Developing CCSS Math Transitional Units of Study: Patterns

I had a good time this past year playing around with new resources and teaching methods as I worked with my Intensive Algebra students.  The Formative Assessment Lessons (FALs) were a great help to me, and I used a number of them both as-is, and as inspiration for developing my own materials.

This year, I will again be teaching two sections of Intensive Algebra 1, along with three sections of IB precalculus.  Because the IB curriculum and assessments are pretty clear and established, this will really give me the time and flexibility to focus on the needs of my lower-skilled freshmen, which I am happy about.

My district has been partnering with the Silicon Valley Math Initiative (SVMI), and they provide numerous resources, such as the MARS tasks and Problems of the Month.  They have also created a format, based on the work of Phil Daro, for generating units of study to be used as we transition from California State Standards to CCSS, before high-quality curricula are commercially available (if ever!) and districts adopt new materials.

The basic unit structure is as follows:

1. Introductory lesson for engagement, to spark curiosity and interest.
2. Several conceptual development lessons, after which you can expect student understanding to still be fragile.   These lessons are what we typically think of as inquiry-based or constructivist lessons.  (I think this is a good eye-opener for me, because it has always felt frustrating how my students would still not "get it" after we engaged in, what I considered to be, really powerful learning opportunities.  Setting my expectation that students' understanding is going to still be fragile at this point will be key.)
3. One or two getting precise lessons, in which the teacher attends to precision, definitions, conventions, symbols, etc.  This is often going to be a more traditional "I-We-You" direct instruction approach.
4. One or two getting general lessons.  The goal of these lessons is not 100% clear to me, but some ideas for generalization are to use concepts across different contexts, generalize with variables and parameters, use different types of numbers, operations, functions, or structures in the same context, and so on.
5. A formative assessment lesson (which often takes multiple days).  These are intended to be done about 2/3 of the way through the unit.  They all start with a pre-assessment, and a well-defined set of tasks to help students further develop their understanding of the concepts.
6. Additional concept development lessons, as needed.
7. One or two robustness and differentiation lessons.  This is an opportunity to do re-engagement lessons with students who are struggling, as well as enrichment for students who are showing solid understanding.  The goal is to move all students from a fragile to a more robust understanding via a variety of rich problem solving opportunities.
8. An expert task assessment, in which students engage with tasks that have them operate at levels 3 and 4 on Webb's DOK.
9. A closure lesson to revisit and organize the unit goals and outcomes.
10. A summative assessment to see if students really have mastered the unit goals.

Clearly, this is intended to create true depth of understanding of a concept, and it sacrifices breadth to a certain extent.  But I am totally fine with this.  After years of focusing on breadth over depth in CST land, and seeing students never really reach mastery of concepts, and need to be retaught concepts again and again, I am happy that we are finally being encouraged to use a new approach.

I did a lot of work with quadratic patterns in the last month of school with my Intensive Algebra 1 students, using this FAL as inspiration:  Manipulating Polynomials.  It started off rough, as I underestimated the complexity of the task.  But I kept at it, I re-scaffolded and pushed my students, and by the end of the year, most of the students could see how to generalize a quadratic pattern.  These students were still struggling with basic integer and multiplication facts, but they could look at a complex pattern and determine that the rule was f(n) = n^2 + 2n + 1, for example.  Moreover, they could clearly explain why it worked and how they got it.

I decided that, instead of ending the year with this, it would be a great way to start the year, and use it to set the context for much of the other learning to follow.  Last year, I also started with patterns (as you can see in a previous post), and at the time, I felt it was successful.  However, I realized later that I made a fundamental mistake in the way I did it.  I only had students work with linear patterns, and they were pretty quickly able to see that the increase after each step was the slope, while the 0-th step was the y-intercept.  This was great!  But, once they made this generalization, they stopped ever looking at the structure of the pattern.  They would go right to making an in-out table, and from there, generate the function rule.    When we got to the quadratic patterns at the end of the year, they had absolutely no access to them.  They kept trying to make in-out tables, would see that the growth was not linear, and then hit a wall.  I had to un-teach them the concept of making a table and, instead, try to understand the structure of the pattern.

