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The Exponential Curve



The purpose of this blog is to help generate and share ideas for teaching high school math concepts to students whose skills are below grade level.



Updated: 2018-02-02T10:32:56.063-08:00

 



Developing CCSS Math Transitional Units of Study: Patterns

2014-06-11T13:53:45.145-07:00

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:Introductory lesson for engagement, to spark curiosity and interest.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.)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.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.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.Additional concept development lessons, as needed.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.An expert task assessment, in which students engage with tasks that have them operate at levels 3 and 4 on Webb's DOK.A closure lesson to revisit and organize the unit goals and outcomes.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 multi[...]



Constructing and Deconstructing Equations

2013-10-02T15:59:05.939-07:00

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

2013-09-27T23:21:11.853-07:00

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

2013-09-21T23:18:16.448-07:00

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.


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.





The Exponential Curve, Phase II

2013-09-14T20:29:38.299-07:00

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 h[...]



Where have I been?

2011-08-05T21:22:28.620-07:00

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 contac[...]



Some fun(ish) worksheets

2010-04-13T09:08:20.746-07:00

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

2010-03-20T10:51:55.672-07:00

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

2010-02-10T20:45:17.798-08:00

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

2010-02-08T20:07:53.612-08:00

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). [...]



Both Flattering and Disturbing

2010-02-02T19:26:40.086-08:00

Students in our Numeracy support class have been working on plotting points to help them with graphing in Algebra 1.  The students just finished a connect-the-dot cartoon graph assignment.  One of my students apparently decided to dedicate her drawing to me.  Like my Speedos?



Algebra 1: Representations of Linear Equations

2010-01-10T18:23:23.072-08:00

Increasing my students' representational fluency has been something I've been working on for a while.  Our second semester started last Monday, and to start my Algebra 1 students off easy, I had them do a four-fold poster of a linear relationship to review what we did last semester: situation, equation, table, and graph.  They did the work fine overall, but quite a few students had more troubling questions than I had expected (i.e. "how do you make a table?").   I guess it just shows that we have to keep going through these different representations and their connections again and again.

We have started the new unit - working with linear equations - in which students have to write the equation of a line given its slope and a point, or two points, or a point and a parallel line.  In the past, I have done this only algebraically (except for the initial explanation of concepts); this time around, the students will have to practice the problems both algebraically and graphically.  And, more importantly, the skills tests will require them to show mastery with both methods.  Let's build those connections!

Here are the first few lessons in the unit.

Lesson 1 (Representations of Linear Functions)
Lesson 2 (Graphing Practice)
Lesson 3 (Write the Equation of a Line)
Lesson 3: Keynote / Powerpoint

And some snippets from the worksheets to illustrate what I am talking about:





Introducing Linear Inequalities

2010-01-08T19:24:07.429-08:00

To show that a line is a representation of an infinite number of points, I like to give my algebra 1 classes an equation, like y = 2x + 3, and then give each student a couple of different ordered pairs - some that are solutions and some that aren't.  I have them each work out their points, and then go to the board to plot an open or closed circle, depending.  Once all the students sit back down, we look for patterns and see that all the closed circles fell on a straight line.  Discuss, and voila.

This extends nicely to linear inequalities (and systems of equations and inequalities).  On Tuesday, my algebra 2 students were reviewing linear inequalities so I did this activity with them.  I really like it, because it is engaging, and it helps build a mental picture that they can rely on later on when they are struggling through graphing problems on their own.  My students often get stuck on the "pick a test point" part of the process; but now, I ask them if they would have plotted a closed or open circle based on their result, and to think about what the picture on the board looked like.  This usually helps them see which side of the boundary line to shade, and to be able to explain why.

Here is what the board looks like after students plotted their points:




Then, we looked for patterns.  Usually, a student will come to the board and draw some sort of line after getting frustrated with trying to explain it in words.  Then I reveal the shading:


And there are usually some audible "ahhs" and such.  Another great benefit of this is that the string of open circles on the boundary helps students see what the dotted line is all about, and why changing the inequality to include an equals sign would create a solid line - a string of closed circles.

Here is my lesson that goes with this.  And the keynote.



Algebra 1: Graphing Lines Practice

2010-01-07T19:51:26.344-08:00

I just used this worksheet from Mr. K for the first time the other day.  I thought it had a pretty cool setup, but I didn't realize just how effective it would be until I used it in my first class.  The "solve the joke" aspect of it helps draw them in, but the hidden beauty is in its self-checking properties.  Since each line must pass through exactly one number and one letter, a line that doesn't do this must be graphed incorrectly.  Students started realizing this and would go back and find mistakes without having to check with an answer key.  The only bad part (sorry to say) is that they had absolutely no idea what the answer was supposed to mean (see earlier post).

I made up a "balloon pop" homework to go with this that was inspired by Green Globs.  I wish I had the tech access for my students play that game.



