moments you shouldn’t be?” It really calls into question how you can sit in your privilege and not address something that really matters, and shows you the people who don’t have that choice. This is one I think everyone, but especially every teacher, should read.

Homegoing – Yaa Gyasi

Homegoing begins with the stories of Effia and Esi, half sisters who have never met and don’t know of each other’s existence, at the start of the slave trade in what is now Ghana. It follows the trail of their descendants to present day.

This is one of the most beautifully written books I have ever read and I keep raving about it to everyone who will listen to me. Every chapter gives you a brief glimpse into the next descendant down the line and in every chapter I found myself wanting to read a whole book about that character. In the end, it’s not a spoiler to say the two lines find a way to intertwine again and it is lovely and wonderful. It also gives a very interesting glimpse into different perspectives on the slave trade and the history of some of the ancient (and modern) Ghanaian people. But wow, is this book lovely to read.

The Grapes of Math: How Life Reflects Numbers and Numbers Reflect Life – Alex Bellos

It’s basically various stories about the histories of different math concepts – including how Kepler and Galileo used to send each other anagrammed riddles of their new discoveries, a giant survey to discover the world’s favorite number, and more!

It was just a really fun read. I had previously read and loved Bellos’ Here’s Looking at Euclid and also loved it. As is evident from those titles, he’s great at puns, and I also think he’s pretty great at explaining the math from a layperson’s perspective – so if you find math fascinating, but don’t feel super confident in your skills, this one’s for you! It’s just really interesting and will make you think about things you never considered. I know a lot of people don’t love nonfiction, but this is a pretty easy read for nonfiction.


So there it is – my top 5 books of the first quarter of 2017. Please, if you’ve read anything great, leave me a comment and hype it up for me – I love recommendations! Also, if you end up checking any of these out, report back and tell me if you loved it as much as I did!

The Pride Bubble (#DCSDBlogs Week 1)

!!! It’s the first week of the district wide blogging challenge that I created !!!

I’m really excited for this blogging challenge – to see teachers in our district who have wanted to blog have a reason to finally get started.

Anyways, the theme for week one is One Good Thing – to share something good that happened in your classroom in the past week and explain how it was celebrated.


It feels like on a daily basis, I’m carrying around my pride in my students within this bubble. I’m always proud of them – I teach a population of students who are in a daily fight with the low expectations the world has placed upon them, and every day that they show up in my classroom gives me pride that they haven’t given up yet. So the bubble always exists.

Sometimes it’s very fragile and small, and sometimes it inflates more and more.

And then sometimes it bursts, because it’s just too full of pride to be contained anymore.

My pride bubble burst on Friday.


My Algebra 1 students came into my class at the start of the year with little to no mathematical success in their histories. The challenge at the start of the year was to get them to even try. Throughout the year, we’ve slowly started doing some explorations/investigations at the start of new material in an attempt to expose them to the ‘real mathematical world’ where you aren’t just told a rule or formula; you discover it. When we first started doing these, most of what I would get were complaints like “how are we supposed to do this, we haven’t learned it?” and the like.

On Thursday, we began an algebra tile exploration on solving quadratics by completing the square. We’d already learned to solve quadratics by factoring and using inverse operations, and I’d alerted them to the fact that by the end of the year, we would have FIVE different methods for solving quadratics. I even warned them that this particular method would possibly be the least favorite for many of them, because that’s been my experience in the past with students.


We began looking at some problems together as a class – I explained that our goal was to make one side of the equation into a perfect square of algebra tiles, and we reminded ourselves that if we add extra tiles to one side, we must add those same extra tiles to the other side to keep the equation balanced. My pride bubble started swelling when we reached the point where we wrote the factored form of our first example as (x+2)(x+2) and one of my students offered, with no prompting, “couldn’t we write that as (x+2) squared?”


Then another student noticed that the problem suddenly looked like the ones we had been solving the previous week using inverse operations, and asked if we could solve it like those.


The next day, they were off, using their algebra tiles to complete the square and solve quadratics, on their own or in pairs. As I circulated, I kept hearing things that made my pride bubble swell more and more.

“No, remember, you have to split the x tiles evenly because we’re making a square”

“Wait, in this one the ones tiles are with the other tiles to start. Don’t we want them separate? Can we just subtract them to move them to the other side?”

“We’re always going to add positive ones tiles, right? It’s either negative times negative or positive times positive.”

“I don’t think we even need to use the tiles for this one. I know what’s going on.”


And these kids, who fought so hard against these investigations when we first started doing them in first quarter of this year, started asking me and each other extra questions that weren’t even part of the written instructions.

“Wouldn’t it be cool if there was a set of algebra tiles with an x cubed tile? What would that look like? It would have to be 3D, but how would you decide which side should be red and which side would be the other color, because there would be more than two sides but only two colors.”


“Hey, all of these have an even number of x’s.”

Me: “Would it be harder if there were an odd number?”

“Yeah, because you have to split them evenly”

Me: “We’re going to talk about that on Monday”

“Oh, man, that’ll be cool!”

(This was a student who just last quarter frequently sat in class mumbling under his breath about how pointless the class was and how much he wished it were lunchtime and failed over half his assignments)


“Are there problems that you can’t do like this? What would those look like?”


At this point, I was sitting at a table grading the previous class’ investigations because they were moving along so well without any prompting from me. The pride bubble was pretty huge at this point, and I was just sort of smiling to myself in the corner.

They started to get to the last two questions of the investigation, which asked them to look over all the problems they’d completed and try to find the relationship between the number of x tiles in the original problem and the number of extra tiles they’d had to add to complete the square. These types of questions have always defeated them in the past – I don’t think they’ve ever been asked to generalize before they got to me, and so they just fight against having to do it. They also hate to actually read instructions, so I was expecting all sorts of questions just because they didn’t want to read the fairly large block of text of the question.

Instead, they started to read the instructions aloud to each other. They started flipping through their packets to look at examples. They read the instructions line by line and paused to consider each piece.

All of them at least found the pattern that we were splitting the x tiles in half.

Many of them found the whole pattern and were able to use it to correctly solve one last problem without using the tiles.

One student, considering all his examples, asked, “Miss Mastalio, what’s the word for the answer to a division problem?”

This was his final answer:


This was when my pride bubble burst. I wanted to cry so many happy tears. These kids have fought and fought and fought thinking about how math works this year. Somehow, the dam has broken and they’ve worn down.

For some reason, this investigation wasn’t a fight. It was a triumph.

It was a great reminder to just keep trying. That they need practice grappling with new ideas, with finding patterns, with expecting math to have logical conclusions. That it will eventually pay off.

These kids are getting a school wide shoutout on Monday – these are read over our announcements and I individually named each student in the one I wrote after class ended. They were my #teach180 tweet for Friday and I’m so excited to do our formal notes on completing the square tomorrow and be able to say, “I know several of you already found this pattern; what was it?”


I love fourth quarter, when everything starts coming together.




(Here is the investigation I used, which is adapted from the exploration from section 9.4 of the Big Ideas Math Algebra 1 curriculum)