Are My Rules Mutually Exclusive?

Awhile back, I read Sarah Carter’s post about the Guess My Rule activity she used at the start of the school year to build her group work norms for her classes. I really need to do a better job of building these in my own classes next year, but that’s another post. More recently, I saw her post about using these rule cards again with Venn Diagrams

My Algebra 2 students were right in the beginning of their last unit on counting principles and probability when I read this post, and I knew it would be perfect for us to practice the definitions of mutually exclusive / inclusive and the probability addition rules that go with each of those cases.

I really like doing activities that leave some parts of the problem up to chance, so that the answers don’t always come out “nice” or “normal” (I hate those terms, too. In this case I mean that some of the probabilities were often zero or 100%). I think that addressing these sorts of problems in their practice helps better prepare them for any sort of problem that may come at them, even when tests and other assessments normally have “nice” answers. It also forces them to really consider definitions as they work.

So, the first part of their task was to draw two of the rules cards and set up a Venn Diagram of the intersection. They had to decide if the rules were mutually exclusive or if they were inclusive, then use the proper probability rule to calculate the probability of rule 1 OR rule 2 being true out of the deck.

You can actually see the student worksheet that I made to go with this really well in this first picture, to see how they recorded their work.


Here’s one that was mutually exclusive:


We were also working on complements of events. For this one, they were asked to set up a Venn Diagram with THREE rules from the deck. Then, I asked them to describe the complement of one of the rules being true, and to find its probability. In retrospect, I should have changed a few things about this part of the activity. First, the labels for the rules need to go by the circles. My students really struggled with transferring their Venn Diagram numbers to their papers because they couldn’t keep track of which rule was which circle, especially if they had drawn their Venn Diagram in a different rotation than the one on their paper.

Second, I made the descriptions of the events too vague, so students really struggled with describing the complement.


I do think this activity was really helpful for my students to practice definitions of mutually exclusive, inclusive, and complement, and to practice using some probability rules!

I’ve uploaded the original file I used here, and a new version with the changes I made after doing the activity here. You can find the Guess My Rule cards in Sarah’s original post here.


Looking Back on the Things We’ve Done…

Yes, that title is from a Backstreet Boys song. I actually think the song works pretty well for the idea of reflecting on our year.

At the end of each school year, I like to have my students write end of year evaluations/reflections on their learning. I feel like it gives them a designated time to really sit and think about the massive amount that they have learned, and to be kind of impressed with themselves!

You can read my post from this experience last year here.

The questions:

  1. What did you learn in this classroom this year? What math concepts did you learn, what study strategies did you learn, what other random things did you learn through the course of our discussions and time together?
  2. What did you like and dislike about the COURSE (Sports Statistics, Algebra 1, Algebra 2 or Intermediate Math).  This should be specifically about the mathematics in the course, your textbook or assigned worksheets, etc.
  3. What did you like and dislike about Mid City in general, Miss Mastalio as a teacher, and the environment of the classroom? (You can be honest here, I will not be offended)
  4. Is there anything else you would like to tell me that doesn’t fit into one of the above questions?

I really enjoy sharing snippets from their responses because it’s a rare moment when they feel proud of themselves, and also because I always cry reading them. My students are the most genuine, most caring people I have ever met and it’s great to see behind the curtain a bit. It’s also nice to look back and have memories of each group of students I’ve taught (I’ve kept all the reflections from every year of teaching so far)

Our building has been through a lot this year, so I think it’s extra important to dwell on the good things that have happened within our community this year and to see the positive impact we’re having.

All answers are unedited from how students wrote them.

Question 1:

(written in highlighter to emphasize) “2 is a PRIME #. Always use parentheses!”

“I feel like the most important thing I learned is that math CAN be fun you just have to have the right approach and group of people to keep you engaged.”

“I learned that if you practice something enough you don’t need ‘studying'”

“You can have friends in unexpected classes. You can make jokes and games out of everything. You can have fun with everything as well.”

“I learned how to do math that I never knew existed. I also learned how to be a good student and listen to the teacher when they explain things”

“I think I might even like math better now”

Question 2:

“I got a lot more comfortable asking questions which helped me do a LOT better”

“I liked the question stack activity where we just solved the problems in the pile”

“Algebra 1 is great I really like the hands on learning and the fun activities in this classroom it really helped me learn” (wait, but did you read that? ALGEBRA 1 IS GREAT?!?!?!? my heart!)

“I sucked at graphing, I always have but with the help I got I figured it out and it was easy. I’m definitely gonna use these concepts next year.

Question 3:

“I like Miss Mastalio as a teacher and now the people in this class and just everything in this class. The aura, the vibes, the feelings.”

“Miss Mastalio offers a really comfortable, judgment free zone, and when paired with small class sizes it has been perfect for me to start truly understanding and even liking math.”

“I liked that Mid City has good teachers”

“I like that there are soo many people who actually care + are there to help”

“I like that no one bullies me here”

“I thought Ms. Mastalio was very funny and nice because she accepted my artwork even when it was bad” (it wasn’t bad, I have pictures of Baymax all over my room because of this student)

Question 4:

“I freaked out a few times in my head but this has been the most comfortablest I have ever been at math.”

“I like the way the room is set up, it’s easy to get around”


I think next year I would like to add parts to this reflection that are more specific to each class – ask what their favorite standard was and list all the ones we covered, for example.

We Know a Lot about Quadratics!

Algebra 1 is really a class of insane growth for students, if you think about it. Most of them come into the class still pretty uncomfortable with the thought of ‘x’ and what it means, unable to solve a one or two step equation consistently.

They leave knowing how to solve quadratics and convert them to other forms. It’s crazy.

This year, I assigned my Algebra 1 students an end of year project to review and mesh together everything they learned about quadratics in the last quarter and a half of the year. This included:

  • Solving by factoring
  • Solving by completing the square
  • Solving using the Quadratic Formula
  • Rewriting functions in factored form
  • Rewriting functions in vertex form
  • Rewriting functions in standard form
  • Graphing quadratics using key features like zeros, y-intercept, axis of symmetry, and the vertex

Each of them was assigned a different quadratic equation and directed to do that entire list of things with it. It was a very hard task for some of them, but it also helped them realize how much they’d learned this year. Working on the projects was a really good review for their final exams next week, since it reminded them of some things we hadn’t done in awhile. They also ended up working with quadratics that didn’t necessarily have “nice looking” (I hate that term) solutions which always makes them slightly uncomfortable.

Here are some action shots of them working on their projects, including use of several of our dry-erase templates for the Quadratic Formula, rewriting in vertex form, and factoring.


The finished products look really great on the bulletin board outside my classroom! (Along with my “ways to be mathematically smart” poster that I stole from someone on twitter that I can’t remember!)


The one that’s a different color from everyone else’s was my example so they could figure out which work they were supposed to show on the poster.IMG_1074IMG_1075IMG_1076IMG_1077IMG_1078

After everyone finished, I had them complete a scavenger hunt to find quadratics that had certain features: a negative axis of symmetry, for example. This ended up generating a TON of really awesome math conversation. I stood there listening to them have this discussion on the last Friday of the school year, during a shortened schedule, using vocabulary terms and pointing out examples to each other, and kind of got a little choked up. A group of them even were trying to see if they could all get different examples for each task from each other! It was pretty amazing to listen to them and think back to when they couldn’t solve a one step equation in August.



Files for the project prompt and a rubric for grading that I used are here. Also, files for cards for each quadratic that the students chose are here. I cut these out and had them draw one out of a bag. I completely randomized level of difficulty but you could separate them into levels of difficulty and have certain students draw from separate bags.


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.