Algebra 1: Skills List - Spring Semester

2010-01-04T22:22:50.828-08:00

I spent a good deal of time right before break trying to figure out exactly how far I can push my students for the second semester of Algebra 1.   These skill items will be broken down into chunks for the skills tests, and MC-ized for the benchmarks and final exam.  I regret how many concepts I had to leave out due to time pressures; and still, the list seems daunting and endless.

If you're interested, this is what my students will be doing over the coming months.

doc / pdf



ELL Joke Worksheets?

2009-12-17T17:06:59.377-08:00

(image)
Kate wrote a great post about the value of a well-structured worksheet last month.

I agree that there are huge benefits of having a unified task, with some type of self-checking or affirmation. And a little fun and/or creativity doesn't hurt. Joke worksheets do that pretty well. However, my students (who are generally not native English speakers) hardly ever get the joke. They tackle the sheet with excitement, but there is usually that little moment of disappointment at the end when they don't get the punchline. Instead, of course, of the expected groan and eye-roll that accompanies a quality pun.

"What do you get when you mix prune juice with holy water?"
"A religious movement"

After two minutes of explanation, that loses some of its original zing.

So my question is if anyone has or knows about these kinds of worksheets developed for ELL students? I'm kind of doubting that there are any, but it never hurts to ask. I think I will probably end up creating some next semester, with jokes solicited from my students. Then I can publish the DCP Spanglish Algebra Joke Book.



Algebra 1: Situation Graphing

2009-12-15T20:53:21.150-08:00

I learned a few years back that jumping right into graphing slope-intercept equations never worked. This is one of those concepts that, before I became a math teacher, I never would have guessed would be so hard for students to master. Start at the y-intercept, use the rate of change to plot the next point, and you're done - right? Yeah, not really. So after a couple of years of teaching, reteaching, re-reteaching, and tearing my hair out, I decided to try some other things. Eventually, I realized that a ton of scaffolding of the concept of slope was needed, along with firmly rooting linear functions in situated contexts.One of the constant problem areas is deciding which way to draw the line for a negative slope. To graph something like y = -(2/3)x + 5, students would often move down 2 and left 3. My old attempts at correcting this focused only on the mathematical explanation: -(2/3) = -2/3 = 2/-3. So, you either go down 2 and right 3, or up 2 and left 3. If you go down 2 and left 3, that means -2/-3, which is 2/3. This is a perfectly reasonable way to explain it, but it didn't really provide much of a lifeline to my lower-skilled students, as it hinges on mastery of the division rules of signs, as well as remembering that a fraction also represents a division problem.The other common problem was for students to correctly identify the starting point number, but to plot it on the x-axis instead of the y-axis.The way I run the unit now is to provide numerous opportunities to graph and describe situations, both with and without numbers, in just the first quadrant of the coordinate plane. Distance, income, height, and so on. The quantity being measured is always on the vertical axis, and the horizontal axis always represents time. When we eventually generalize to y = mx + b equations on the full coordinate plane, I use the situated contexts as memory anchors. If a student doesn't remember where to start, I say something like, "Where do we show that the Hare got a two foot head start? On the feet axis or on the seconds axis?" In these situations, a positive rate of change always means "moving up" and a negative rate of change always means "moving down", while time is always passing to the right. This is a much more helpful way for my students to think about how to graph their decontextualized lines. Suddenly, there is a reason for the direction the line is moving in, instead of just a sign rule.Another benefit to this approach is that my students are now a lot more flexible with the form of the equations. My situated equations typically are in the form y = b + mx, which seems like a more natural connection to the preferred method for graphing. Once they grasp that the number without the variable is always the starting point, then they can handle both y = b + mx and y = mx + b relatively interchangeably. Also, it really helps them to understand the difference between equations like y = 2 and y = 2x. The first shows a starting point of 2, with zero rate of change. What does it look like on a graph if someone is not moving, but time is still passing? Exactly - a straight line! (I'm still working on that one - even my highest skilled students still say straight when they mean horizontal. My "all lines are straight" response doesn't usually clarify the way I'd like it to.) And in the second, the rate of change is 2. Ahh, it's like a graph of someone running 2 feet per second... but where did he start from? Zero? Where is that?This approach takes a ton more time, of course, but I can't see any way around it for my students. I hope that I am providing them with a long-la[...]