So, my plan this year is to start with a general patterns and functions unit, where the key concept will be to focus on the structure of the patterns.  After the first unit, I will then move on to a linear function unit, where we will look at linear patterns as a subset of patterns in general.  Each time, I will ask students not to just give me the rule, but to also explain the connection between the rule and the visual pattern.  For the third unit, we will work on linear equations, which will be motivated by giving them a linear pattern, telling them how many items are in a particular step number, and then having them try to work backwards to figure out what that step number is.

I've already made most of the content for this unit, but please note that none of the following lessons have been taught yet.  They are just my initial stab at creating a unit following the SVMI transitional plan.  Please feel free to look at, play with, and comment on these lessons.  If you see any problems or areas for improvement, please comment.  The introductory lesson is not there, as it is still under development.  If you have ideas for how to excite students about patterns, please share!  Also, the expert task is still under development.  The idea is to first have students work on the Table Tiles FAL, and then create patterns of their own using paper cut-outs, and write functions for the patterns they developed.  I still need to create the project description and rubric.

Here is the link to the box folder with the materials:  Patterns Unit

I hope everyone is having a great, relaxing start to their summers!

Constructing and Deconstructing Equations

I am ready to move my algebra 1 students into solving equations, now that they have done such a good job with patterns.  I looked through the different FALs available on the MAP site and Building and Solving Equations 1 caught my eye, as I experimented with using this method to teach equations in the past.  I don't think I did a great job of it last time, and I think I can definitely improve and hopefully make it work for the students.

The most common problem I see with students who struggle with solving equations is the order to do the steps in.  Usually, they can figure out that, in a 2-step equation, you always get rid of the subtraction/addition part first, but multi-step equations are a real issue.  And, when they get the dreaded "reverse style" equation, like 8 - 2x = 2, then everything can fall apart.

The idea of the deconstruction method is to really have students focus on the skill they do know - order of operations - to figure out what steps to take to solve.  And, it goes beyond just saying "reverse PEMDAS", to really concretely having students walk forward and backwards through all the steps.

Yesterday, I had students practice simply constructing equations.  I gave them x = 6, as suggested in the FAL, and we built one-step equations off of that using different operations.  Then, we took those new equations (like x + 5 = 11) and added a second step with a different operation.  Eventually, we built through 4-step equations.  And, we practiced checking by substitution.  The FAL includes sample student work and we analyzed that.  One student's work is correct, but can be improved by clearly showing the steps.  The other student's work is incorrect, and checking by substitution at each step reveals where the error is.  (The error has to do with fraction addition, so I didn't really discuss it with most students, since that would have been a bit too overwhelming in this lesson.)

At the end of the lesson, I could see that students were starting to get it, but still not totally clear.  So today, as a class, we built two more equations.  Then, I began the process of deconstructing them.  Here is how the board looked after those two problems:

After this, I continued with the activity in the FAL.  Each student picks two solutions, and constructs a four-step equation around them.  Then, they exchange problems, and have to deconstruct each others' equations to see if they could find the original numbers.

Students were working really hard on it, but were definitely struggling with a variety of issues - order of operations, basic calculations, how to organize their work, and so forth.  But, on the positive side, most students knew which operations they should be undoing at each step.  So, progress!  I'm going to keep working with them on this for a few more lessons.  Then, I'm going to give them a set of two-step equations, and my hypothesis is that they will be able to blaze through them, like a batter taking the donuts off the bat.  Hopefully, pushing them really hard at the outset will pay off later.  I'd appreciate hearing other teachers' thoughts on this method for teaching equations.

