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.

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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.

 

“The 35 Game” for Compound Inequalities

This week in Algebra 1, my students completed the first part of their inequalities unit with much success, but now storm clouds appear on the horizon – compound inequalities, where english class meets math class with talk of conjunctions – those pesky and’s and or’s. A dice game helped my students make sense of these compound ideas.

The 35 game: 3 students, 3 dice, 3 rounds.

mickeyIn each round, a player rolls the 3 dice and records their sum. The goal: by the end of 3 rounds, to get as close to a total of 35, without going over. After round 2, each player has the choice to stop if they like, but highest score, closest to 35, wins the game. To help students understand the game, I gave the class time to play in their groups, record results, and think about strategy. The next day in class, we selected 3 students to play in front of the class. Players took turns rolling, and results were recorded on the board after each roll.  After round 1, here is how a game between Mickey, Sam and Kim was shaping up:

score1

Kim has taken a small lead. Round 2 rolls then go in order. We record them, then look at the round 2 sums.

score2

Still pretty close, Sam now leading. It’s Mickey’s turn to roll. Mickey probably needs to roll in round 3, but what is he hoping for?  Some rolls will cause him to go over. Will any rolls cause him not to take the lead?  All students in class were equipped with number lines going from 3-18 which I made using Number Line Generator.  Class discussion quickly yielded consensus that 3 was the lowest roll for Mickey, and 14 was the highest. How do we write these as inequalities, and how do these inequalities “play” together. The key word here is “and”, and all students recorded the possibilities:

numline1.jpg

After we agreed on the interval of possible “safe” values, Mickey made his round 3 roll – and was safe!

score3

A total of 31 – not bad, but 2 other players yet to go. Moving on to Sam, students discussed her possible “safe” rolls, and I was surprised how quickly we were able to generate the inequality. Note, for ease of discussion, we made ties “safe”, as a tie would keep a player in the game (we’d do a new game after to break any ties).

numline2.jpg

How did Sam do in round 3?

score4.jpg

Too much! And Mickey is still in the lead.

Moving to Kim’s turn, I changed the focus from the player to her opponent.  Rather than find rolls which are advantageous to Kim, I asked students to think about Mickey: what is HE hoping for her to roll?  Which rolls would cause him to win the game?  This small twist took a bit more time in groups, and provided rich discussion of the difference between the conjunctions AND and OR.  In this case, Mickey would be happy if Kim rolled less than 11 OR if she rolled more than 15. Shading these on the number line revealed a solutions set which looked different from the previous 2:

numline3.jpg

In the end, this simple game allowed for group discussion and a natural discussion of the conjunctions. In class the following day, we started once more with the game, and I stopped the game now and then to have students sketch solution sets of the game from differing perspectives.

One last note: there is a clear discussion of discrete vs continuous variables to be had here, and I brought it in when it seemed like the class could handle it. In our game, it’s not possible to roll a sum of 9.5, yet we shaded values between integers on the number line. A chance to bring in domain discussion here, where the domain of the game is limited to integers between 3 and 18 versus the real number line we often use, is welcome here – grab the opportunity to highlight the precise mathematical language.

Activity Builder – Classroom Design Considerations

This past summer, our forward-thinking math-teacher-centric friends at Desmos released Activity Builder into the wild, and the collective creativity of the math world has been evident as teachers work to find exciting classroom uses for the new interface. Many of these activities are now searchable at teacher.desmos.com – you’re welcome to leave now and check them out – but come back…please?

Its easy to get sucked in to a new, shiny tech tool and want to jump in headfirst with a class. I’ve now created a few lessons and tried them with classes which range from the “top” in achievement, to my freshmen Algebra 1 students. In both cases, I’ve settled upon a set of guiding principles which drive how I build a lesson.

  • What do I want students to know?
  • What path do I want them to take to get there?
  • How will my lesson encourage proper usage of math vocabulary?
  • What will I do with the data I collect?
  • How does this improve upon my usual delivery?

It’s the last question which I often come back to. If making a lesson using Activity Builder (or incorporating any technology, for that matter) doesn’t improve my existing lesson, then why am I doing it?


