Interpreting Graphs

I decided my Algebra students need a break from factoring.

(Actually, I decided that I need a break, because if I have to grade anymore quizzes that continue to show no understanding of a concept that we have been working on since January, I might explode.)


Our school currently has a grant funded by NMSI, the National Math and Science Initiative. The goal is to increase our enrollment in AP classes, as well as our performance on AP exams. The part of this that applies to me, as a non-AP teacher, is that we have training and resources geared toward teachers of lower levels to help introduce concepts that will be vital in calculus and statistics. These activities are usually pretty interesting, and well-scaffolded, so I’ve enjoyed using the few that I’ve done with Algebra students. They also provide AP-style multiple choice and free-response questions that are scaled down for the appropriate level.

I found one activity on interpreting distance graphs and another on interpreting rate (speed) graphs and decided to give them a try. We started Distance Graphs two weeks ago. Because of the nature of the assignment, I’m requiring students to answer using complete sentences (like you would on the AP exam) and provide explanations for their answers (like you would on the AP exam).

The actual interpreting of the graphs went surprisingly well. I was impressed by the accuracy of their answers to the interpretation questions. They know how to read graphs and they can use a fair amount of that knowledge to make inferences about the situation. For example, no one tried to convince me that because the graph went up and back down again it meant the person was going up and down a hill. They also made the connection between slope and speed really well.

The explanations for the questions (explain how you know, or explain how you figured this out), however…didn’t go well. At some point, I said if anyone else tried to tell me “because the graph shows it”, I was going to strangle them. (Because they are teenagers, of course the immediate reaction to that was to spend the next 10 minutes ONLY saying “because the graph shows it, Miss!” and then laughing like my impending aneurysm is the most hilarious thing in the world. The main culprit only stopped when I asked the class, “So when I kick A out of class in a minute because he’s DRIVING ME CRAZY, will his distance from me be increasing or decreasing?”)

We completed the first assignment where we analyzed distance graphs, and they turned them in. I scored them out of 2 points for each question – 1 point for the correct answer and 1 for a good explanation. My class average was 54%, and the highest score was 83%. (I was unpleasantly surprised by the number of students who didn’t even get the questions right that I had written out on the board.)

“So guys, I graded your Distance Graphs worksheets yesterday. They’re pretty terrible. I mean this in the most loving way possible, but your explanations suck.”

They laughed.

I acknowledged that I know why their explanations were terrible – they’ve never really had to do this before. Most of them have never been asked to write explanations for anything in math class before. They nodded. I said this is why we’re working on it. I know you’re not good at it, so let’s learn how.

Tangent: I was once again reminded of how much my students appreciate it when I acknowledge these kinds of weaknesses openly and make it clear that I’m not trying to hold them to an unreasonable expectation. I’m saying that I knew you weren’t going to be good at this, and I knew it’s not because you’re just refusing, it’s because you haven’t learned it. Without that moment of honesty regarding their side of this situation, there’s no way I would have been able to get them to complete the next activity we did.

So I passed back their graded worksheets and we quickly skimmed through the problems so I could describe what kinds of explanations I had been looking for. Then I passed out a 10-question multiple choice “quiz” (another source from NMSI, so the problems are intended to be similar in style to AP multiple choice questions). I gave them a few minutes to go through and answer the questions and then we checked answers. I was clear that their score on the assignment was in NO way related to the number of questions they got correct at this moment.

After we reviewed the answers, I instructed the class to choose 5 of the 10 problems and write an explanation for the correct answer. I encouraged them to check with me before turning in their assignment so I could make sure their explanations were good enough.

I was impressed. Granted, not all of my students actually had me check their work before turning it in (probably not even half did), but the ones that did were surprisingly good. In most cases, I make suggestions to improve their explanations – make them more clear or more specific – but the foundation that was there was solid.

Yesterday, we started an activity on interpreting rate graphs (as opposed to distance graphs). We talked about how the interpretation of this graph is different, since the y-axis is speed not distance. I had to remind the students of this fact over and over again, which I expected because they are used to reading distance graphs and not rate graphs. Honestly, I think I expected to have to do it more than I did.

At the end of class yesterday, I told two of my classes this:

“You guys are doing really well with your explanations! I’m really impressed by how much improvement I’ve seen from that first assignment a couple of weeks ago. This makes me really happy for two reasons. First, I’m proud of you – I get to see you learning and improving, and that makes me happy for you. The second reason is actually a selfish reason – it makes me feel like a crappy teacher when you aren’t learning. That’s what’s been going on the last 13 weeks of school – you haven’t been showing me that you’ve been learning anything, and I’ve felt like a bad teacher because of it.”