Review game: Trashketball

2009-12-14T18:52:25.567-08:00

I know that many teachers out there play some form of Trashketball, so this isn't really groundbreaking. However, I always have problems with these kinds of review games. Structuring them so that the higher-skilled students don't dominate or pressure the other students can be quite difficult. Or, looking at it the other way, there are plenty of lower-skilled students who are happy to sit back and let others on their team get the work done for them. I developed Tic Tac Toe Battle Royale a couple years ago which addresses some of these concerns pretty well. But you can only do the same game so many times. My experiments with Trashketball in the past haven't been that successful, and so I thought about how I could improve it to work more effectively in my class. This is what I came up with:Break students into groups of 3 or 4 - for me, this yields no more than 6 groups in my Algebra 1 classes. Give each group a letter, and each person in the group a number. Write these in a grid on the board. (If there is an unfilled spot in a group, that spot becomes a wild card - any person can take that number.)For each round, create 6 separate problems that all target the same concept, but that are slightly different. This prevents the copying problem found in board races.Hand out a template for doing the work on. My freshmen need an organizer for everything. "Get out a sheet of paper" just doesn't fly.Show the 6 versions of the problem, giving the class enough time to get it done.Call for silence. Block the projector. Randomly (or not) call a number between 1 and 4. The student in each group with that number comes to the board - all 6 at once. Have the board sectioned off so they know where to write. They are allowed to bring their own graphic organizer up with them, but no one on the team may offer help at this point. The idea here, of course, is that students must make sure that all group members have done the work. Students who tend to slack off have to at least write down the work that others in their group are doing. Not ideal, but it's better than spacing out.Have the trashketball basket set up. As students complete their work on the board, tell them if they are right or not (make sure to have answer keys ready!). Right answers get a point, and they get to take a shot for a bonus point. There is less waiting around time this way - some students will still be writing their problems as others are already lining up to shoot.Record the scores and move on. Winning team gets a whatever.How it looks:I did this for the first time today, and was amazed by how well they did. There were only 2 students in the class that I couldn't get totally engaged. The rest did all their work, were excited to take their shots, and so on. It takes longer to make this activity due to the multiple problems, but it was really worth it. Man, do they love tossing paper balls into the recycle bin. I know it kind of breaks my respect class norm, but it really warms my heart to hear a kid (who I can usually barely get to sit down, and who really wanted to win) say to his teammate who hadn't done his work on the board carefully: "Fool! I told you it was negative eleven!"Trashketball Problems (Keynote) (Powerpoint)Answer Template (Word)[...]



Math Department Photo: 2009-2010

2009-11-23T20:25:04.775-08:00

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Michelle Longosz, our former photo teacher, always takes amazing department photos for us each year. Here is the radical math department.



XKCD based lesson: The Coordinate Plane

2009-11-15T18:28:02.944-08:00

Ever since I first saw this xkcd cartoon, I wanted to use it in a lesson. I finally put that together this year. I used the cartoon as a way to help convey the idea that points on a coordinate plane are a way to easily visualize the relationship between two different variables. The purpose of the numbers is simply to quantify those relationships, if such a quantification is necessary. I then had students make their own graphs for homework, with variables of their choice. If I had more time to spare, it would have been nice to do this in class (and the outcome would have been better, I think).This lesson (Unit 4, Lesson 5) and others can be found in my box.com widget to the left. I recently updated Algebra 1, Units 3 and 4.Here are some examples of the students' work.[...]



Story of 1

2009-10-31T12:13:50.296-07:00

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I was clued in to the existence of The Story of 1 a couple of weeks ago from my twitter PLN. I had my sub show it to my algebra 1 classes when I was out of town, and it seemed to go well. Then, one of my colleagues was sick this week and did the same lesson. Her sub said that the students were really engaged with the movie. I couldn't find a question guide on-line for it (though I didn't search all that long), so I made one up.



Algebra 1: Solving Equations Puzzle

2009-12-18T19:53:17.812-08:00

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Here is a puzzle activity for reviewing equation solving. I found that it worked better when I made an answer mat for students to put their pieces onto (I indicated a couple of pieces on the mat to help them align the rest of their pieces).

Here are two files in Pages and Word that you can work from to make your own.

Edit:
A comment from David Wees in a previous post with a similar puzzle I did for quadratics:

Yeah your puzzle is cool. So cool that I've created a random generator in Adobe Flex.

See my algebra puzzle generator.

Awesome!

Edit 2:
There is an app called Formulator Tarsia that will do this, but it only works for Windows (which I don't have access to) so I haven't tried it out. Give it a try!