One recent lesson I built for my algebra 1 class asked students to make discoveries regarding slopes and equations of parallel and perpendicular lines. Before I used it with my class, a quick tweet 2 days before the lesson provided a valuable peer-review from my online PLC.  It’s easy to miss the small things, and some valuable advice regarding order of slides came through, along with some mis-types. The link is provided here in the tweet if you want to play along:

The class I tried this with is not always the most persistent when it comes to math tasks, but I was mostly pleased with their effort. Certainly, the active nature of the activity trumped my usual “here are bunch of lines to draw – I sure hope they find some parallel ones” lesson.

As the class finished, I called them into a small huddle to recap what we did. This is the second lesson using Activity Builder we have done together.  In the first, the students didn’t know I can see their responses, nor understand why it might be valuable.  In this second go-round, the conversation was much deeper, and with more participation than usual. In one slide, the overlay feature allowed us to view all of our equations for lines parallel to the red line:

parallels

We could clearly see not only our class successes, but examine deeper some misunderstandings.  What’s happening with some of those non-parallel lines?  Let’s take a closer look at Kim’s work:

Parallel 2

What’s going on here? A mis-type of the slope? The students were quite helpful towards each other, and if nothing else I’m thrilled the small group conversation yielded productive ideas in a non-threatening manner – it’s OK to make errors, we just strive to move on and be great next time.  The mantra “parallel lines have the same slope” quickly became embedded.

The second half of the lesson was a little bumpier, but that’s OK.  Before questions regarding slope presented themselves in the lesson, storm clouds were evident when the activity asked students to drag a slider to build a sequence of lines perpendicular to the blue line.  Observe the collective responses:

perp2

So, before we even talk about opposite reciprocal slopes, we seem to have a conceptual misunderstanding of perpendicular lines.  I’m glad this came up during the activity and not later after much disconnected practice had taken place.  In retrospect, I wish I had put this discussion away for the day and come up with a good activity for the next day to make sure were all on board with what perpendicular lines even look like, but I pressed ahead.  We did find one student who could successfully generate a pair of perpendicular lines, and I know Alexys enjoyed her moment in the sun.

perp1

What guiding principles guide you as you build activities using technology? How do they shape what you do?  I’m eager to hear your ideas!

 

 

It Took Me 2 Years to Get This Approach to Imaginary Numbers

This past week the NCTM annual conference was held in Boston, and what an enriching epxerience! What made it so special this time around was meeting and hearing from my PLC of Twitter friends, many of whom I had admired from afar for some time. I’ll discuss the power of the MTBoS (Math-Twitter-Blog O’Spehere) in a later post.  Today I want to focus on a powerful session I attended in Boston, and how a new persepective developed – even after a 2 year delay.

The story starts 2 years ago at Twitter Math Camp in Philadelphia.  At that conference, I participated in an Algebra 2 small group, facilitated by the super-creative Max Ray, from the Math Forum. Splitting into smaller groups, I worked with a team to think about rational expressions – a unit which is often dry as sand in Alg 2 courses, and where I thought we could make some head-way. While we worked on our slightly-less dry, yet safe lessons, Max and a small group were discussing complex numbers on the board. There were mysterious circles, transformations, and discussions I didn’t understand.  I suppose I was taught about complex numbers the “traditional” way – we need them to solve certain quadratics and memrize some wierd rules about their behavior. We perform strange operations on them, and we definitely don’t ask why. I suppose I could have simply wandered over to the group and found out more, but the mathematical intimidation factor was high – I’m sometimes too proud to admit what I don’t know.

Fast forward 2 years, and I see Max is presenting a session with Michael Pershan. This is a must-attend. Two engaging speakers whom I appreciate for their ability to use students’ natural curiosity to facilitate math conversations.

Here’s the set-up: Michael finds a handful of volunteers to stand at the front of the room, standing on a hypothetical number line (Max stands at zero). The participants are then asked to consider the following transformations to their value, and move accordingly, returning after each move to their original position.