Here are the handouts we're going to practice with tomorrow.   (doc)   (pdf)

Linear Patterns in Algebra 1

I've been working with my Freshmen on linear patterns for the last couple of weeks.  I've been really amazed by how well they've taken to them, and how quickly they were able to start figuring out the function rules.  I've been using lesson ideas from The Pattern and Function Connection by Fulton and Lombard.

First, I gave students matchstick patterns and boxes of toothpicks, and had them build the patterns.  This was important, because often students would draw the picture incorrectly (not seeing where toothpicks were repeated, for example) and would not quite see how the pattern was growing.

Then, I gave them other types of patterns like this one, and asked them to describe how they were growing, and to determine the number of items in the 30th step and in the n-th step.  I think that building a concrete connection between the variable "n" and the "step number" will be invaluable as we move into solving decontextualized equations.  I was really excited when the first student figured out that this pattern could be thought of as a center square surrounded by "groups of n squares" and explained it to the rest of the class.

After this, we moved into organizing the steps of the pattern into a t-table, and then plotting them on a set of axes.  We defined "n" as the step number, and "f(n)" as the function value, meaning the "number of items at step n".  I worked through one example with them, but then I found that most students took off on their own and blazed through the whole set of patterns I gave them.  During the following lesson, we repeated this work, but I introduced the concept of "rate of change" and "starting point" and we talked about how these values are seen in the different representations of the pattern.

I really liked the following diagrams from the book:

I decided to build a poster project around it.  So in the following lesson, student pairs picked a new pattern to work on, figured out all of the representations, and then made a poster to illustrate what they learned.  I asked them to indicate in one color how the rate of change was portrayed in each representation, and in another color the starting point.  Students worked for a lesson and a half, and did a great job.  Here are a few examples of their work:

Today, I gave them some function rules, and they had to then work backwards to generate a pattern that could fit that rule.  They struggled with this quite a bit at first, and it showed me that, although they are doing really well, they are far from having fully internalized the connections between the representations.

On Monday, I will give students a pattern, and tell them how many pieces are in the last step.  From there, they will have to figure out what the step number is.  I think this will be a good scaffold for understanding how to solve equations.  For now, I am going to treat it like a problem solving activity and am interested to see what they will come up with.

Below are some of the handouts I made to supplement the excellent materials in the Fulton book.
Pattern Project handouts (doc) / (pdf)
Working Backwards handouts (doc) / (pdf)

IB Math SL - Functions

I'm teaching IB Math SL for the first time this year.  For those unfamiliar with the IB program, it is a two-year program that covers pre-calculus and calculus topics.  I'm doing year one right now, which means my juniors will be with me again next year.  I feel bad for the students I had last year in Algebra 2 - they'll have me for 3 out of 4 years in high school!  I only teach one section of the class, but so far I am really enjoying it.  The students are very motivated and positive, and the textbook for the class is actually pretty decent, so it is a good resource.  I do still supplement it with other resources, however.  This past week we've been reviewing functions (operations, inverses, etc.) and I modified some of my old work to use with them.

One activity that I use every year is the Functions Relay Race.  It really pushes students to use all of the different representations of functions at the same time, and it helps them see "f(x)" as a single value that can be operated on.  I also target typical problem areas - absolute values, negative exponents, etc.  I give each team the main handout that has all of the functions on it, and then they get the problems one round at a time.  I usually give a small prize when they finish a round.

• Functions Relay Race (doc) / (pdf)
• More Functions Practice (doc) / (pdf)

And then here is a handout that helps students review/explore rational functions.  I especially like the point-by-point division that shows how a hyperbola is generated when you divide two lines.  Although, students are definitely confused by the directions and I always have to show them what I mean.  I've rewritten the directions several times, and I can't figure out a way to phrase them that works.  I think the problem is probably not in the directions themselves, but that students don't really have a solid understanding of the connection between the graphical representation and the symbols used.  In any case, feel free to check it out and let me know what you think.

