Category Archives: Algebra 1

Three Good Things (I’m an overachiever)

At the moment (as in, this week) three things are going really well in my classes. I considered writing about one of them, but I’m an overachiever so I’m going to write about all three.

Also, my masters degree is going fine. Being an overachiever helps with that too.

Problem Solving with In-N-Out

On Friday, the kids started the In-N-Out problem (see link for Robert Kaplinsky’s take) and we finished it yesterday. I love this problem. This year, I hung up my group-sized (2ft by 2ft) whiteboards at the beginning of the year, put the kids into random groups (different groups on the two different days actually) and told them to get cracking.

They did great. They discussed with each other (as much as freshmen who aren’t used to collaborating on things could be expected to), they wrote their ideas, some of them labeled their work really well, and some of them looked around the room from time to time to get ideas or see if they were on the right track.

Today, since they finished the problem yesterday, I used it as an example in each class to move into discussing slope/rate of change and the y-intercept in an equation. When I brought it up, the kids told me they really enjoyed working on it and requested more problems like that.

So now I have to find more problems that are good for vertical non-permanent surfaces, and preferably are related to standards that I actually have to cover in Algebra 1.

This is an amazing problem to have. I am so excited right now. I told the kids that I had a great time watching them solve the problem, and it’s true. I had so much fun. I definitely want to do stuff like this more often, and would be very into the idea of moving toward a much more problem-based curriculum.

Evaluation

Yesterday I also had my formal evaluation, during one of these Algebra 1 classes. My administrator sat in my room and listened to the kids solve this problem, which is not exactly an EEI-structured lesson format but I don’t care. It’s so much fun. And it’s so great to do for an evaluation where the biggest thing they are judging me on is student engagement because I probably had nearly 100% engagement, nearly 100% of the time.

The administrator told me a funny story afterward, and is planning to put this quote in the write-up of the formal evaluation. Apparently the group he was sitting nearest had a student who just kept insisting “It’s 90 cents. It’s 90 cents.” Another student in the group kept asking him how he knew. (!!!) He just kept insisting that it was just right and couldn’t explain his answer, and she finally told him, “You know that when she [meaning me] comes over here, she’s going to ask us! You have to be able to explain it!”

Woo hoo!

I nearly jumped for joy in front of my administrator when he told me that. This is amazing – we’re seven weeks in and some of them already know that being able to explain it is more important than the right answer (as least, it is to me).

This is a great sign, and I’m so excited about what this means for the rest of the year.

KenKen Puzzles

Clearly my theme for this blog post is about explaining reasoning, because this point is also going to touch on that.

So I do a bellwork routine, where we do the same type of bellwork question every week. Mondays are visual patterns, Tuesdays are estimation questions, Wednesdays are Which one doesn’t belong, Thursdays are a writing assignment because our school is doing a writing initiative this year, and Fridays are KenKen puzzles.

Mondays and Fridays are my favorite. I’ll probably swap out the estimation and WODB days for other things as the year goes on, but I usually keep the patterns for the entire year. This year I think I’ll keep the KenKen puzzle all year, because it’s going REALLY well.

I do this bellwork routine in Algebra 1 and Geometry, and since I had a handful of the Geometry kids in my Algebra 1 class last year, the Geometry class has been trained very quickly and very well. They can do harder patterns and are ready for harder KenKen puzzles than the Algebra 1.

The Algebra 1 kids are definitely picking it up, and getting the hang of how to fill in cells, what’s helpful and what’s not helpful, and how to think about the clues. We moved up from 3×3 to 4×4 a couple of weeks ago, and I bet they’ll be ready for 5×5 after fall break. (6×6 will be longer because moving up to an even number is harder than moving up to an odd number.) Every week, I make them do as much as they can on their own, and then we go over the puzzle as a class. I ask the class “What do we know now?” after each step and write down pretty much whatever they tell me. I spend a lot of time paraphrasing and restating what kids tell me, because I want to focus on the logic of the puzzle.

So the KenKen puzzles are going well, but some really cool things happened on Friday.

First, Algebra 1 – I had kids raising their hands and participating in the KenKen discussion who have NEVER raised their hands in my class before. They can tell me what goes in a cell and how they know. At this point, pretty much all I have to say is “Ok, how do you know?” the entire time. They are really getting into it.

Now Geometry is a whole different ballgame. On Friday, we’d been doing proofs for about a week. They’ve done examples of proofs, they’ve used CanFigureIt.com to explore proofs in a structured environment, they’ve written reasons for proofs that have all the statements written, and they’ve attempted writing their own proofs.