  • Add 2 to your value – participants all move to the right 2 spaces.
  • Multiply your value by 3 – participants all move to the left or right accordingly, depending on whether their original value is positive or negative.
  • Multiply your value by -1. OK, now the plot thickens.  While we can find our new position, Michael does a materful job in having participatins reflect upon the transformation. The first two moves required left and right shifts; here we need to consider a rotation about the origin. This rotation provides a rule for multiplication by a negative.

Photo Apr 16, 12 36 08 PM

The table has been set, the silverware polished. and now we need some new volunteers. We have a new number line, and some new transformations to think about.  BUT this time around we want to complete our movement by using the same transformation twice.  Let’s roll!

  • First, add 4 by using the same transformation twice.  This is a nice appetizer – let’s move 2, then 2 more.
  • Next, multiply by 9. This is a little trickier, as some folks almost crashed into the next presentation room. But two multiply by 3 moves do the job.
  • Now, multiply by 5. Oooh….we have an entry point into radicals. Some quick discussion, and we two moves – multiplying by a little mroe than 2 each time.
  • Finally, multiply by -1…in two moves…..

WAIT!  This is the stuff Max was talking about 2 years ago that I didn’t get.  The bulbs have gone off.  I GET this now!  We do a 180 degree rotation do perform a multiplication by -1, so now we need two 90 degree rotations.  And now we have an entry point into imaginary numbers, without the scary-sounding term.

What I appreciate most here is that we don’t need to wait until deep into algebra 2 to think about the imaginary unit.  These concepts are accessible to younger students, and we have a responsibility to achieve some conceptual buy-in before just thrusting abstract ideas in front of our students. You can find Michael and Max’s shared files here on their Teaching Complex Numbers page.

I get it now…I think….and I’m not ashamed to say it took me 2 years.

UPDATE: You need to immediately run to check out the fun summary Ashli has provided of this session. Her notebook sketches are unreal (in the non-numbr sense)!

Inverse Function Partner Share

We’re working through functions in my college-prep pre-calculus class; meaning a more rigorous treatment of domain, range, and composition  ideas than what students experienced in earlier courses. As I was about to start inverses last week, I sought an activity which would provide some discovery, some personalization, and less of me rambling on.

These are the times when searching the MTBoS (math-twitter-blog o’sphere) leads to some exciting leads, and the search for inverse functions ideas didn’t disappoint – leading me to Sam Shah’s blog, and an awesome discussion of inverse functions which I turned into a sharing activity. A great list of blogs and MTBoS folks appears on this Weebly site.

To start, I wrote a function on the board, and asked students to think about the sequence of steps needed to evaluate the function:

The class was easily able to generate, and agree upon a list of steps:

  1. Square the input
  2. Multiply by three
  3. Add 1

From here, I asked the class to divide into teams of 2. Each partnership was then given two functions on printed slips (shown below) to examine: list the steps of the function, and provide 3 ordered pairs which satisfy the function.

THE FUNCTIONS:

inverses

Notice that the functions are arranged so that A and B in each set are inverses.  Partners were given two different functions, but never an inverse pair. So a team could get 2A and 4B, but not 3A and 3B.

My plan was to complete this entire activity in one class period, BUT weather took hold. They day we started we had a two-hour delay, and the next two days were lost due to snow, then a weekend. SO, the best-laid intentions of activity, sharing and resolution became activity…..then 5 days later.

As we started the next class day, I asked students to review their given functions (and re-familiarize themselves), then seek out the teams who had the other half of the function pair and share information. So a team which had 2B sought out 2A, and so on.

After the sharing, a classwide discussion of the pairs was then seamless. Students clearly saw the relationships beteen the inverse pairs and the idea of “undoing” steps, and we could now apply formal definitions and procedures with an enhanced understanding. Also, by sharing ordered pairs, students saw the domain-range relationship between functions and their inverses, and this made graphing tasks much easier. I’m definitely doing this again!

Finally, notice that pair 2A / 2B features a quadratic / square root. While we didn’t dive right in at the time, this set the trap for a discussion of one-to-one fucntions and the horizontal line test the next day.