Category Archives: First Year

Exploring Graphs of Quadratics

(Link to worksheet file is in the middle of the post. TL;DR summary at the end.)

The textbook we use at our school (Glencoe 2005) has a lot of things that I don’t like about it (why am I teaching the Quadratic Formula before simplifying radicals?) but when I started looking through the chapter on quadratics I found something interesting. After the text introduces graphing a quadratic and identifying the axis of symmetry and vertex, they devoted two pages to this “Graphing Calculator Activity”. The idea was that students would explore families of quadratics by using a graphing calculator to graph the parent graph – y=x^2 – and various manipulations of the graph to see how changing values affected the graph.

I liked this idea, since my students have not had nearly enough time this year to explore graphs and their equations, but I don’t have graphing calculators in my room. Most of my math colleagues have class sets of calculators, but I’m new this year. (This is also the reason my students have not been able to explore equations and their graphs this year.) I haven’t taught them how to use graphing calculators and anytime I’ve needed to graph something in class, I’ve done it on the projector using Desmos.com. I’ve also encouraged them to use Desmos to check their work and even required them to use it to do their homework on Solving a Quadratic by Graphing. (Their graphing-by-hand skills are abysmal.)

A quick comment here: I have about 4 students with their own TI-83 or 84 graphing calculators, including a visually-impaired student whose calculator talks to her. I forgot what a GIANT PAIN IN THE ASS they are to use! Seriously – 2nd > TRACE > scroll down to ZERO or press 2 > scroll across to LEFT BOUND > scroll over to RIGHT BOUND > GUESS > then it gives you the answer Then you have to do it all over again for the second solution?! After spending the whole school year using Desmos (where you just click on the graph and it gives you the coordinates and it doesn’t cost $100), I’m just not doing it anymore.

Where was I? Graphing worksheet. Right.

So instead of using graphing calculators, I booked a day in the computer lab to use Desmos. I created a worksheet that uses all the same concepts from the activity in the textbook, only I edited my instructions a little so my students would understand what I meant. I informed the kids that we would be in the lab, and passed out this worksheet when they got to the lab with basically no explanation. (I probably should have clarified expectations better – see “The Bad” below.) Most of them finished it, and a few finished very early. (Also see below.) I refused to let them turn it in until I was happy with their explanations and observations, which was very good. I made them answer Question #3 the way it was supposed to be answered, and ensured that they were noticing the things they should have been noticing.

Download the Word document (.docx format) here:

Graphing Quadratic Functions Exploration

The Good:

In general, I think my students noticed the effects that different changes to the equation can have on the graph. Their written explanations and observations tended to show a fairly solid understanding of this. Because of our weird schedule that week (standardized testing on different days for different grade levels), I didn’t get a good chance to discuss their findings as a class. I’m also not sure if they truly generalized their observations, or if they retained the information.

My students generally seemed to understand and enjoy the activity. Considering how often my class feels like I’m pulling teeth, I consider this a huge advantage. I will absolutely consider activities like this anytime I introduce graphing in the future.

After spending the whole class period graphing on Desmos, a few kids were fascinated by everything Desmos can do. A couple of students graphed a bunch of quadratic and linear functions on the same coordinate plane and used a variable to animate their graphs. This turned out really cool, and I made them save their work on their accounts. A few kids got to explore all the graphs that Desmos contains in the menu, including crazy polar graphs. Some even noticed the Staff Picks graphs on the Desmos homepage, opened them up, and started playing with them.  This was so awesome – they kept calling me over to show me stuff on Desmos. I’ve never seen them interested in and exploring something that has to do with math before. At all.

The Bad:

I probably should have given my students a better idea of my expectations for the worksheet. I should have explained that I wanted them to make discoveries for themselves; that I hadn’t explicitly taught them the answers to the questions on the worksheet yet. I also should have explained that I was looking for them to use the vocabulary terms they learned in class (parabola and vertex, mainly).

Question #3 wasn’t phrased very well for my students. (In my defense, this is partly because they didn’t read the whole question. They tried to graph y=ax^2 and were confused when Desmos either wouldn’t graph it or immediately defined a=1 and there was no change to the graph from the parent graph.) I should have made it more obvious that I was looking for a general explanation of what happens when you substitute a number in for a.

I didn’t realize how quickly some students would finish, and I should have had an alternate activity prepared for when they did. It didn’t occur to me until my very last period of Algebra 1 that Function Carnival would be a perfect extension for this lesson.

TL;DR

I loved this activity. Almost every student was engaged almost every moment. I have a couple of improvements for next time, most of which were caused by my own inexperience (first-year teacher). I’ll be using similar activities to introduce graphing lessons a lot more next year.

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I’m Tired and Frustrated, and Just Finished Grading Tests

My students took their first test today.

In reviewing for the test and then grading the tests today, I have learned:

  • They don’t understand that there is a difference between 24 divided by 12 and 12 divided by 24 (both if these are 2, apparently).
  • They get so freaked out by the idea of a fraction in the problem that many of them skip the whole problem or just ignore the fraction part. (I’m referring here to a problem that asked them to multiply 1/2 by 8. Asking them to multiply 1/4 by 12 just about caused heads to explode.
  • I have absolutely no idea where to start here. As a first-year teacher, I consider it a good day when I can spend 30 seconds thinking about the next day. Forget trying to figure out how to teach the Algebra curriculum and remediate basic math skills at the same time.

Also, this weekend I learned that 2/3 of my students did not pass their Arizona standardized test last year in 8th grade. The test isn’t that great-even students who pass it may not have a fantastic math foundation. It does explain why so many of them aren’t understanding what goes on in class, but again, I have no idea how to deal with this.

