Key Features of Polynomial Graphs BINGO

My Algebra 2 students have been working on describing key features of polynomial graphs. Our reporting standard for this unit is F.IF.C.7:
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Specifically, it is part c of this standard:
Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

We focus on the key features in this standard because later in the year, we have another reporting standard where the main focus is using factorized polynomials to sketch graphs without a calculator. So in this unit we are using Desmos and working on vocabulary. The key features listed in our district’s SBAR (standards based assessment and reporting) rubric for this standard are:

  • x-intercepts
  • y-intercept
  • end behavior
  • extrema
  • domain
  • range
  • increasing/decreasing intervals

We have described key features previous to this for piecewise functions, absolute value functions, step functions, logarithmic functions, and exponential functions. From those, they are pretty comfortable with intercepts, domain and range. New to this function family are end behavior (which in Algebra 2 we describe informally without limits as rising or falling left/right), extrema, and increasing/decreasing intervals. They really struggle with types of extrema (local/relative vs. absolute) and with increasing/decreasing intervals.

Before their quiz, I wanted to give them a bit of a challenge to help them process some of these vocabulary terms more. In this activity, the numbers don’t really matter as much as the features themselves, so it really promotes understanding of the concepts. Originally, my thought was to make actual BINGO boards for each student with pictures of graphs on them, and then call out a key feature. They would be able to mark off a graph with that key feature. I started to do this, and made a Google Slides presentation with all the key feature descriptions I wanted to use, and then I started trying to make the BINGO boards and realized how much work it was going to take to make what I wanted. I wanted everyone to have a different board, and I was going to have to take screenshots of graphs that met all the features I had listed…so I went back to the drawing board. I did find this cool site though, where you can make randomized BINGO cards, so if you were doing this with vocabulary or something easier, check bingo baker out! I am realizing now that this may have worked if I had made the prompts a graph, and the spots on their boards the key features…next time!

I decided to reframe it as more of a challenge than a typical BINGO game, but with their goal to still get 5 in a row. I made up a Google Sheet BINGO board, with a blank cell beneath each key feature description. As the instructions state, students are to use Desmos to try to make a polynomial graph that meets any of the described key features. When they do, they copy and paste the Desmos graph’s link in the blank cell, then highlight that cell to show they’ve completed it.

As soon as they started working, my students asked, “can we use the same graph for more than one box?”, which for some reason I had not considered yet. I made the decision on the spot that they could, because I figured if they could make a graph that fit a whole bunch of the requirements, then good for them. Depending on what you want from your students, you might tell them that each box has to have a unique graph. My students used the strategy of making a random polynomial and then seeing how many of the boxes it worked for and putting the link in all those squares. I think they enjoyed the challenge of trying to make a polynomial that fit a large number of the squares. If you wanted them to focus more on the individual squares, then you may want to force them to have a unique graph for every square.

You can see that this student used the same link for several boxes:

The students had a fun time with this, and it was a perfect length activity for yet another 2 hr late start day that we had (our winter has been miserable – 7 days cancelled and 5 days shortened!). I think I will be more strict on encouraging them to fill as many squares as possible once they get a BINGO next time to push them further, although many students did that.

A student also let me know that I had two of the same end behavior squares – I fixed this so that they are now different, but you may notice the duplicate in the screenshots! (This is what happens when you are trying to come up with 24 different key feature descriptions and not paying careful attention and there’s only four possibilities for the end behavior of polynomials)

You can make a copy of my Google Sheet for this activity HERE. This will make a copy of the file to your own Google Drive which you can then edit as you wish ūüôā


2019: Multiplying Me, Adding You

Those of you who have been following this blog for a few years (?!?! weird that that’s a possibility at this point) know that I have largely given up on New Year’s Resolutions. You know, those ones that you try for January and then forget about. I do resolution type things for Lent every year which is more effective for me, and I prefer to start the New Year with a theme instead of a goal that’s going to make me feel bad about myself.

For 2018, the theme was Unashamed, Unafraid, Unfinished.

Unashamed: I have found myself defending the things I like less this year while contemplating this word. I can more confidently say, “Yeah I just really didn’t like that” or “My favorite album of the year was 5 Seconds of Summer’s Youngblood” and not feel like I have to explain it or put myself down for it. I’ve also been working on self-deprecating myself less – I’ve realized that it’s a reflex for me a lot and I really don’t like that because I do like myself.

