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.

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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

books

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.

2018-2019 Goals Check In: September

Before the school year started, I wrote this post detailing three goals for myself for the year. I promised to check in on this every month, and it seems that it is already the end of September, so here we go….

Goal 1: Grade Less Papers

Maybe I have done too well on this goal? I have not taken things home to grade a single time this year, and I have left school within 45 minutes of contract time ending every day. I feel pretty great about that part of it, but I don’t know that I am hitting the bullseye on providing the most useful feedback to students.

I don’t think that my feedback is WORSE than it was when I was grading physical papers and handing them back every day, I just don’t think it’s BETTER either. I’m really focusing on the students self-assessing, as I mentioned in my original goal post that’s my focus for my action research paper for the completion of my master’s, and I think I’m improving on that front. I’ve spent more time making clear answer keys this year than grading, so that students can check their own work.

The things I still want to work on here are the accountability for truly checking your own work and giving a real, honest self assessment, and I want to add in a component of them reporting on the types of problems they had to try more than once on. I want them to work on recognizing the specific skills they need to improve on. Then I would love to experiment with giving a problem set and individualizing which problem numbers different students work on based on this reporting, to make my feedback really connect with their practice. I also do want to find a way to still give them written feedback of some sort even if they aren’t handing in physical papers all the time.

Goal 2: Experiment with non-traditional assessments

The goal here was at least one “alternative” form of assessment every half term, and we are at our first midterm and I….almost did this?

In Algebra 2, many students needed a full two days to complete our first traditional assessment (these are common for the district and aren’t currently required for every teacher to use but I know they will be in coming years so I’m trying to get in the habit of using them). Students who finished in the first day, I gave an additional assessment opportunity. The standard we were working on was F.IF.7b: Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. 

I had students roll a die to determine if they would create their own piecewise, step, or absolute value function. Once they created one, they wrote a paragraph explaining how one would graph it – step by step instructions. This gave them a chance to show that they knew how to process the components of the function even if they maybe made mistakes when actually graphing it. I said they could talk it through with a partner to revise their steps. I really did like this assessment option but only a few students actually completed it, so I would like to have the whole class complete this and also nail down my instructions more because some of the descriptions I got were not as fleshed out and detailed as I would like.

An attempt was made, but I don’t think I would say I quite hit this one for this half-term.

Goal 3: Class time to discuss completed work

I said specifically that I wanted to bring misconceptions from practice work to the whole class for discussion, and that I wanted to provide time after assessments to discuss and ask questions.

I think I’ve been doing okay at the misconceptions thing, except for the most part I’ve been doing it at the start of the next class period instead of the end of the one they’re doing the practice in, because that’s just worked out well for me and I think it kicks off the next lesson nicely. I really wish I had a document camera because at the moment I’m mostly just reproducing student work I saw and I would like to actually SHOW the students’ work itself, but that isn’t technology I currently possess. Would like to develop a workaround for this.

The time after assessments I have done super well on for Algebra 1 – we spent an entire class period comparing their tests to the rubric and having them analyze their mistakes on each problem, and they could ask their classmates or me to help them put their mistakes into words.

 

I thought this was so beneficial for them and I’d like to do it more often. My current issue is that my Algebra 2 students have struggled with absences this year…so it’s often almost a week later that enough of them have actually completed an assessment for us to be able to talk about it as a whole class. And then I forget. So I’m thinking about what to do there. My main goal here is to really work with my Algebra 1 students on how to learn from mistakes and work to improve their understanding. I’d like to do this more with mid unit quizzes also.

Improvising…

Today, I did one of my old favorite lessons with Algebra 1 – Inequality Speed Dating. By this time of the year, students are starting to think that they either have mastered solving equations or have no idea how to solve equations, and if we do any sort of straightforward practice they will be either bored or won’t try, based on which group they’ve sorted themselves into. So I like to use this activity to kind of jostle them into life a bit and force them to take more accountability for their work.

On Friday, we completed the first portion of the activity, which is for students to “become experts” at one single inequality. This year instead of gluing them into their notebooks, I gave them each a post-it with an inequality written at the top. Once they’d solved it and had it checked by me, they stuck it on the most recent page in their notebooks.

Today, we took out the post its and I explained the rules. Find a partner, write your inequality at the top of their whiteboard, 3 minutes to solve, then 1 minute for each partner to “coach” the other on how to fix their solution. If you read my post about this activity from last year, I changed several things about how I ran it.

And it fell apart a bit.

1st hour, I had two girls who flat out refused to participate, plus a small class, so those who were participating ran out of problems quickly after they had matched with only a few partners. Luckily, I had extra inequalities because my other classes are larger, so I gave them all new sticky notes with new problems and we managed to continue. Improvisation #1.

