Category Archives: Problem Solving

Algebra Notebooks: One Year In.

Although I have been thinking of this for a long time, last year for the first time, I guided students to create notebooks for Algebra 1 and Algebra 2. The major design influence for these notebooks was from Sarah Carter(@mathequalslove ). Her notebook dividers inspired me, and gave me exactly the structure that I needed to put these together. Feel free to skip down to the end of the post for the files. In this post, I’m sharing much of what we used for Algebra 2 last year.

I am convinced that this process really helped us in some key ways. We used them as a structure to organize the year’s work, a format to help relate one idea to the next, and a compact guide to prepare for skills-based assessments. The process of curating the notebooks was very clarifying for me. Textbooks have so much information, that it can be overwhelming for students. I wanted to include the most important ideas and examples in our notebooks so they would be useful and clear but not overwhelming.

Although I do admire some of the more “crafty” notebook pages, I didn’t include any foldy parts, or mini-booklets or that kind of thing during this first year of implementation. These are still “interactive” notebooks (INBs) in the important sense of that definition; that students use these as a tool to interact with math. I would think that part of the purpose of the foldy parts is to support students in using their notebooks not to just read over their notes, but to self-quiz, so because we did not do this, we did some explicit work on how to use their notebooks to study, even if they did not have to unfold a part to find the answer. We built in at least three meaningful interactions with new content: during class to fill in the big picture, at home to complete the notes, and then in class to use the notes to solve new problems (…a lot of open-notebook entrance tickets).


Each section has a unit divider, which includes summary learning targets, honors-level extensions, and essential questions. Just about everything in the notebook asks students to create their own notes – although in a very few places, I just made summary notes for them. These notes are generally filled in after we have done some discovery and worked examples together. They are one of the culminating parts of each lesson.


Here are some reflections on the Algebra 2 notebooks. I’ll share the Algebra 1 notebooks soon. As this was the first year, these are definitely incomplete. However, this is a substantial start, and I would be happy if it helps another teacher to get started. The files are .pdf for ease, but please do let me know if you want editable versions or have questions or suggestions.

A few lessons from year 1:

  • Students take FOREVER to glue things
  • Students love gluing things – even high-schoolers
  • Students appreciate organizational help – both the ones who really need the help and those who would be fine on their own.

Things to add to the mix for next year:

  • Some tweaks to a few of the graphic organizers (eg. the factoring pages – I didn’t really like these, and the method is a bit cumbersome. I learned a better technique from a colleague this year that I think I‘ll use next year).
  • Include a complete vocab page for each unit
  • I want to do a better job with helping binder organization (…the companion to the math notebook – everything that doesn’t go in the notebook goes in the binder).

In some cases, in the files below, I’ve included both a blank and a filled-in version, which generally includes teacher notes. Unless the time pressure doesn’t allow for this, I would suggest always giving students the blank versions so they can make their own notes. Having said that, summary notes can be useful if they are given at the end of a topic to make sure that the notes are correct – I tried to do regular teacher or peer notebook checks, but some mistakes slip by, and we wanted to make sure that the notebooks contained correct info.

These are not meant to be a stand-alone. As with any textbook, these notes are always meant to be guided by the teacher.

Attributions/Notes for Notebook Pages: Although I created a fair amount of this from scratch, I definitely borrowed a lot as well. I have made every effort to give credit for everything I’ve used in these notebooks. If I slipped up, I truly apologize. Please do let me know and I’ll add an attribution. This is just one year in for me, and should not be seen as the entirety of the courses. In a few cases, I found someone else’s graphic organizers and just used those. If I didn’t manage to keep track of the sources, I didn’t want to take credit for this work, so didn’t include those pages.

 Unit Dividers – major design inspiration from Sarah Carter(@mathequalslove )

Unit 0

  • Algebra Learning Agreements – we create these together, and I print a poster, which everyone signs and is posted in the classroom. I printed copies of these for kids to glue into their notebooks so we could refer to them when needed.

Unit 1

  • The Key Feature Cards were adapted from the New Visions Curriculum
  • Visual Patterns Guidelines – I had mixed feelings about this one, as the last thing I want to do is to do the explicit thinking for students and rob them of the best part of math – but I decided to include these pages so students have at least one or two worked examples. Upon reflection, I don’t think that this got in anyone’s way.

