Posted in Math II, Quadratics

My undying love for Desmos

For the past two years, I’ve been out of the classroom working as the high school math specialist for my district, so I’ve been totally jealous of everyone using Desmos classroom activities with students. I’ve presented trainings on how to use Desmos classroom activities and Activity Builder; I use the Desmos app on my phone every day; I bug my teachers to use Desmos at every opportunity; I am convinced it’s the most powerful math learning tool there is; I just haven’t gotten to use it with my own students … Until now. And it was every bit as glorious as I thought it would be.

First up: Marbleslides Parabolas. I have never seen students so intent on a problem in math class before. I mean, I know the students find me FASCINATING and everything, but seriously, the amount of learning going on in the room was amazing. I started taking pictures and they didn’t even notice! I set students up in pairs on chrome books and let them go. I was a little afraid that students would start changing numbers randomly without paying attention to the cause and effect, but from every pair I heard some version of “change that number so we can move it over to the left.”

Only a couple of sticking points: I know it shouldn’t, but it still surprises me that students weren’t willing to just hit the “Launch” button and watch what happened; they were trying to get each graph perfect before launching. I need to work on growth mindset with this group. This population struggles with the need to look “smart” in front of their classmates and aren’t willing to put themselves out there if it means being wrong. Additionally some groups got hung up on the domain; they wanted to “move the parabola to the right more” when they really meant that they wanted to change the domain. It took a lot of questioning to coach them to see how the domain restrictions were affecting the graph.

Next day, we followed up with another Desmos activity, Quadratic Transformations, that I borrowed generously from Mary Bourassa‘s Quadratic Transformations part 1 and part 2. Basically I loved her activities and wanted to condense them down to one day to solidify what we had learned from Marbleslides the day before. Students were engaged and thinking deeply, and they were able to apply what they learned to graphing and writing equations in vertex form the following day.

As a teacher, I never sit still while students are working (actually u have trouble sitting still no matter what… ADHD anyone?). I love hearing students’ conversations, seeing what they’re doing, and challenging their thinking (read: bothering them). So I walked around the classroom as they worked and didn’t really take advantage of the teacher dashboard. I think that’s going to take me some time to get used to. Maybe if I accessed the teacher dashboard on my iPad, I could still walk around the room and feel comfortable doing so. Does anyone have any tips for using the teacher dashboard effectively?

I’m looking forward to my next Desmos activities; I have two planned for next month: Building Polynomials #1 and Roots of Quadratic Functions: Looking for Special Cases.

I love my husband. I love my two daughters. I love my friends and my colleagues and my students. And I love Desmos.

Posted in Math II, Quadratics

My favorite lessons so far

Wow, I forgot how busy working both jobs is! Blogging every day has taken a backseat to grading homework, creating assessments, math placement, data analysis, meetings, answering emails, first weeks of school minutiae at the school site and district level… Oh yeah, and then actually having a life outside of work. I want to make a commitment to blogging, though, so that becomes a natural part of my reflective process.

So what have my students been up to? We’ve been learning about quadratic functions in multiple representations, key features of parabolas, and transformations of quadratics. Right now we’re wrapping up our first unit after doing some really great learning tasks.

Last week students worked on What are the connections, a task borrowed from CPM’s Core Connections Integrated II. The situation is a water balloon toss where different data is given: first an equation, then a graph; then (oh no!) the water balloon hits your computer and you only have time to scribble down some data points in a table before it fizzles out; then (since your computer is broken) you have to do the final measurements by hand, so you only get the launch point, the maximum height, and the point where the water balloon lands on the ground. They have to graph and compare the parabolas given the different representations. This task had students working in teams in a way they hadn’t before. They were asking questions, verifying their work, saying things like, “the function has the same values for x=8 and x=9, so the axis of symmetry must be x=8.5!” The follow-up questions had to do with the domain, range, intercepts, and vertex in context of the situation. They learned so much from this one task about the shape and key features of a parabola, not to mention the valuable skills of working together as a team. This is a must-do for all Math II (or Algebra 1) classes!

We spent a few days working on vocabulary, interval notation, and sketching parabolas given key features. Then we spent a day working on this application, Yearbook Sales, that I created. I ended up extending the task to two days because I loved how the students were working on it in teams. The situation: students have to find the best price of the yearbook, given that for every $5 increase in price, 20 fewer students would buy the yearbook. I was hoping for them to apply everything we had learned so far: connecting multiple representations (table, graph, equation), writing an equation for a quadratic equation given a situation, understanding the meaning of key features (intercepts, vertex).

What I loved most about this task was the arguing! I went to check on progress at one table, where two girls were really running the show, leaving their other teammate to his own devices. The girls explained their reasoning: they had written two linear functions (one for the price of the yearbook and one for the number of students purchasing the yearbook), and when they multiplied the two together to get the total profit, they thought that the two values should be as close together as possible to maximize their profit. Then their teammate asked, “so how much was the total amount of money?” and they responded confidently, “about $10,000.” Their teammate looked a little puzzled, then said, “but I found a way to get $15,000.” The girls’ heads snapped toward him, “Whaaaaaa…?” and I just backed away. It was perfect. I didn’t even have to challenge their thinking because their teammate did, and it got them working collaboratively toward an improved solution. After a successful day of work on this task, I gave students additional time the next day to complete a final draft of their solutions with questions to help guide their thinking.

We also spent a couple of fabulous days playing Marbleslides and transforming quadratics, but I’ll profess my undying love for Desmos classroom activities another day.