How can I teach my students the Algebra they have to know for their semester final in December and teach them how to not panic at the very sight of fractions? We’re already behind in our pacing, and it’s only the 4th week of school. I’m probably only going to get further behind, because I can never get through a lesson in the time I think it should take.

How am I supposed to find time to even think about this when I have to write lesson plans and PowerPoints and quizzes and tests from scratch?

Two Weeks In

Yes, I probably could have posted at some point in the last two weeks, but I was tired.

First Day

I had a bunch of stuff I planned to do when I got to school, including unloading my trunk, re-hanging the posters that insist on falling down off my wall, printing off all the stuff I wanted to say on the first day, and making sure all my technology was working the way it was supposed to. Then, on my last trip moving stuff from my trunk to my classroom, I locked myself out of my room. It took two phone calls to the front office and almost 30 minutes for someone to finally come unlock my door and let me in. (I’ve spent the last two weeks triple-checking my pockets to make sure I have my keys every time I leave the room.)

So my first day started in a bit of a panic.

First Week

I only got through about half of what I intended to get through with my students, and had to finish up my first-day stuff on the second day. My students took a test for me on the prerequisite skills they need for Algebra (integers, fractions, decimals, etc) on Tuesday, and I didn’t get around to getting them graded until the end of the week.

On Monday, I was informed during 5th period that the counselors had a senior with nothing to do during 4th period, so would I like a TA? I jumped all over that. Then a student walked into my room during 6th period saying she was my TA. Within about an hour on the first day of school, I went from having 0 TAs to having 2 TAs, and no idea what to do with them. I’ve spent a fair amount of time this week coming up with things they can do. I am loving the fact that I can have them grade, except when I discovered the answer key was wrong after they had graded 3.5 of my 4 periods’ worth of stuff.

The first week went fairly well. The kids weren’t used to the school environment or each other yet, so they stayed pretty quiet and at least pretended to listen while I was teaching. Most of them did their homework. My only issue is my last class of the day – they just won’t stop talking! I rearranged seating charts this week because I couldn’t take it anymore (and because I wanted my students to be sitting in groups, not rows) and that helped a little bit. What really helped, however, was calling the parents of about 4 of the worst offenders and letting them know I’m “really concerned that [your kid] isn’t getting everything he/she could be out of this class”. The parents I talked to were great, and the class was much quieter for the rest of the week.

Second Week

I started my second week feeling panicked and frustrated because I’d had a to-do list a mile long over the weekend and hadn’t gotten much done. Specifically, I hadn’t done lesson plans for the week. (At this point “lesson plans” actually just means “powerpoint presentations”, not all the stuff I’m supposed to be doing.) I went to bed unhappy on Sunday night, and woke up still a little panicked on Monday morning. I didn’t lock my keys in my room this time, and managed to be ready for 1st period.

Now, here’s the trick: my planning period is 2nd, so all I really have to be able to do is get through 1st period on Monday morning, and the rest of the week will be fine.

I’ve started teaching material from the textbook this week, which has provided some much-needed structure to my plans. I’m struggling to get through everything I need to in the amount of time I have, although I’m getting better. I’m sure in a few months I’ll be pacing my lessons much more consistently. Pacing has always been a big issue for me – I tend to talk too much and not move on to the next thing when I should.

This week, I got an email asking how things were going with my mentor, and if I’d had a chance to talk to him yet, which I thought was funny. My mentor’s classroom is across the hall from mine, and I talk to him almost every day. Clearly, not everyone is lucky enough to have an easily-accessible mentor like I am.

Another teacher from the department, my instructional coach, commented on Friday that I looked a lot more relaxed this week than I did the first week. I was. Honestly, I’ve learned the hard way that even if I don’t get everything done that I want to over the weekend, I’ll still survive, and that has gone a long way toward my mental well-being. (In the future, that may not be such a good thing for me to know, but for right now, I needed it.)

My instructional coach also came in and observed my last-period class (at my request) to see what I can be doing better to discourage the constant talking. She had a lot of positive feedback on my teaching, including telling me that she couldn’t tell that I was completely unprepared for the class. (I had been typing bellwork questions for the students during the passing period when I realized I had forgotten to finish my powerpoint for that class.) She also had a few helpful suggestions for me to focus on in that class. The one I’m going to focus on first is how I position myself in the classroom. On Friday, I had the kids working on homework for a while, and then a review worksheet. As they did so, they had lots of questions, so I ran around the class answering them. My instructional coach recognized that when I do that, I often position myself with my back to half the class, making it very obvious that I’m not paying attention to them, and they take advantage of that. This week, I’m going to work on where I stand and how much I appear to be aware of what is going on in the class.

My students have their first quiz tomorrow and their first test on Friday, and then we’re finished with chapter 1 of the text! I can’t believe how fast this has already gone by. As the teacher, the day never feels like it’s dragging – usually I don’t have enough time to say everything I need to say. I can’t even remember how many times the bell has rung and I’ve nearly growled in frustration. “What?! Class can’t be over – I’m not done!” And because it’s high school, I have to let them go so they can get to their next class.

It’s only Sunday morning, and I feel much more prepared for this week. I have a couple of finishing touches to add to my powerpoints when I get to school tomorrow morning, but I have them written for the first 3 days of the week. I may even write up formal lesson plans this afternoon so I can say I’ve done it. I would work on writing my chapter test, but it’s saved on the hard drive at school, not in the cloud, so that will just have to wait.

I have a funny story to post from one of my classes on Friday, but I have to get ready to leave, so I’ll do it later.