On Friday, we went over the KenKen puzzle on the board, solved the whole thing, and I stopped them. “You guys. Everyone. Did you hear what you just did?” They all stare at me. “Did you hear what just happened as you solved that KenKen puzzle?” Now they’re really staring at me like I’m a crazy person. They’re probably sitting there like, yeah, we solved the puzzle, just like we do every week. So I said, “You just did a proof.”

Now they’re 100% positive I’m crazy.

I said, “For every cell in that puzzle, you made a claim, and you justified it. You told me what should go in that cell, and you told me why. That’s a proof. You just did a proof to solve the puzzle.”

Blew their minds. It was amazing. It took a solid few minutes to calm them down again, but the sudden realization of the connection between proofs (which they currently hate) and something that they are generally enjoying was totally worth it.

Also, I enjoy blowing the minds of high school students. It’s a major part of the reason why I teach.

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Systems of Equations Introduction

Last year, I was really struggling to teach my Algebra 1 students how to solve a system of equations using substitution. This struggle completely flabbergasted me, because substitution always made so much sense to ME and I found elimination to be harder, while my students LOVED elimination and couldn’t comprehend substitution at all.

“What do you mean, ‘substitute’?” “(Seriously? We’ve been substituting things into equations all year! I can’t even say the word properly anymore because I’ve said it so much!)

“So, I put the other equation here?” (Then they replace the y with the whole equation y=3x so now they have two equal signs.) (Their other favorite thing was to leave the original variable in the equation, so now their new equation says y3x.)

They adored elimination. It was their favorite thing ever. They started adding equations together on problems that didn’t even have a system. Of course, they really struggled to remember to multiply the equation before adding when they didn’t have equal coefficients…but their equation-solving skills were so atrocious that they could somehow still manage to lose the variable they had not actually eliminated…and now I’m just getting frustrated all over again so I’m going to go back to my original point.

Ahem. So I was looking around online last year for a different way to approach teaching substitution as a method for solving systems, and I ran across two ideas, both of which I’m using this year. I’m only going to discuss one of them in this post, because I need to go to bed. I’ll write up my thoughts on the other one later, after I actually implement it next week or the week after. Continue reading

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.

Baseball Statistics Project

I wanted to post a project my students will be doing next week in class. As a first-year teacher, I’ll appreciate any feedback more experienced teachers want to give.

The Situation

I teach mostly Algebra 1 and one period of Algebra Support. This is a class in which students were enrolled for extra support if they failed the AIMS test last year in 8th grade. They are enrolled in both Algebra 1 and Algebra Support – it’s not an alternative to the normal Algebra 1 class.

Our school doesn’t do any kind of block scheduling, so the students who are in Support have two math classes a day. The Support class was intended to be more than just a math-related study hall, but since this is the first year my school is doing these classes, we weren’t given much guidance as to the scope and structure of the classes. I spent much of the first semester floundering resorting to using the class period for review of the concepts covered in Algebra 1 that day, or in the last week or so. Since my Support class is the last period of the day, this was easy. I didn’t plan ahead very much, except that students were in the computer lab doing remedial assignments on Tuesdays and Thursdays and wrote reflective journal entries for me every couple of weeks or so.

I’ve decided I want to move this class toward a more project-based structure this semester. (My students will hate this, because they don’t know how to function when they’re given something more complicated than a worksheet full of math problems with straightforward answers, but that’s the point.) I started off by looking through the Problem-Based Curriculum Maps on the Emergent Math blog, and found an activity about the NFL passer rating. It includes a ridiculously long, complicated formula to calculate the statistic. This looks like fun, but I’m well aware that if I put that formula in front of my students, their brains would short-circuit and they’d be out for a week. (Kidding, kidding. We do have pretty significant confidence issues in that class, however, which isn’t surprising.)

Anyway, the NFL activity got me thinking about statistics in sports. I’m not a football fan, so I have a really hard time understanding football stats and how they are used and applied. I do, however, understand baseball stats. I grew up watching baseball and listening to the commentators talk about the game, players, and stats. I know, off the top of my head, how most of the stats are calculated and what they mean, so I thought it would be a lot easier for me to teach a lesson using baseball statistics. Most of my students are football fans, so we may have some struggles to overcome on that front. I also knew that I didn’t want a formula that was so complicated, as my students don’t really understand how to use and manipulate formulas effectively, and a lesson like this would require me to basically do the whole thing for them on the board.

