Category Archives: Algebra

Golfing with Linear Equations

For the first time in many years, I am teaching a College Prep Algebra 1 class with a fantastic group of 9th graders. Nearing the end of our linear functions unit, my colleagues and I discussed a desire to have some sort of culminating activity. And while I have used drawing projects often in some courses, in algebra 1 such tasks have often left me feeling unfulfilled. Too many horizontal and vertical lines for my liking I suppose.

I recalled reading about a potential golf-related task on twitter. To be honest, I don’t recall whose exact post provided the inspiration here (note – I am thinking it was Robert Kaplinsky or John Stevens, but I may be wrong. If anyone locates a source, I’ll edit this and provide ample credit), but it felt like a game-related task could provide by the strategy and fun elements which tend to be missed by drawing tasks.

HOW THE GOLF CHALLENGE WORKS:

The goal – write equations of lines which connect the “tee” to the “hole”. Use domain and/or range restrictions to connect your “shots”. Try to reach each hole in a minimal number of shots. Leaving the course (the green area) or hitting “water” are forbidden. All vertical or horizontal likes incur a one-stroke penalty.

On the day before the task, the class worked through a practice hole. Besides understanding the math task, there are also a few Desmos items for students to understand:

  • Syntax for domain / range restrictions
  • Placing items into folders
  • Turning folders on /off

practice hole

golfFor the actual task, a shared a link to a Desmos file with 5 golf holes. I tried to build tasks which increased in their difficulty. In practice, the task took an entire class period (75 minutes), and students worked in pairs to discuss, plan, and complete the holes. All students then uploaded their graphs to Canvas for my review, and filled out a “scorecard” which included “par” for each hole.  It became quite competitive and fun!

hole1hol4

hole5

In the end, there is not too much I would change here. Perhaps add some more complex holes. I’d also like to provide opportunity for students to design and share their own golf holes, and study the “engine” which built mine.  I hope your class has fun with it! Please share your suggestions, questions and adaptations.

Last Week I Refused to Teach Factoring

The students in my Freshman Honors class have certain expectations for how a math class works – a teacher lectures, there’s lots of drill practice, and then a test. Breaking this mold, and causing them to think of themselves as reflective learners, is one of my many missions. So this past week, when confronted with factoring, I simply refused to lecture.

My 9th graders have seen factoring before, but it was back in 7th grade, and it was only a surface treatment. So after a brief opener where we discussed what a “factor” means (both numerically and algebraically), I dropped the bomb –

  • I’ve posted your learning targets online
  • I’ve posted videos, resources and practice problems if you need them
  • I’ve set up online practice if you need it
  • You have a timed quiz on Friday (we started on Tuesday)

And….scene!

Panic….apprehension….incredulous looks….

So, you’re not going to teach us?

Nope.  Now get to work.

Here are some details of what I posted:

LEARNING TARGETS

  • F1: I can identify and factor expressions which involve greatest common factors.
  • F2: I can efficiently factor trinomials of the form ax2+bx+c, where a = 1.
  • F3: I can factor trinomials of the form ax2+bx+c, where a does not equal 1 (or zero).
  • F4: I can identify and factor perfect square trinomials.
  • F5: I can identify and factor “difference of squares” expressions.
  • F6: I can factor expressions which may represent a combination of F1 to F5.
  • F7: I can factor expressions “by parts” (or “by grouping”) when necessary.
  • F8: I can factor expressions which are the sum (or difference) of two cubes.

RESOURCES

Each learning target featured a video – some from Khan Academy, and some from other sources I searched for – but I attempted to provide a variety of methods. Some featured grouping as a primary means, others demonstrated the box method or the diamond.  This was the most important aspect of this learning experience: I wanted students to experience a variety of approaches, evaluate them, and make a personal decision about what worked best for them.  The students did not disappoint.

I also posted other online resources, such as worked examples and flowcharts.  One of my favorite resources – Finding Factors from nrich, was also included. Finally, I created an assignment on DeltaMath for each learning target, and a final jumbled assignment. The end of each day featured an exit ticket quiz and recap, to assess progress and provide “next steps” during the week.

SO WHAT HAPPENED?

Some students latched onto factoring by grouping for every quadratic, and explained their reasoning to their peers.  Many of these same students later in the week found more confidence in their number sense and chose to group only for “tricky” problems. One student was particularly insistent that the box method was the best was to go for all problems. Others found the diamond method helpful – which led to deep conversations about number sense and how to make searches more efficient. And in one fascinating conversation, a student discovered a “trick” he had found online. The group debated the merits of the method, tried some practice…but as nobody in the group could figure out why the method worked, they quickly dismissed it.  Good boys!!!

