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.

Posted in Math II


My favorite thing about teaching is that every student is unique in how they think about and approach solving a problem, and each has different areas for growth; for me as a teacher, figuring out how to meet each student where they are is the awesomest puzzle ever!

So here are two puzzles I’m working on with my new class:

  1. I’ve got some super high achievers who are clearly bursting to share connections to rich math topics that are tangential to our lesson and may take us too far off point. I need to find a way to affirm their knowledge without allowing it to take us too far off of the objective. I definitely do not want to dampen their enthusiasm.
  2. In a class full of “A” students, how do I take every kid to their next level? I need to look for areas in which each student can improve their work products.

For the first puzzle, I’m thinking I’ll have each student present one problem that piques their interest. It can be related to our coursework or not, and I’ll have them pose the problem in our online classroom so that their peers can comment and participate in online discussion. Has anybody done something like this before? I need to structure this task enough so that students know what they’re doing, but leave it unstructured enough that students have the freedom to express themselves.

For the second, I’m already working on finding areas for each student to work on. I’m going to create a Google form for goal setting so that I can see what the students want to work on this year. From our interactions in class, I have prompted individual students to try multiple representations, to explain their thinking more precisely, and to solve the problem in as many ways as they can, as well as giving individual feedback for improvement on homework.

This is where I’m planning to start solving these puzzles. I’m always open to trying new things, so if you have suggestions for me, I’d love to hear them!

More about my lesson sequence tomorrow… I’m tired and zero period will be here very soon!

Posted in First day, Math II

My first blog post!


This year I’ve decided to try blogging as a means for reflection on my daily practice. I’m attempting to blog every day, which I realize is a huge undertaking on top of everything else, but I really want to make this a priority. I’ve never been good about reflecting in the moment, so here is my chance!

Today was my first day of school. I’m just teaching one class during zero period. It’s our accelerated Math II course, and I teach a super accelerated population of 8th graders who come to the high school to take this course. This teaching assignment is on top of my full-time job as my district’s High School Math TOSA.

I started with this quick and mathy way to group students, inspired by and adapted (read: stolen) from CPM. Once students found their teams, we read this article and we talked about some of the ways that we could promote “Koinonia” in our classroom. I like the message of this article, but it fell a little flat; I needed to start with something lighter to get students talking first. I really tried to hit home the idea that we can get further by working together and building on each other’s ideas, rather than tearing each other down and promoting your own ideas at the expense of others’… I’ve seen this too many times in honors level courses.

Then we hit the main event: Painted Cube, a low-floor high-ceiling task used in the Week of Inspirational Math from youcubed. Students figured out the 3x3x3 cube pretty quickly, so I asked them to move on to the 4x4x4 cube, then they moved themselves onto 5x5x5, 6x6x6, and nxnxn, depending on their motivation. They looked for patterns, they built models from sugar cubes, they explained their reasoning (with lots of nudging and “minus two? where did that come from?” from me). I tried to focus the conversation not on the task itself, but on communicating their ideas clearly; sure, you figure out the right answers, but why should I believe you? Prove it! I told one student, “you can’t just know…”

I asked three students to share their work in front of the class. The first had thought about the problem completely differently from the rest of the class; he had sliced the cube into the “front and back slices” which were the same, and the “middle slices” which were all the same. This forced every student to think about a solution method other than just their own and their teammates’. The second student’s solution method was similar to the rest of the class; to show his work, he had just drawn one face, then labeled each with a number: 1, 2, 3. This prompted the questions, “Can you all understand this thinking?” “If someone from outside the class looked at this, would they understand this thinking?” and “How can we be more precise in our symbols and vocabulary in order to make this understandable by everyone?” The third had drawn an abstract sketch of one face of the nxnxn cube, with one square in each corner, then “n-2” labeled on each side, and showed an expression to represent each of the parts of the nxnxn cube.

It was a successful morning overall. I will update later with student work… didn’t have time to take pics this morning, as I already kept them over by 3 minutes today (it’s only the first day!!!).