• Rational Functions (doc) / (pdf)

The Exponential Curve, Phase II

I haven't posted anything for a couple of years now, and I think I'm finally able to start up again.  I'm at a very different place in my life now, compared to when I first started this blog, and, thankfully, compared to when I stopped writing.

I am really excited by the move toward Common Core, and the abandonment of the CST tests.  I've felt my spirit and teaching practice withering and dying with each new modification I made to try and cram more standards in faster and more efficiently.  When students would ask why they need to learn math, I would give a standard reply about problem solving abilities and critical thinking in all areas of life, and this felt more and more cynical as I compared what I was saying to the actual content I was delivering.

This year, in my intensive algebra 1 class (for students who are well below grade level), I've quit worrying about skills lists.  I'm focusing on using high-quality problems and resources.  I'm trying to actually do what I say, and engage them with activities that require thinking, explaining, justification, problem solving, and persistence.  And since the new Smarter Balanced tests, according to David Foster, will be only 31% material at levels 1 and 2 of Webb's DOK, and 69% at levels 3 and 4, I feel like I can justify my new approach to any skeptic.

Just yesterday, I spent nearly 30 minutes playing the Game of 21 with them, a quick and easy misere game.  (The first person says 1, 2, or 3.  The next person increases by 1, 2, or 3.  Alternate turns.  You can't go past 21, and the first person to say 21 loses).  It was great to watch them struggle with it.  Of course, I made them start, and so I won game after game.  But then some students started realizing that I was going to win as soon as I said 16, and after a while longer, they realized I would win when I said 12.  Some students were frustrated, some continued to challenge me blindly, and some were clearly paying close attention and trying to develop a strategy.  I love the moment when the first kid tells me that, no, *I* need to go first.  And then when they finally beat me, it's a great moment!  At this point, I stopped the game without discussing the strategy, and told them to play it against a friend or family member for homework.  We'll pick it up again on Monday and see if more can beat me.

We've been working on finding out the rule for patterns, using only visual examples, building them with toothpicks, etc.  So I decided to start them on the non-linear Growing Staircases POM from SVMI for the second half of the class (A pattern where you start with one square, then add two, then three, etc).  First, I spoke with them about what perseverance means, and why it's so important.  I told them that we were going to all get through level C of the POM, but we were not going to finish today, or even by the next class.  That real math problems take longer than an hour to solve (or the 15 seconds that they are accustomed to).  They mostly completed level B (figuring out how many squares it would take to build a 10-step staircase), and they used counting and other patterns to do so.  Level C (find a rule for the number of squares in an n-step staircase) is going to be a lot harder, and I'm looking forward to seeing what they come up with.

I decided to do a lot of patterning work with them before we even talk about solving equations, using their visual experience with developing a rule as leverage for understanding.  When they see something like 4n + 3 = 51, I want them to think something like "That's 4 groups of n-blocks, and then 3 blocks more, giving a total of 51 blocks.  So the 4 n's would have to make 48 blocks.  So each one would be 12!".   Right now, their pattern work ends with coming up with a rule.  Soon, I'll start asking them to figure out what step number it would have to be, given a specific total.  I'll treat this as a problem-solving exercise now, and eventually formalize the steps into the standard solving equations algorithm.

I've also been drawing heavily upon the excellent work found in the Formative Assessment Lessons at the Mathematics Assessment Project .  These are really meaty lessons that incorporate excellent group tasks, and really help push students to use the CCSS Math Practice standards.  So far, in this algebra class, I have used Positive and Negative Numbers in Context and Interpreting Algebraic Expressions.  The one caveat I have is that the amount of time it takes to run one of these lessons is well underestimated in their notes.  The expressions FAL took my intensive class four lessons to complete, not the optimistic 1 lesson with a 20 minute follow-up!  But of course this will vary from class to class.