In the end, the quiz scores were great.  But beyond the scores, I feel confident that the students have made choices about their learning, assessed and revised their thinking, and can move forward using their new tools.

WHAT DID THE STUDENTS THINK?

Today I asked students to reflect upon their learning experience, and provide me feedback.

What was your overall feeling about last week’s learning method?  (1 = “Please never do that again”, 5 = “I loved it – do it more”.)

chart

Describe something you LIKED about last week’s classes, and why you liked it.

  • I liked being able to choose what i wanted to do. I could focus on my weaknesses and do less problems on what i was good at. I also appreciated the practice problems.
  • I liked that if you knew a topic you could move on and didn’t have to wait for someone else or the next day of class.
  • I liked that I could learn and do problems at my own speed.

Describe something you DIDN’T LIKE about last week’s classes, and why you didn’t like it.

  • I did not like that you did not explain how to factor
  • I didn’t have as much instruction from the master of factoring. {note – I suppose this is me?}
  • the teacher wasn’t involved

This last comment intrigues me…and I’m not sure if I should be bothered by it…I don’t think I should be.  In many respects, I feel I worked harder during the classes, as students were all over the place.  But I also realize students don’t see all of this going on around them.  I’ve become intrigued by how I can be less of a teacher and more of a facilitator in my classes, and this was a solid step forward I feel.

Now, off to plan to not lecture tomorrow….

Activity Builder Reflections

We’re now about 9 months into the Desmos Activity Builder Era (9 AAB – after activity-builder). It’s an exciting time to be a math teacher, and I have learned a great deal from peeling apart activities and conversing with my #MTBoS friends (run to teacher.desmos.com to start peeling on your own – we’ll wait…). In the last few weeks, I have used Activities multiple times with my 9th graders.  To assess the “success” of these activities, I want to go back to 2 questions I posed in my previous post on classroom design considerations, specifically:

  • What path do I want them (students) to take to get there?
  • How does this improve upon my usual delivery?

 

AN INTRODUCTION TO ARITHMETIC SERIES (click here to check out the activity)

My unit or arithmetic sequences and series often became buried near the end of the year, at the mercy of “do we have time for this” and featuring weird notation and formulas which confused the kids. I never felt quite satisfied by what I was doing here.  I ripped apart my approach this year, hoping to leverage what students knew about linear functions to develop an experience which made sense. After a draft activity which still left me cold, awesome advice by Bowen Kerins and Nathan Kraft inspired some positive edits.

seatsIn the activity, students first consider seats in a theater, which leads to a review of linear function ideas. Vocabulary for arithmetic sequences is introduced, followed by a formal function for finding terms in a sequence. It’s this last piece, moving to a general rule, which worried me the most.  Was this too fast?  Was I beating kids over the head with a formula they weren’t ready for? Would the notation scare them off?

plotsThe path – having students move from a context, to prediction, to generalization, to application – was navigated cleanly by most of my students.  The important role of the common difference in building equations was evident in the conversations, and many were able to complete my final application challenge.  The next day, students were able to quickly generate functions which represent arithmetic sequences, and with less notational confusion than the past.  It certainly wasn’t all a smooth ride, but the improvement, and lack of tooth-pulling, made this a vast improvement over my previous delivery.

DID IT HIT THE HOOP? (check out the activity)

DAN.PNGDan Meyer’s “Did It Hit the Hoop” 3-act Activity probably sits on the Mount Rushmore of math goodness, and Dan’s recent share of an Activity Builder makes it all the more easy to engage your classes with this premise. In class, we are working through polynomial operations, with factoring looming large on the horizon.  My 9th graders have little experience with anything non-linear, so this seemed a perfect time to toss them into the deep end of the pool.  The students worked in partnerships, and kept track of their shot predictions with dry-erase markers on their desks. Conversations regarding parabola behavior were abundant, and I kept mental notes to work their ideas into our formal conversations the next day.  What I appreciate most about this activity is that students explore quadratic functions, but don’t need to know a lick about them to have fun with it – nor do we scare them off by demanding high-level language or intimidating equations right away.

The next day, we explored parabolas more before factoring, and developed links between standard form of a quadratic and its factored form. Specifically, what information does one form provide which the other doesn’t, and why do we care?  The path here feels less intimidating, and we always have the chance to circle back to Dan’s shots if we need to re-center discussion.  And while the jury is out on whether this improves my unit as a whole, not one person has complained about “why”…yet.

MORE ACTIVITY BUILDER GOODNESS

Last night, the Global Math Department hosted a well-attended webinar featuring Shelley Carranza, who is the newest Desmos Teaching Faculty member (congrats Shelley!).  It was an exciting night of sharing – if you missed it, you can replay the session on the Bigmarker GMD site.