Category Archives: Algebra

“The 35 Game” for Compound Inequalities

December 2019 Update: find a full lesson plan for this game here

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)!