Putting students in control of their learning

2009-10-31T11:47:25.185-07:00

In the last couple of years, I've worked to really clarify exactly what skills I expect my students to learn. The assessment system makes it crystal clear what skills students know and don't know. And then I realized: Oh wait - it's only crystal clear to me. Students focus on their test scores, and come in to retake and improve tests, but they really don't think about what mathematical content they need to develop - only what test number they need to retake. I still have a few students who insist on retaking skills tests even though they haven't done any work to learn the skills that they got wrong the first time. Even when this fails to produce the results they want, they still resist actually working with me to learn the skill.I think that helping students really understand what the individual skills consist of, and what their personal ability level is on each skill, is really the next step. I want students to understand the connection between their level of numeracy and their success in mastering algebraic concepts. I also want students to make connections between their behaviors in class and their growth (or lack of growth) in the lesson's objectives. Finally, I want to provide students with greater differentiation so that all students can both feel challenged and successful.So, I put all of that together into a new plan for beginning and ending class. Students will start class with a 10 minute Do Now that has three parts. Part 1 is a Numeracy Skill Builder that targets a specific elementary math concept that is either key to the specific lesson, or something that students have been struggling with. Part 2 consists of one or two algebra concepts that are the lesson objectives. These are broken into basic, proficient, and advanced levels. The proficient level is the form in which the concept will be tested on a skills test. Students are told to solve only one problem in each concept, at the level they feel most comfortable at. Part 3 is a multiple choice test prep question. The purpose of this is obvious, as we need to get students ready for state tests, ACTs, placement tests, and so on.Students have 10 minutes to complete these problems individually and silently. No helping is permitted here (in general), because the purpose is for students to really get a sense of what they know at the beginning of class on their own. At the end of the 10 minutes, I show the answers so students can see how they did, but we don't spend time actually reviewing these specific problems. I quickly collect the papers.We have the lesson. Ok.Now, in the last 5 - 7 minutes, I hand back the papers. On the back, students complete the Exit Slip / Reflection. They are supposed to go back to the Do Now problems, pick one algebra concept, and try a higher level problem. The idea is for them to see how much they can improve in an objective over the course of the class period. So, even if they are only able to accomplish the basic level (when they couldn't before), they can see growth in themselves and feel good about that. Students who already could do the advanced concepts at the beginning of the class have a shot at doing a harder challenge problem, so that they too can push their thinking (my advanced students really like this).I just started doing this today, so I don't have too much to report about it yet. It seems to have gone well, though it took longer than the 10 minutes because I needed to explain the process a few times u[...]



Distributive Property and Multiplying Binomials

2009-09-21T23:30:21.517-07:00

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I made a review lesson for my Algebra 2 students on these topics, to make sure they are really ready before we start performing operations on complex numbers.

Some instruction, some board races, and there you go. Hope you like it.

Lesson
Keynote
Quicktime


I also updated my Algebra 1 box with unit 2 files and the first four lessons of unit 3.



Algebra 1: Solving Equations

2009-10-27T21:36:16.981-07:00

I am beginning the planning stages of our unit on solving equations in Algebra 1. In my past experiences, some students pick this up very quickly, no matter how you teach it, while other students struggle mightily. I want to try some alternate approaches this year, to really reach those students who have not been able to learn this skill in the past. I remembered an order of operations approach that I read about in the NCTM magazine a few years back. I can't recall the name of the article, but a little google searching found me this document that is even better than what I remembered.Our students in Numeracy already work with bar modeling to solve word problems, so this seems like a natural extension to solving equations. I like this approach because it helps focus on the idea that the variable is a given quantity that must be determined, instead of focusing on the steps that isolate the variable. It also might help with those difficult to master "converting verbal sentences to algebraic equations" problems. Here are a few examples of how this might look. I know the diagrams are a bit confusing at first, but I think they would make more sense to students as they watch them get created and do them by themselves.I also like the other representation discussed in the article. This is the original order of operations process that I had been searching for. I like this because it gives a very clear framework for solving equations - reversing the order of operations.When you look at each stage, you can draw equal signs between the boxes. These would be equivalent to the intermediate statements in the traditional "do the same thing to both sides" approach.So for the unit, I am thinking that we would spend two or three lessons on bar models to build the concept of what we are actually trying to do (find the value of the unknown amount). Then, spend a couple lessons on the order of operations representation to build an understanding of the process for isolating the variable. Finally, transition to the traditional approach, which is clearly the fastest and cleanest way to solve an equation of the three. This would take more time, of course, but the hope is that it would build a more enduring understanding.Has anyone tried these methods with their students?Edit:I wonder now if it would make more sense to start in with modeling sentence/word problems with the bar model method, and not start by saying that we are "solving equations". That way, more students would be engaged with the material, and we could eventually use the bar models to develop the equations. This way, the unit doesn't start with the problem "solve (3/5)x = 45", which will stop most kids dead in their tracks, but maybe with something like "It took Sandra 45 minutes to finish 3/5 of her homework. How long will it take her to finish it all?", which kids might have more of an entry to. After we solve it, we can then discuss how to represent it as an equation.Edit 2:I also need to think about how to incorporate the balance idea and preserving equality... Kids don't always know what the equal sign really means. Maybe in the transition time from the box method to the traditional method?Edit 3:(Written on 10/27 - at the end of the unit)On reflection, the problem was not having enough time to really devote to the two alternative methods. Both did show a lot of promise, but we weren't able to really practice eith[...]