I’m looking forward to finishing this activity with two of my classes tomorrow, so I can tell them what I got to tell one class today – you are doing calculus! Using the slopes in a distance graph to find the speed, and using the rate and time in the speed graph to find the distance – this is what the AP Calculus class has been doing this year! Granted, they are using harder graphs and equations, but this is how it starts!

I love blowing their minds with this kind of information.

After we finish these, I need to find a good activity for them to do more independently and summatively. They might be doing a poster project. Just need to find a good problem set.


At the Bottom of the Roller Coaster Again

Last weekend, I wanted to sit down and write a post about how I’ve realized how much my students trust me and know that if I’m teaching them something, I have a good reason for it. Maybe I’m being a little bipolar, but that’s not how I’m feeling today.

I’m frustrated a lot this semester again. I feel like I’m constantly on edge, waiting for the next moment that some student is going to “take it too far” and I’ll blow up. I haven’t actually blown up much, considering I’m so tense all the time, but the feeling is there. (Actually, it’s more likely that I’ll snap at one of the kids who never irritates me. I’ve had so many negative interactions with some kids that I’m more careful with them than the ones who don’t usually act out. Poor kids.)

Some of my difficulty this semester stems from some emotional things I’m going through in my personal life, that I don’t want to get into detail about in such a public forum.

Some of my frustration is my own fault. I wanted to teach this Support class this year with this doe-eyed dream of turning math around for these students. I wanted to be that one teacher that would make a difference, that would help them understand math and show them the beauty behind all these things that they have to learn in Algebra 1 (and I LOVE the content in Algebra 1). It’s only my third year of teaching and I’m still prone to the teaching-like-I’m-in-a-movie line of thinking, where my class is going to be the most amazing thing that’s ever happened to these kids.

So when they come to class and refuse to do any work for the first 4 weeks of the semester, or chat with their best friend instead of watching the movie that I’m using to teach about the Standards for Mathematical Practice, or repeatedly whine about how they “don’t get it” but never do anything to help them “get it”, I get frustrated. (Seriously? You won’t even watch a movie? What can I possibly do to support you if you’re this stubborn about not doing what I’ve asked you to do?) And I get disillusioned.

Yes, I knew teaching was going to be hard. I knew I wasn’t going to get paid much. I knew that I was going to have more work than one person could reasonably be expected to do in a day. I knew that I was going to have bad days, bad weeks, bad months, and bad years. And yes, I know that this is just a bad week, and maybe it’s just a bad year.

Living it is different from knowing it. I’m watching my seniors attempt to write a research paper and realizing that they have no idea how to construct a coherent sentence, how to use research effectively (actually let it inform their writing without plagiarizing their entire paper), and in some cases, how to work together. I’m watching my freshmen watch “The Martian” and realizing that not only are their reading comprehension skills terrible, their movie-watching comprehension skills are also terrible. It seems like they have no idea what’s going on in this movie. If they have no idea what’s even happening, they certainly can’t relate the situations in the movie to the 8 Mathematical Practices, which is their assignment.

I don’t know what to do. I feel helpless and lost.

Some of my frustration is related to the lack of motivation and critical thinking skills and life skills and ability to not give up that I’m seeing in my students. I’m re-reading Chase Mielke’s What Students Really Need to Hear again and it’s resonating again, just like it did my first year of teaching. I hate watching my students throw away their education and refuse to learn the most important lessons they could ever learn about how to live life.

On the other hand, I’m seeing some progress this year. Students A and B are paying attention in class, participating in positive ways, and actually doing their work sometimes. Student C has finally started speaking to me politely every time he interacts with me, even though he’s not doing much in the way of work. Students D and E continue to demand my attention, but their purpose is to ask for help with their work. Students F and G are my top-performing students across all of my freshmen, and no one is more surprised by this than they are. Student H is finally learning Algebra and demonstrating that (when she pays attention to what she’s doing) she can solve problems without mixing up her integer operations. Students I, J, K, L, and M all asked for paper sets of Algebra Tiles so they can use them in their Algebra 1 classes as well (unlike the rest of their classes who resist using the tiles unless I physically move the tiles around for them).

I love these kids. They can be so much fun to teach. And then other days, it’s just so draining to deal with all the drama and the lack of motivation and the lack of skills. I can’t figure out if I need a day off or if I need some better coping skills. Or if I just need to be a “better teacher”.

I had a caffeinated beverage this morning, so it’s not that.

Grading and Practice in Algebra Support

I was thinking earlier today about how I grade in my Algebra Support (math-intervention-type) class, and reflecting on how I think it was really effective for the kids last semester. So I figured I’ll write a quick post about what I wound up doing.