2nd hour, my students were super grumbly once they could no longer find a partner who was within their immediate vicinity and it became clear that they would have to move, God forbid. Cue Improvisation #2, where I put 5 new inequalities on the board and asked them to attempt all 5, then consider those 5 and the ones they had been coached on by their partners and tell me: 1 thing they felt they understood, 2 things they needed to work on, and 1 action they were going to take to improve. The last thing is inspired by that class period’s unwillingness so far to take responsibility for their own learning – they still have very fixed mindsets so we’re trying to work on them taking actions when they don’t understand.

4th hour, students were more receptive and almost all participated willingly and appropriately. We ran out of new partners with about 15 minutes left in the period, so in came Improvisation #3: I put two new inequalities on the board and asked them to solve them, then tell me 1 thing they felt good at and 1 thing they needed more practice with. This group, I encouraged them to write an example for their “one things” because I realized in glancing quickly at 2nd period’s that most of them did not have words to describe either what they were good at or needed to work on.

I was kind of upset, because this activity is usually one of my favorites and it…didn’t go well today, and I had many students so reluctant to participate. So, part of this post is taking credit for my own teaching improvisational skills. I don’t know what I would have done if this had happened 2-3 years ago, because I didn’t have the background knowledge to realize that pressing on was not the right choice, or the skill and capacity to come up with an alternative on the fly. So I’m proud of all three improvisations and I think they helped increase the effectiveness of the activyt.

 

But mostly, I want to reflect on why it didn’t go as well this time and how I’m going to regroup.

  1. Since my room isn’t set up in rows or a U shape of tables this year, I decided to have them move around to find their own new partners. This was clearly a mistake because it was where almost all of the refusal came in. “I don’t want to work with any of these other people”, “I don’t want to get up”, “I only want to work with this person”….there are less excuses if I set it up like I have before and set it up with chairs across a table with an inside and outside ring, and have one ring rotate each turn. Definitely worth rearranging my classroom for half a day to organize this rotation.
  2. Last week had a lot of absent students, as about three different viruses were going around. A few of those that were here on Friday weren’t confident about their solutions, and I had several students that I just had to hand a worked out solution to today so they would have a problem to exchange with their partners. Not really much I can do about that, but I’m acknowledging that it contributed to some reluctance to participate and low confidence today.
  3. I think the recording sheets I’ve used previously are necessary to this activity. As much as I love nonpermanent surfaces (whiteboards), and as much as our district is pushing us to make less copies, I think the accountability of “I actually solved the problem my partner gave me” and “I actually checked and coached them through their solution” are needed here.
  4. Good change: I think specifying 3 minutes to ONLY SOLVE, then 1 minute to coach the first partner through their solution and 1 minute to coach the second partner through theirs was really helpful. It led to less rushing than in previous years because they weren’t allowed to look at each other’s solutions until the first timer went off. I think I need to be even more explicit about this next year though, with NO MOVING ON to the next stage until the timer GOES OFF. We need to cultivate more time for thinking.
  5. Good change: problems on post-its instead of glued in notebooks. No need to bring the entire notebook with you around the room unless you wanted your notes. No constant flipping back to the right page because you shut it.
  6. Good change: we had a talk about “coaching” beforehand, along with an example conversation you might have with someone whose solution was incorrect.
  7. Idea: One thing students always get confused on in this activity is multiple solution pathways. That’s part of why this is a great activity, because it starts those discussions, but I get students coaching their partners that their solution was incorrect because they moved their variables to one side before they undid adding 7, or because they moved the variables to the opposite side that their partner had. I think it would be nice to solve an inequality like 2x + 5 < 4x  – 7 several different ways on the board to start class – perhaps something like 3 correct solutions, and 1 incorrect one, and then discuss how the coaching period would look for each of the solutions.

I want to remind myself that I still really like this activity – and that some of my favorite benefits of it did still happen for some students today, but I also want to acknowledge that I did something that didn’t go so well publicly, and I want to reflect on those failures to continue making my teaching better. I already feel better after typing those reflections out! And see, still some good coaching and solving going on in these pictures I took!

2018-2019 Teaching Goals

Oh my goodness, this summer has FLOWN BY?! I had grad school classes for the first five weeks, then did some curriculum writing with my district for two weeks, mixed in with a lot of concerts and a trip to NYC to see Harry Potter and the Cursed Child and visit the Museum of Math and a really awesome MC Escher exhibit. And now teachers report back next Monday, students come back next Thursday, and I am NOT READY but also very ready.

I miss students in my life, which as always reminds me that I’m in the right profession. I miss making dumb algebra jokes with them, I miss singing Toto’s Africa or Sia’s Chandelier with them (what will the theme songs of this school year be???), I miss laughing and learning and loving with them. I’m stoked to have a bunch of new freshmen in Algebra 1, a ton of kids I had two years ago in Algebra 1 for Algebra 2, and the entirety of last year’s Algebra 2 class that was one of my favorites of all time is going to be the first ever Pre-Calculus class in our school!