Aside from this, I am also teaching Algebra 2, and I am using CPM instead of whatever NCLB book we have sitting in our library.  This will be the subject of other posts, as my students and I wrestle with a completely different way of teaching and learning math.

And, as if this weren't enough, I am teaching the first year of IB Math SL (which is roughly equivalent to the first year in a pre-calculus/calculus sequence).  This class is filled with really positive, engaged, motivated students, and it's a joy to teach, so I shouldn't complain too much about the extra prep!

So hopefully this post will be the first of many more to come.   Being part of the mathtwitterblogosphere is invaluable, and I want to start adding my voice to the mix again.

Where have I been?

I've been putting off writing this for a long while, but it's finally time.  Since January of 2010, I have gone through some life-changing experiences.  I posted already about my sister's death last February; that, plus some difficult health issues (mine and other people's) have made for an extremely trying time.  While many things about me have remained the same, overall, I feel like a different person now.

For the past eleven years, I have worked full-throttle at DCP, at a minimum of 50 hours per week, but often more like 60 or 70.  I poured all of my energy into improving my curriculum and instruction, providing extra support to my students and their families, and basically eating, sleeping, and breathing school.  Doing this was entirely my decision, though the realities we face at the school (students' low skill levels, ever increasing demands for success on testing, fewer and fewer spaces in the CSU system, and continually decreasing state funding that whittles away our program) create an environment in which the passionate teacher feels that working like this is necessary.  I've lasted longer than any other teacher there (I was awarded the Lobo of the Decade award!) but I realized earlier this year that I needed to make some changes and run my life differently.  I love teaching, and I did not want to leave the classroom, but I needed a new environment where I could finally have some balance.

This past year has not been all bad.  In fact, my girlfriend and I decided to get married last fall, and we ended up having the ceremony on July 3rd in Sunnyvale.  It was a beautiful wedding - we had a traditional Jewish wedding ceremony as the foundation, but we personalized it to make it fully egalitarian (as opposed to the traditional man's acquisition of the woman), and lots of our friends helped out to make the day amazing.  The wedding canopy was designed by two of our friends (one is the art teacher at DCP) - they presented us with an amazing quilted canopy covered in pictures of our family members.

The cake was created in our own kitchen by my wife's friend.  The kitchen was a disaster at the end, but the cake came out really well.  It fit right in with our rainbow theme.  And the cake topper was created by another friend of ours - based off of some pictures of Jen and me!

Besides the Chuppah and the cake, we had lots of other friends help with running all the details of the day. We had a fantastic time, and the whole weekend flew by incredibly quickly.  We just got back from our honeymoon in Mendocino - neither of us had been there before, and we really loved it.  Very peaceful and beautiful, just what we needed.

My wife Jen is a teacher also, and she started work at a school in San Bruno (near the San Francisco airport) last year.  She loves it there, and they had a position open for this year, which I was hired for.  I'll be teaching IB Math Studies, a CAHSEE prep class, and Intensive Algebra 1.  I think this is a school environment in which I can do a good job for my students, but still be able to pursue outside interests, exercise, and take care of my health.  I've met most of the math department there, and they are all nice people, so I am looking forward to starting this new stage in my career.

I don't yet know what I'll do with this blog.  I need to find out if there are any school or district policies against blogging.  I don't intend to blog under a pseudonym.  Which reminds me - instead of my wife taking my last name, we combined our two names (Greene and Wekselbaum), and we've both legally changed our names.  So you can now call me Dan Wekselgreene.

In any case, I will leave this blog up indefinitely, since I know people are still finding it and downloading resources.  I enjoyed being part of the virtual math blogging community, and I've pretty much lost contact with everyone.  Hopefully, I will be able to post more in the future, and also find my way back into the fold (though not too far in - balance is the new name of the game!).  Good luck to everyone with the new school year, and may you also seek and find balance, so that you can stay strong both in and out of your classroom.