Unit 2

  • Unit divider: family of functions poster – not sure from whence I got this graphic, but I didn’t make it. If you know, please let me know so I can give credit.

Unit 3

  • Exact Trig Values Chart from Don Steward

Unit 4

Unit X

Back Cover

  • Sweet math poster taken from

Here is the file. Enjoy, and please let me know if you get some use out of this!

2017_Alg 2 Notebook

Polynomial Guess



I found a really nice number guessing game several years ago, and I’ve used variations of this puzzle several times over the last few years in my Algebra class. Kids can’t help but want to know the answer to a logic riddle like this, and this year it occurred to me that I might be able to leverage this “want to know the puzzle answer” to motivate some more focus on understanding quadratics or higher degree polynomials.

The idea is that you choose a “secret” number, and give clues one at a time until students can narrow down the possibilities to a single answer.  I decided to try the same idea with more complicated expressions, so I created a couple of quadratics puzzles, and a polynomial version. I had these posted on the Sweet Math wall, and would add clues roughly one each day. I ran a simple number version, alongside the more advanced ones to allow entry for middle schoolers, and extension for high schoolers. I definitely noticed kids lingering at the clues as I added them. Some kids even asked me what time of day I would be adding another one. In an unexpected turn, it was a history teacher who submitted the correct guess for the first number puzzle. In your face Algebra students!

Although I haven’t done this yet, I like the idea of creating examples for sequences, and I think I’ll try this next year. Is the glory of being the first to guess correctly important enough to take the chance of guessing when there might be two possibilities for the common ratio of a geometric sequence? Do you team up with another student when you’ve narrowed it down to two possibilities so one of you is guaranteed to be victorious?

Here are a few of the puzzles I made in case you’d like to try them out. Please do let me know if you find them useful or if you think that I should sequence the clues differently or if you have other ideas for how to make them better.


Number Guess 1

Number Guess 2

Polynomial Guess


Put Your Marks Where Your Mouth Is

I was struck recently when reading Robert Kaplinsky’s post, Why Are You Using That Problem?.  In this piece, he articulates 3 different reasons that we might choose to use a particular problem: to introduce a concept, for productive struggle, or for problem completion – each of which has its own value.  Kaplinsky convincingly argues that we should be purposeful not just in which problems we choose, but in why we choose a particular problem.  I have been considering his thoughts in reference to productive struggle, and specifically in how to best to support students when we choose a problem with this as our goal.

I have always talked about encouraging and nurturing a classroom culture in which it is safe to take risks, and to experiment.  When I introduce a novel problem, I tell students that my expectation is that they try – that making an attempt is what I value.  My marking scheme always always includes credit for getting started, and for each step of a problem.  But in some ways, my marking scheme has been generally geared toward a sequence of steps leading toward a correct answer.  In awarding credit for specific answers, I have been communicating to students that I value those answers.  If I want students to put value on productive struggle, I need to demonstrate that that is what I value – and by extension, that is how they can earn marks.  (For now, I’ll leave the question of whether earning marks in general is productive at all…)

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I worked with Dr. Andreas(@DrAecon), my excellent colleague in the chemistry department, to create a set of guidelines and a rubric to support students in becoming independent and confident problem solvers.  In our guidelines, we tried to give students some explicit ideas to try when they are presented with a novel situation, and in our rubric, we tried to express criteria that values the process more than the answer.  We want students to make valid arguments, to justify their reasoning, to persevere in problem-solving,  and to feel that they can take risks, including taking a path that might not lead to a correct solution.  I hope that our guidelines and materials communicate these values.  I want to be explicit and transparent with students about this.  We’ll need to set aside time where we are not focused on learning specific content, but instead are focused on becoming strong and reflective problem solvers who know that they can earn respect and grades with this focus.  I want to put my marks where my mouth is.

Here are the rubric and problem-solving guidelines, based on Polya’s four steps.  I’ve included both Word and PDF versions, and a blank version, which is intended for students to use as an organizer for their work.  These are written with 10th grade students in mind, but I plan to create simplified versions of these for middle and elementary school students.  Please let me know if you can use these, and if you have any suggestions for improvement.


PDF: Problem_Solving_Rubric_Grade_10

Problem Solving Protocol_Grade 10

PDF: Problem Solving Protocol_Grade 10

Problem Solving Protocol_Grade 10_Blank

PDF: Problem Solving Protocol_Grade 10_Blank