Unafraid: I have gone out on a lot of limbs this year. I’ve met a lot of new people and gone to a lot of social events and put myself in situations that I never would have before. I’ve talked about my feelings to friends and family more with the motivation that, you know, if these people are really important in my life, they’re not going to run away or be offended by me saying how I really feel about something. I gave a speech to all the teacher leaders in my district where I almost cried onstage! I’ve fallen back on the old phrase my college roommate and I used to use all the time: “what’s the worst that could happen?” – we would explore the worst case scenario about something we were afraid of and realize that, actually, that wasn’t so bad. I’ve done a lot of things this year where the worst case scenario was just like, “I won’t have very much fun at this thing” which…is not a great reason to not try something.

Unfinished: Pushing myself to be better this year has been really rewarding. In addition to working on less self-deprecation, I’ve worked on less insult based humor towards others – realizing that I’ve used that to cover up actually talking about how I feel. I’ve become a better teacher by trying new things and learning a ton in grad school. I worked so hard in our real analysis course in particular but also in other courses and am very proud to have maintained my 4.0 GPA for the master’s program so far!

So now, where am I headed for 2019?

Multiplying Me, Adding You

Okay, so it’s cheesy and mathy. Yes. But. Here’s what it means to me.

Multiplying Me

I’ve read a few things as 2018 ended talking about how New Year’s Resolutions are too often about changing ourselves. It got me thinking that I don’t really want to change who I am in 2019. I am really, really¬†happy with my life right now, and proud of a lot of things. So I want to multiply the things I like about myself, including some of the positive changes I started to make in 2018.

I think I’ve become a pretty good baker and a pretty good cook and I want to continue trying new recipes and new skills and do it more often, including sharing what I make with others.

I’m really excited about finishing my master’s degree this summer and I am ready to be super proud of myself and to celebrate the accomplishment – along with working really hard on my last two classes and my action research project paper. (My last two classes are also both branches of math that I really, really love, discrete math and statistics, so I am psyched).

I want to continue to work on my self confidence and read good books and keep my house clean and celebrate teaching moments I am proud of. And yeah, I want to make myself better in a few ways, but do it by emphasizing characteristics I already have but don’t embrace fully instead of trying to change who I am.

Adding You

I haven’t always been great at letting other people into my life. I am not always a great communicator. In 2018, I struggled a bit with my family relationships but developed a lot of friendships and started dating someone incredible.

So in 2019, I want to isolate myself less. I pushed myself a lot last year as I mentioned to go to more social things that I wouldn’t normally, instead of shutting myself in my house every school night and pretending I had things to do.

I love my trivia team and spending time with them outside of trivia nights as well – one of our team members got a new job so she no longer works with me, so it’s even nicer to see her every week (or most weeks).

I love spending time at the local brewery where my boyfriend spins records once a week, and getting to know all the regulars there. I love that I am a regular there.

I love spending time with my niece and nephew and seeing them learn and grow, even if my nephew is going through a phase of meltdowns and attitude right now. I realized last year how much I miss spending time with my sister without them, though, so my goal is to invite her to do more things just us.

I am bad at communicating with my parents regularly. They’re both retiring this year, and I want to include them in my life more and not have it feel like an obligation. I want to figure out how to bridge the gap between the relationship my mom wants us to have and the one I want us to have.

I want to be a good girlfriend, to not shut down when I’m upset or when I get stressed or when I’m scared of how I feel. To find new and interesting things for us to do together, but also go to lots and lots of concerts and like, make dinner together and watch four hours of Great British Bakeoff in a row.

I want to go to concerts with my friends and go visit them on random weekends and not have to have a specific event to see the ones who live far away. (That will also be easier when grad school wraps up!) I want to develop the new friendships I’ve made this year with incredible people that I really treasure as a part of my life, and I want to stay in touch with my grad school cohort after we graduate.

In all these relationships, I want to be more present in the moments that I am with other people. I want to use my phone as a defense mechanism for my introversion less and be more attentive to conversations. I want to not worry about tweeting about everything or posting about everything right away, but I also want to stay close to my internet friends whom I love dearly. I want my social media posts to feel important because they aren’t too frequent.