I’ve been thinking a lot this summer about changes I want to make in my classroom, as always. I learned so much in my grad classes this summer (real analysis, an assessment course, and Geometry pedagogy) and am excited to apply those ideas, plus talking to other teachers and reading books and continuing to expand my worldview have me itching to try some things out.

I want to keep my goals attainable – we talked in grad school about not changing too much at once, for several reasons. You don’t want to burn yourself out trying to totally overhaul your entire teaching style at once, and you also can’t tell what effects the changes have if you make a bunch at one time. So I’m going to set three larger goals for things to change this year, and then goal 3b is to check in on those goals regularly throughout the year on this blog (midterms and end of terms, possibly?)

Goal 1: Grade less papers

My assessment course and our district’s transition to standards based grading have made me think a LOT about how I grade things. I am a big believer in giving feedback in a timely fashion, because how are students supposed to improve and learn if they don’t have feedback on what they are doing well or what they could work on until two weeks after they did something? Last year, this had me grading papers for almost every class I taught almost every night. I’m going to go ahead and admit that this was not that bad because I just did it while I watched tv in the evenings, and I did not have much of a social life on most weeknights. I seem to have developed much more of a social life over this summer and I…well I really don’t want to be stuck in my house grading papers every night.

I had the realization during the assessment courses we took the last two semesters in my grad program that feedback =/= grading. Wow! Some of you might be thinking that that’s obvious, but I had been thinking of them as synonymous in my mind for the past five years. We talked a lot this summer about oral feedback, checklist feedback during work time in class, partner feedback, and other ways to get students the feedback they need without them necessarily having to turn in a piece of paper (or digital work) daily. With the population of students in my building (at risk students), I try not to assign homework, so there are plenty of opportunities to give them feedback on their work during class without me taking a bunch of grading home. I will still grade paper assignments sometimes, I am sure, but I want to develop checklists of problem solving skills / math skills (based on our SBG rubrics for each standard) that I can quickly fill out during class as I walk around observing student work and I can carry my chromebook around the room also to input proficiency scores for practice assignments, which don’t count towards their grades anyways.

The goal is to take grading home two days per week or less.

Goal 2: Experiment with non-traditional assessments

I started doing this a tiny bit at the end of last year, with poster projects for both Algebra 1 and Algebra 2 asking them to synthesize the information they had learned about sketching polynomials or solving quadratics. They got to choose problems from a list, or draw random problems from a hat. They got to consult with partners. They were much less anxious and stressed and I felt that the results were a much more accurate depiction of what they knew than the paper-and-pencil tests I typically give.

Those aren’t going to disappear, but I want to try more different approaches to help decrease some of the test anxiety I see. I’ve read a lot about giving timed consultation periods at the start or end of a testing period – a few minutes, long enough to discuss something you need to clarify or get validation on, but not quite long enough to have your partner explain how to solve the whole problem. When talking about our SBG implementation, our district curriculum specialists have mentioned combining several individual skill quizzes into one assessment score that encompasses an entire standard. I want to think about having students make a mini-portfolio by choosing problems from the textbook with certain criteria to demonstrate their knowledge.

Basically, I want to incorporate more collaboration and choice. Fundamentally, this is based on two beliefs I have about mathematics. In the “real world”, 1. my students are rarely if ever going to be forced to complete math work in isolation. There will always be an option to ask someone for help or use a resource. 2. they will rarely ever be told they have to solve a specific problem. The problem will present itself in a non-math way, and they will have to select the math that will best help them solve the problem and that they feel capable of doing.

The goal is to have at least one “alternative” form of assessment every half term in at least one of my classes.

Goal 3: Class time to discuss completed work

I think this is one of my biggest teaching weaknesses, honestly. Since we do so much of our work in class and also have issues with attendance, I tend to leave this kind of task to happen individually during our intervention period or during work time. So two things here. One, I want to bring a common misconception or work done by a student to the whole class’ attention at the end of class on a practice day to have a more solid conclusion to the work. Using the checklists I described earlier will help with this effort since I will be able to see boxes that I didn’t check for many students and find things to address easily.

Two, I want to really commit myself to taking ten minutes the day I hand back assessments to discuss them and have students ask questions. My students often don’t want to discuss their work as a class, so I’m thinking I will have to format this as a challenge: I want one question about why something you tried didn’t work, I want one person to share a solution method they think was unique, etc. Open to ideas on that!

My action research project to complete my masters degree will take a look at how self-assessment impacts student achievement, so part of this is also making sure to USE the SBG rubrics that I use to grade with my students and having them assess their own work using them, or rating their understanding of learning targets, etc. Mostly, I want my students to reflect more on their own work and not just turn in and move on.

The goal is to have some sort of activity that forces my students to actually look at their work every time I hand a summative assessment back.