I started looking around online to see if I could find a baseball project that would fit my needs, and quickly found a page suggesting an idea for a project in which students would be given a roster and a variety of statistics, and asked to determine how much each player should get paid. I liked this idea, but would have to create the materials myself.

This project doesn’t really address any algebra concepts, as there isn’t really any blatant math to be done. It does, however, ask students to evaluate information and make and defend decisions based on that information. I decided to use this project during the first week back to school, to work on their critical-thinking skills without including any of that terrifying math. I’m hoping it will be a good way to get brains re-engaged after the winter break.

The Project

Instruction Sheet

Major League Baseball has just terminated all long-term contracts, and every team in the league now has to re-negotiate the salaries for each player. The new salaries will be re-negotiated every year. You are the board members of your team, and your job is to decide how much each player on the team will be paid this year, based on their performance last year.

You have a roster of 15 players, their statistics from last year, and a team salary budget. The players include 10 position players and 5 pitchers. You have been given hitting statistics for the position players and pitching statistics for the pitchers. You will need to decide how to use these statistics in order to figure out how much you want to pay each of your players, keeping in mind the constraints of your budget.

The class may choose to agree on certain restrictions, such as a cap on salaries or a base salary. We will make this decision before everyone begins working, and all groups must follow the restrictions that the class agrees on.

You will turn in your list of players and their salaries, an explanation of how you made your decisions about those salaries, and all the work that you did to complete the project. You may type or hand-write your list and explanation. Your explanation must include a description of how you decided how much each player was worth and which statistics you found most useful in making your decisions.

Groups will be graded using the following rubric:

  • Player Salaries: 15 points
    • Player salaries reflect the player’s value to the team: 10 points
    • Player salaries use the full budgeted amount: 5 points
  • Group Participation: 15 points
    • Each member of the group contributes meaningfully to the project: 10 points
    • Completion of Group Participation Analysis: 5 points
  • Written Explanation: 20 points
    • Explains how the group decided the player salaries: 15 points
    • Explains which statistics were used: 5 points
  • TOTAL POINTS: 50

This project is due on Friday, January 10, at the beginning of class. The Group Participation Analysis will be completed in class on Friday.

[Instruction sheet available to download at end of post.]

Project Logistics

Students are being assigned to groups of 3, and will be given a spreadsheet with the names and statistics of 15 players on the team and a total salary budget for those players. (I took the actual total salaries for each team and divided it in half to make the numbers a little closer to the actual amounts.) They are also being given a glossary of baseball terms relevant to this project, which is included on the last page of the Excel file [linked below].

As the instruction sheet says, I want the students to work together to decide how to spend their budget. They will turn in a list of players and their salaries, as well as an explanation of how they decided how much each player will be paid. I tried to set up my grading system so that the written explanation (and group participation) is more important than the actual answers they give me for salary amounts.

I’m emphasizing group participation for a couple of reasons. First, this class has a really bad combination of students who don’t work well together. They either don’t get along or get along too well (and will spend the whole time talking instead of working).

Ideas for Raising/Lowering the Level of the Project

I don’t expect to have any need for raising the level of this project (I’m going to consider it a huge success if they just do what I’ve assigned) but I did have a couple of thoughts.

Leave out some of the statistics (batting average, ERA, on-base percentage, etc.) and have the students calculate those themselves given the raw data (number of at-bats and hits, earned runs and innings pitched…). I chose not to do this for this project because I didn’t want to give students hints as to which stats might be best for evaluating the worth of each player.

Have students find the statistics themselves online. Honestly, they’re easy to find – this wouldn’t be too much of a challenge.

The level of the project could be easily lowered by adjusting the statistics included in the spreadsheet. I included a lot of stats here – a lot more than my students should need. Part of my expectation is that I want them to evaluate the usefulness of the stats and decide which ones are more important than others. Younger students may get overwhelmed by the amount of data I’m providing.

Similar projects could also be created for other sports, of course. As I said, I believe my students tend to watch football, not baseball, so I may re-do this project later in the year using football stats instead.

Resources and Download Files

Baseball Statistics Project – Instructions Sheet

Rosters and Stats Lists

I used the following websites for the statistical and salary information:

Baseball-Reference.com – statistical information for all teams and players. Can copy CSV data into an Excel file easily.

ESPN – Arizona Diamondbacks Salary Information – This page lists the salary information for the D-backs, but the right side of the page also lists salary information for every team in the league. As I mentioned above, I divided these numbers in half for the project.

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?