I want to reach out to the #mtbos and #dcsdpln communities again, but to not burn myself out with requirements about how often I post.

I want to celebrate all the incredible people I have in my life, including myself. Happy 2019.

Book Recommendations (Vol. 08)

We’re almost at the end of 2018, which has simultaneously seemed like the longest year in existence (remember that the Olympics were THIS YEAR?) and also like I don’t know where the last few months have gone. I’ve done less reading than the past couple years, a combination of grad school and just being busier life-wise, but still an amount that I am happy with.

I’m currently curled up with my cat sleeping adorably next to me on the couch and Niall Horan’s album on my record player, and I’m ready to tell you about my top 5 books for the end of this year, AND the top five books for 2018 overall!


My total at this point in the year is 59 books (with probably one or two more getting finished before the 31st ends), which puts me at 13 for this quarter.

An Absolutely Remarkable Thing – Hank Green

April May is heading home from work late at night (early in the morning?) when she stumbles across a giant….statue? She makes a video of Carl (the statue) and posts it on youtube, for kicks. When she wakes up in the morning, she realizes that she’s gone viral – because there are DOZENS of Carls, across the globe, and they all just appeared there. No one knows what is happening. And then everyone starts having the same dream – a dream you can control – a dream that is an escape room like puzzle. What.

If you’ve read my previous book posts, you know I am a huge John Green fan. I’m also a fan of Hank’s other work like all of the video series he produces on YouTube. I was very nervous when he announced he was publishing a book because, well, his brother is the author. But this book is a DELIGHT. It’s hilarious, and such an incredibly timely commentary on internet fame. April May is one of the realest characters I’ve read in awhile, in a not entirely likable but incredibly relatable way. The mystery and suspense of figuring out what’s going on with the Carls is so gripping and the ending is INTENSE. I cannot wait for the sequel to this, have already forced another teacher in my building and my mother to read it, and haven’t stopped thinking about it. And listening to Queen and Carly Rae Jepson (relevant to the book, I promise).

Chaotic Good – Whitney Gardner

Cameron’s cosplay is starting to get attention. Not all positive attention. Guys seem to be upset that she might be a “fake fan”. And then the guy at the comic book shop by her new house (her family just moved, and right before senior year) tries to direct her to the “girl section”. She borrows her twin brother’s clothes and is treated entirely differently when she walks into the shop as a boy. In fact, another employee invites her to play Dungeons and Dragons with them. But can she handle making a portfolio of original costumes for a huge opportunity AND pretending to be a boy playing D&D, especially when feelings get involved?

This book was truly just a delight. It captured the sexism of nerd culture very well and the girl character was well rounded – more than just a “not like the other girls” girl. I thought her twin brother was a really interesting juxtaposition to her character and the story definitely blossomed with the interactions between them. The Dungeons and Dragons storyline is perfect for little reveals about each of the characters by using the characters they play in the game, and there are incredible cartoon panels interspersed throughout the book that depict the D&D story. It’s definitely a take on a fandom culture book like ones I’ve read before, but it wasn’t tired and definitely brought new and inventive ideas to the table.

If You Come Softly – Jacqueline Woodson

Ellie and Miah run into each other (literally) on their first day at Percy Academy. Miah is struggling to fit in with his new basketball team, and Ellie is struggling to fit in with…anyone. They find each other, and despite all of their differences, fall in love.

Speaking of John Green, he’s started a book club called Life’s Library. This is the first book for the book club and I am so, so glad it was chosen. This book is absolutely TIMELESS. It was written over 20 years ago but feels like it could have come out in 2018. I absolutely sobbed reading the ending. As soon as you start reading it just feels like you’re reading a classic. Love, love, love.

The Great Alone – Kristin Hannah

Leni’s dad changed in the war. It’s 1974, and he can’t stay in one place for long. He gets fired, packs them up, and they move. This time to the middle of nowhere in Alaska. They’re going to live off the land – even though they don’t have any clue how to do that. Leni just wants to fit in, and for her dad to be happy again – and stop hitting her mom. Can they survive Alaska – and her dad? Will they find a place to be themselves?