Posted in Math II, Quadratics

Homework & Quadratic Growth Patterns

As per my MO, I ran right through the bell in Friday’s lesson. I spent way too long talking about homework, but it was a necessary evil. I love new classes and meeting new students, but I really don’t like having to set procedures and guidelines in the first few days of school. I know spending time at the beginning pays dividends later, but I really wish they could just read my mind…

Here are my thoughts on homework:

  1. Students should never work on exactly what they worked on in class that day. They just learned how to do it 5 minutes ago, and yet we expect them to now be able to do it perfectly? Practice doesn’t make perfect, it makes permanent, and students with shaky understanding will now permanently do it wrong. Homework should be practice with concepts and procedures they are already comfortable doing.
    • If students already feel comfortable doing their homework, parents won’t feel the need to step in and teach their child, reducing home anxiety surrounding homework. AND maybe we can avoid some of the tricks and shortcuts that parents and tutors often teach. Double win!
  2. Homework should be mixed practice. When students get 20 of the same problem, they become robots and stop thinking about what they’re doing. My go-to example of this is when we’re graphing quadratics and I throw in a linear function to graph, they graph a parabola because they’re not thinking. We’ve got to mix it up so they don’t get complacent… I need my students always questioning.
  3. Practice should be spaced out over time. I figure if they’ve really learned it, then it’s fair game. I throw in all kinds of stuff for them to practice.
  4. Students need some kind of feedback on homework. This is close to impossible to get right, I think. On the suggestion of another teacher in my district, I’ve started making complete, detailed solutions for each assignment and doing in-class corrections (5 minute time limit). She said that this was the single most effective thing she’s done to improve the quality of work turned in by students. I really hope it has the same positive effect in my class!

This year, all of my homework assignments are divided into three parts: Ready, Set, Go (thanks MVP!). Ready problems are intended to get students ready for the content in the next unit, Set problems are intended to employ the concepts in the current unit, and Go problems are all content that has been previously mastered. I try to build complexity and conceptual understanding over time. I really like where this is going so far; it’s a ton of work, but it’s totally worth it for my students and if it helps other teachers in my district.

(Can you see why I can never manage to finish a class period on time?!? I could go on about this for days…)

On with the lesson. I needed a quick way for my students to see multiple visual patterns in multiple representations without having to spend a lot of time; a simultaneous roundtable sounded like the perfect way to achieve this.

In teams of 4, each student was given the first three terms of a quadratic pattern (all different), for which they drew the fourth term and explained how they figured it out. Then they rotated their papers so that they had a different pattern, for which they created a table based on the four terms and description their teammate came up with. Then they rotated again, so now they had a different pattern and had to create a graph based on the table their teammate completed before them. Finally, they rotated one more time to find the number of squares in the 30th term of the pattern and explain how they figured it out. Each rotation was timed at 3 minutes. When they were finished, each team had four completed patterns, and each teammate had completed a different portion of each pattern.

I gave them 3 minutes as a team to look for similarities between all four patterns and connections between representations within the same pattern, then two minutes independently to write it all down. Then we shared out one observation from each table: the second differences were constant, they all had similar curves when graphed, they all had a degree of 2, …

I wanted to focus in on the second differences and writing the explicit equations. We were able to do this for two patterns before I ran out of time. 😦 That’s ok, it gave me time to make this. In a simultaneous roundtable, each student still only has one paper in the end, and thus keeps only part of the team product; this summary page will allow them to record their observations for all four patterns. It’s not perfect, but it will get to the points I want to make on Monday.

I, Kristie Donavan, do solemnly swear to finish my lesson on time on Monday! (I bought a new timer… no excuses!)

Posted in Math II, Quadratics

Quadratic Functions

To begin our unit on quadratic functions, I wanted to lead with a problem that would grab students’ attention and would allow them a peek at many different features of a quadratic function, so I started with Connected from Mathalicious. On their new “Unit Organization” Mathalicious suggested this was a “middle” of the unit task, which is probably better, but the students still got into it.

I had them start with just a blank sheet of paper so that I could see how they organized their thinking. Most students drew diagrams of people/dots connected with lines, several wrote explicit equations for finding the nth term, two created recursive equations. I drew a table on the board and filled in the inputs (# of people), and asked two students from each team to fill in the outputs (# of connections). I asked if the table was linear, and they explained it was not since the rate of change was not constant.

“Oh, then it must be exponential!” I joked, which they did not laugh at (they don’t get me yet). But we showed that there was no constant ratio either. So I asked what patterns we do see, and a student noticed that each term was the sum of the term and the term number before it; in other words, the 5th term was 10 = 4+6, where 6 was the 4th term. This helped us all write the recursive formula.

Someone else noticed that the terms were being added by 1, then by 2, then by 3, etc., and explained that the rate of change was changing at a constant rate. Woo hoo! I introduced the term “second difference”here.

Then a student jumped in with the explicit formula. I wasn’t quite ready for it, so I asked where he got it. He wasn’t sure how to explain it in terms of the context. So I went back to the context: if 10 people are in your social network, they can each connect with 9 other people, 9×10. Then he jumped in, “but you’re double counting, you’re counting the connection between me and you and between you and me as two different connections, so you have to divide by two.”

We wrote the explicit formula f(n)=n(n-1)/2. Then, because I know students sometimes feel uncomfortable leaving expressions “undistributed,” we re-wrote it in standard form and talked about how this expression, while equivalent, didn’t really have a basis in context… 1/2n? Half a person? What does it mean? We decided the first form made more sense.

Here’s where class ended, which was a bummer… I wanted to connect the second differences to the lead coefficient, but my guess is they may not have seen the connection. So instead of my original plan to finish up Connected tomorrow, I’m going to do my next lesson, a simultaneous roundtable of quadratic growth patterns and have them look for similarities in different quadratic functions to get a good feel for the characteristics of a quadratic function. Then we’ll come back to Connected to look for those characteristics.