Some fun(ish) worksheets

I'm going to try to get my box.net materials updated over this coming week.  In the meantime, here are a couple of decent worksheets that you may find helpful.

First, I made one to practice graphing standard form - I just ripped off Mr. K's idea.  Thanks!  And some of my students actually liked the joke (I googled Laffy Taffy jokes).

For tomorrow, students will be graphing systems of inequalities, so I decided to create a little Ohio Jones adventure (Indiana's lesser known brother).  Here is the full lesson and just the activity in pdf form. 

(UPDATE: Here is the follow-up lesson in word form - Ohio Jones and the Pyramid of Power.  Here is the follow-up lesson in pdf form if you're having trouble seeing the word doc).

Here is what the maze should look like after being solved:

Difficult News

I haven't posted for a while now, and I wanted to let people know why.  My younger sister passed away suddenly about five weeks ago.  She would have been 30 next Friday.  I'm back at work now, but it is difficult just to get through the days.  I love my job, but high stress work is not the best thing when going through something like this, and it's been hard just to get the minimum done.  This has also caused me to rethink my priorities and how I lead my life, where I spend my time and energy.  I do plan on continuing this blog, and I will start posting ideas and lesson materials again, but I don't know with what regularity right now.  Thank you all for your support and understanding.

Algebra 1: Systems of Equations

We are finally getting to move beyond basic graphing and finding equations of lines.  It was a long slog, but the skills tests show that the majority of my students are starting to get the hang of it.  I always look forward to the systems of equations unit, because it is a chance for students to synthesize what they have been learning all year - and, in a situated context, no less.  My plan this year is to deepen the emphasis on representational fluency and summarizing, to help build all of those neural bridges we want the students to have.  We started the unit Monday, and I was really blown away by my classes today - all of a sudden, I have students doing algebra!  I had them solving systems in pairs, using mini-whiteboards, where one does the graphical solution and the other does the algebraic solution, and then they compare their answers.  They did a great job, and it wasn't until this activity that many students realized the answers should be the same.  I got a couple of those hilarious, indignant "you should have told us!" comments.  Next week is winter break, which doesn't come a moment too soon; however, I'm worried about how much will be lost over the seven days that nobody is asking them about starting points or rates of change.  No matter, it's worth it to have a rest.  Here are a couple of  examples of what we're doing, and the links to the lesson materials thus far.

Lesson 1 (Intro to Systems of Equations)  doc / GeoGebra files / Keynote / Powerpoint
Lesson 2 (Solving y = mx + b Systems)  doc / Keynote / Powerpoint
Lesson 3 (Practice Solving Systems)  doc

Language and Retention of Math Concepts

I've been thinking lately that one of the reasons my students have such difficulty with long-term retention of mathematical concepts is due to the small number of times I ask them to thoroughly summarize what they have learned.  They do lots of problems, but the language of the problems often does not enter into their brains.  As we learned in Orwell's 1984, without language, there is no thought.  So I am going to start providing more explicit opportunities for the students to summarize and discuss what we are doing in class.

Comic Strips  (Unit 5, Lesson 9:  doc / keynote / powerpoint)
Quite a few students are still struggling with graphing lines.  They know the general process, but don't pay attention to the details - is the slope positive or negative; if a term is missing, is it the slope or the y-intercept, and how does that change the graph?  So, I had all students draw comic strips to summarize the process in these different cases.  I like how this went, but I definitely did not provide them with enough time to do all I asked.  Here are a few good examples.  The first didn't scan that well, but he did an awesome job.

Think-Pair-Share  (Unit 5, Lesson 11: doc / keynote / powerpoint)
This is a tool that our humanities classes tend to use a lot.  I got some advice from them, and will be trying these periodically during the next couple of units.  We did one so far, and it went reasonably well for a first try.  Students need a lot of practice both writing down their ideas and sharing them out.  Here is the handout I gave (it was used immediately after doing a Do Now problem of the type described).