My sister recommended this to me and it is not one I normally would have picked up, but again I found myself sobbing my way through several parts of it. Alaska is truly a character in this, majestic and terrifying and isolating. The characters themselves are all incredible and strong, from Leni to her mom Cora to Matthew and Large Marge and everyone else. It’s a story of resilience and friendship, and of what family means. It will break your heart, and break it again, and break it some more, and then give you some hope and shatter it. Definitely a trigger warning on this one for domestic abuse – very hard to read at times even as someone who does not have any trauma related to that.

A Very Large Expanse of Sea – Tahereh Mafi

Shirin is living in a post 9/11 America as a hijab wearing teen. Her parents move her and her brother around a lot, and she’s learned to put up protective walls and shut all the atrocious comments out. That doesn’t mean they don’t hurt, but she shuts them out – shuts people out. Then she meets Ocean. Annoying at first, he just keeps asking questions. Apologizing when he gets something wrong. But he just won’t give up, and she doesn’t know how to let him in without him getting hurt, too.

This is a fascinating insight into how we as Americans treat those who are different. Throw into that a look into the glorification of high school sports and some breakdancing and this book is truly, truly unique. Mafi writes the high school experience extremely realistically and the romance seems realistic as well. Beautifully written and with some fun callbacks to 2002 culture (dial up internet, anyone?)

**2018 top 5**

If You Come Softly – Jacqueline Woodson

An Absolutely Remarkable Thing – Hank Green

All American Boys – Jason Reynolds and Brandon Kiely

A Memory of Light – Robert Jordan and Brandon Sanderson

Girl Made of Stars – Ashley Herring Blake

Graphing Systems of Equations Word Scramble

My Algebra 1 students this year are¬†struggling¬†with graphing linear functions. Every time we have to graph a line, it’s like they’ve never done it before ever. So now we’re doing systems of equations and we’re looking at graphing them and I knew they would need a TON of support to do that.

I’ve been trying to emphasize that graphing gives us an¬†approximate solution to the system, and that we can’t be sure that it’s the correct solution until we check it in the equations, since that’s how the language of standard A.REI.6 reads and therefore how our SBG rubric reads. So to kind of boost their confidence graphing lines and focus on the “checking the approximate intersection point in the equations” component, I created this activity.

I chose 5 systems of equations, and put the letters in a 5-letter word at each of their solutions on the graph. Then I put some other red herring letters at coordinates where they might think the intersections were if they did things like used a positive slope instead of a negative slope or other common mistakes.

I put the activity in a sheet protector, so students could graph the systems using dry erase markers and erase after each system. I asked them to put on paper the letter of the solution, the coordinates of the solution, and evidence that they checked their solution.

This ended up reinforcing a lot of good things. They knew that their lines needed to go through at least some of the letters, and some of them started just assuming that the first letter their first line went through was the solution. When one of the lines went through two letters, they panicked. This led them to a better understanding that with a¬†system of equations, we need to know how¬†two¬†lines¬†interact. It also helped them to understand that if the point they got originally did not check out with the equations, they needed to regraph their lines and try again, not just say it didn’t work, because they needed the right letter to form their word.

Many of my students didn’t end up getting all the way to the word by the end of class because they were struggling so much with the graphing, but I still felt it was productive work even without that end result of ending up with a word.

If you want to use this activity, you can download the PDF or Publisher files.

The impact.

I’m sitting at parent-teacher conferences. The parent of a student who was not in my Algebra 1 class first quarter but is this quarter comes in.

“I’m so excited to meet you! I can’t believe how much his attitude about Algebra has changed since he’s been in your class! He keeps telling me about how many good examples you give, and how you answer all the questions he asks, and that it makes sense. I just can’t believe the change”

I am, at this point, trying not to cry, because complimenting my teaching is the best way to get an overly emotional reaction out of me. But I’m also thinking, “I can’t believe that student says all those things about me!”

Because this student is not one that really stands out to me most days. He sits in the back, he does his work, sometimes answers questions during whole class stuff. Very rarely asks me questions while he’s working or asks for help. Doesn’t appear to me that he loves math class.

But he’s telling his mom about it at home. So what I’m doing is clearly making an impact on him. I just don’t see it day to day.