I used a standards-based system in that class, although I’ve had to implement it in a school that uses a traditional (percent-based) grade book. In my grade book, the quizzes category is worth 60% of their grade. In that category, the only “assignments” are the standards that I assess on. Every time we assess a standard, I overwrite their previous score for the standard (even if it goes down). This means that, with practice and effort, it’s very easy for a student’s grade to drastically change in my class from one quiz to the next.

We basically go through a teach-practice-assess-practice-reassess-practice-reassess…etc. cycle for each standard. This means we spend a lot of time practicing skills during class, which is basically ungraded (sometimes I’ll enter assignments as a completion grade in the In-Class Work category, but not often). However, as the kids complete each practice worksheet, I do check each problem for correctness. I highlight the ones that are correct and give it back to them to fix. They don’t get to turn in a page until all the problems are highlighted.

This method of checking practice worksheets has a few advantages:

  • Immediate feedback
  • Kids have a chance to see what kinds of mistakes they are making and fix them right away
  • Kids can work at their own pace, and ask me to check at whatever interval makes them comfortable
  • Emphasizes the idea that with practice, they can “get” anything (growth mindset rather than fixed)
  • Emphasizes the idea that they are not done with practicing until they are 100% solid (I don’t let them stop assessing on a standard until they’ve earned three 5s in a row)

Last semester, I know that their skills with integers improved drastically over the course of the semester. (We’re doing polynomials right now and I can ask them what -3x and -7x are and they say -10x!!! It’s amazing!) One of our Algebra 1 teachers says that she sees a difference in her class as well, so I’m pretty excited.

This semester, I’m going to do the same thing with my class structure and grading, but the standards we’ll be focusing on are the same as they’ll be doing in Algebra 1. Right now we’re representing, adding, subtracting, multiplying, and dividing polynomials using Algebra Tiles, and then we’ll get to factoring and completing the square (with tiles and on paper), and using the quadratic formula.

I’m looking forward to seeing the same kinds of progress that I saw last semester. I’m also really enjoying teaching actual Algebra 1 content – I love teaching factoring. So much.

Math Hangman

My students invented a game today. I was busy helping one student learn how to factor the greatest common factor from a polynomial at the back of the room, and some of the kids were goofing off going over homework problems on my Promethean Board at the front of the room. They told me they were playing “Math Hangman” with the rest of the class, and had this drawn on the board:

Math Hangman

They were asking the class to fill in the circles on the outside of the box. (I taught them to use the box method to organize their work while multiplying and factoring polynomials.)

I shrugged, told them I was totally cool with this game, and let them carry on (for the 30 seconds or so until the bell rang).

I think I’ll keep this in mind for next year as a way to introduce factoring. This could be cool – after we learn to multiply polynomials, tell them we’re going to play a game. Fill in the box, then have the kids figure out what we multiplied to get there. Talk about what strategies the kids used to figure out the missing values. Come up with a strategy that works all the time, then introduce factoring as a way to put vocabulary with the strategies that they have come up with.

Could be fun.

Algebra Tiles

I started planning for Algebra Support this weekend, specifically related to their lessons on Algebra Tiles – what they are and how to use them. I needed to start by sketching out what I wanted their notes to look like by the time the lesson is over. I wrote out 5 pages of notes (in a composition notebook) – 1 page each on the following:

  • What each type of tile usually represents and how to create expressions with them
  • What the Zero Principle is and how it works with the tiles
  • How to show adding and subtracting polynomials with the tiles
  • How to show multiplying polynomials with the tiles and a product mat
  • How to show dividing polynomials with the tiles and a product mat

(Writing out the notes like this gives me the added bonus of having a master copy that I can tell students to copy into their own notebooks when they are absent. This has always seemed like such a great idea to me that I never really get around to implementing effectively.)

When I say “dividing polynomials”, I don’t mean polynomial long division. I’m using it as a sort of intro to factoring – if I give you the multiplied expression and one of the factors, use the tiles to find the other factor. I’m hoping that this will help the students make the connection to factoring – when you have to find both factors. Since I’m planning on getting to this concept before one of the Algebra 1 teachers gets to factoring, I’m hoping it will help the students understand what they’re doing when they get to that point. (I don’t know what order the other Algebra 1 teacher is using for all the quadratics skills we are teaching in January, so I’m going to have to find out this week and adjust to figure out how to assist those students.) Continue reading

Things A-Changing…

I’ve got some changes to make for second semester in both classes, and I’m going to write about them in the hope that I can solidify some of my own ideas about them before committing to them. In Algebra Support, I was always intending to make some changes second semester because my plan for the class was for different goals and a different structure in the spring vs. the fall. In Financial Literacy, I have realized that I am being WAY too lenient with my classroom management and expectations, and it has caused significantly lower grades than I had last year. (The average final exam score was 13% lower than last year, and I had 6 students fail the semester as opposed to 2 last year.) Continue reading