So just a reminder to all of us that while it may not seem like we’re having that much of an effect on students…they might be going home and telling their parents, or they might realize it after they leave your class. Your impact is felt.

On Posting Less

Some of you may have noticed that I’ve published a lot less blog posts this year. And a lot less #teach180 tweets. I don’t feel like I¬†owe an explanation, but I feel like an explanation would be helpful for me to reflect on why it’s happening and for you to understand what my whole goal is with this blog.

There are several reasons that factor into why you’ve seen less from me online this school year.

Reason 1: Grad School

I’m in my last 3 semesters of earning my master’s degree in secondary math teaching. The course we are taking this semester, Equity in Mathematics Education, is fascinating and really helpful in improving my teaching. I would love to turn some of my papers for the class into blog posts at some point. However, this class has been a lot of work. Specifically, a lot of really time consuming reading. So on evenings when I may want to sit down and write a quick blog post…I need to read about 5 academic articles instead.

Reason 2: Avoiding Repetition

My most popular posts are the ones that contain downloadable resources, and I definitely know how that feels because those are the ones I love to see from other teachers! Last year, I posted all of my interactive notebook pages for Algebra 1 and Algebra 2 on this blog. This year…I’m reusing most of the same pages so there’s no new resources to post. At some point I will make posts with some of the few pages I change or update for this year, but there’s not a whole lot of new content going on there. My other class this year is Pre-Calculus and it’s the first time my school has ever offered it, so there’s a lot of adjusting on the fly and learning as we go so I don’t feel comfortable sharing those INB pages with the world just yet.

Reason 3: Standards Based Grading

Our math department is piloting standards based grading using Infinite Campus, with the plan to go district wide with this in the next few years. It’s definitely been a situation of having more questions than answers and requiring a lot of communication and collaboration between our math team, and some of that result has been that I’m just not ready to share thoughts on this process yet and some has been less time to write and post things! I’m hoping to have some posts on this experience more in the second half of the school year, as we are starting to kind of find answers to some of our questions and figure out how to do this effectively.

Reason 4: Being More Effective in my Classroom

I realized that the pressure of having a #teach180 post every single day was making me more concerned with how my classroom LOOKED rather than what students were LEARNING. Uh-oh. Not that I was doing bad activities, but just that my motivations were beginning to get a little off. So I’m still “doing” #teach180 this year, and still writing blog posts about activities occasionally, but focusing less on “what can I take a picture of” and more on “how can I communicate this idea most clearly to students”? I’ve found that my pre-calculus students operate really well off of doing practice problems and checking their answers against a key, which is visually pretty uninteresting. My Algebra 2 classes also have been doing this frequently. My Algebra 1 classes this year are really struggling with the learning process – how to move from not understanding something and how to take actions to start understanding it. I’d like to blog about this struggle also, but it’s demanding a lot of my time in the classroom and taking my focus away from taking pictures because every single one of them needs me by their side for every single problem they try to do…I’m really trying to get them to develop more efficacy in their own learning but it has been a SLOW process this year.


So, what do I want to be posting about in the coming months?

  • standards based grading implementation issues and thoughts
  • INB pages that I’ve changed
  • maybe some of the pre-calc INB pages that I’m proud of
  • teaching students how to learn?
  • equity in math education
  • teaching writing linear equations – what I’ve tried, what I might try next year

What struggles are you having in your classrooms this year? I always like to be open about these things because I don’t want to make my classroom seem like a glossy, perfect magazine cover when it definitely is not. Plus, I think it all makes us feel better to know we aren’t the only ones struggling with things in our teaching.

Book Recommendations (Vol. 07)

Well, September ended! My life has been a whirlwind this school year, with a grad school class that involves a TON of reading academic articles and writing a bunch (not my favorite, I’m ready for another math class) and me literally not staying in town for an entire weekend yet this school year. So, I have not had as much time to read as I would like. Or to write this post. Or to clean my house, but that’s another story.

You can read previous posts from this series here:

Vol. 01 | Vol. 02 | Vol. 03 | Vol. 04 | Vol. 05 | Vol. 06


I just said I haven’t had time to read, but that was only in the last month of the quarter. The other two months were SUMMER, so I’ve still logged a fair number of books: 14, to be exact, bringing my total for the year up to 46.

Here’s the best 5 of those 14.

For Every One – Jason Reynolds

This book was originally a speech that Jason Reynolds gave at the Kennedy Center for the unveiling of the Martin Luther King, Jr. memorial. For every person who has a dream.

The artistic presentation of this speech/poem was just stunning. This is a message for everyone who has a dream, but especially for everyone who has ever felt like they can’t accomplish their dream, or that their dream isn’t good enough. It will make you feel good and driven and ready to GO GET THAT DREAM. Or at least keep trying.

Girl Made of Stars – Ashley Herring Blake

This story comes with a huge trigger warning for sexual violence and rape. Mara’s best friend Hannah accuses her twin brother of rape. Mara just broke up with her girlfriend Charlie who has also been her best friend forever. Mara has no one to turn to – she can’t talk to her brother or Hannah, she can’t talk to Charlie anymore, and her parents are pretending nothing happened. Does she side with her family, or with what she knows is right? Are those different things right now? How does she move on from this?

I got a text from my friend Tedi saying “this has to be the next book you read”, and I trust her, so I immediately put it on hold at the library. It is DEVASTATING. It confronts head on all the worst parts of rape culture. Mara is such a real and beautiful character and the reader is tormented along with her about what to do and how to just fix everything, which she can’t. It has LGBT representation with Charlie trying to find their identity as a nonbinary person, it has people messing up with getting identities right. It is so incredibly raw and powerful and vivid.

The Beauty That Remains – Ashley Woodfolk

Shay, Autumn, and Logan are all hit by grief. Grief for different people, from different circumstances, but all tragic. The story follows their grief, and the way that one band connects them all and helps them to cope.

First, on the very first page of this book, someone dies in a car crash. I had gotten in a pretty bad accident about two days before starting this book, so that set me off completely sobbing through most of this book. Maybe not the best book to read at that point in my life. I would put a lot of trigger warnings on this for trauma, death, suicide, drugs. It is, however, gripping and beautiful. It deals with all the raw emotions of grief and how other people around you move on when you don’t. The music element of the book is perfection: the shows, the album reviews, the underlying element of Unraveling Lovely (the band). I cared so much about all of these characters and desperately wanted to see them process their grief in healthy ways and find their new normal lives. The little multimedia bits provide a nice transition through the three character perspectives and I think the subtle tie-ins between the three are masterful.

Ship It – Britta Lundin

Claire loves the show Demon Heart. She even writes fanfiction for it. When she gets to ask the main actor a question at a Comic Con panel, though, he laughs off the possibility that his character could be gay, crushing¬† her dreams. In a PR stunt to try to fix the bad optics of the actor’s answer, Claire ends up going on a tour of Comic Cons with the cast and crew of Demon Heart, meets Tess, and many shenanigans ensue.

This book was just so sweet. I am a sucker for stories that treat fandom as a legitimate thing and not a silly-teen girl hobby (see Rainbow Rowell’s Fangirl, for example). I loved the looks into Claire’s fanfics and the show itself. I loved the overdramatic plotline of them bringing her with them on tour and her interacting with the stubborn actors and awesome actors and show writers. Tess’ character is an interesting juxtaposition to Claire as a fan. This one is not super deep, but it is fun and cute, and does have some good representation going on.

The Miscalculations of Lightning Girl – Stacy McAnulty

Lucy was struck by lightning when she was little. And it gave her extraordinary math abilities. Now, her grandma is insisting that she go to a regular, public middle school for one year before she does college – to get the regular preteen social experience. Make a friend. Join a club. Lucy has no idea how to do any of this, and does not want to. She’s going to make it through the year in one piece, somehow, and move on. Or at least that’s what her plan is.

This book seemed so cheesy at first blink, and then I fell in love with it. It’s a pure celebration of friendship, which you don’t often see and is so beautiful. Lucy’s little math nerd tendencies made my heart happy – even though a lot of them were very surface level things, the use of math to frame her understanding of fitting in was lovely. I loved the teacher character, and the animal shelter inclusion. It’s a middle school read, so it’s quick, but it will leave you in a good mood and I think it would be a perfect read for a middle schooler looking for their place.