Monthly Archives: April 2015

Put(t)ing Rational Numbers in Order

Many of my friends and followers have caught onto one of my guilty pleasures: my wierd fascination with The Price is Right (read about Price is Right and counting principles in this old post).  Here’s how a pricing game made for a fun review activity, and also made my life flash before my eyes (read to the end for that).

Here in Pennsylvania, we use the PA Core Standards.  For Algebra 1, here is a standard under “Anchor 1”:

A1.1.1.1.1 Compare and/or order any real numbers.  Note: Rational and irrational may be mixed.

Seems innocent enough.  Here is a sample “open-ended” task used to assess understanding on our state’s Keystone Algebra 1 exam:

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Exciting….now let me go over here and watch the paint dry….

But during the NCTM conference, a lightning bolt hit. I was checking out a putting game at one of the booths, and I suppose rational numbers were on my brain….Hey – Golf + ordering rationals = feels like Hole in One to me!

In the Price is Right Hole in One game, contestants place groceries in order from least to greatest by price.  The number of items they can order until they are incorrect determines where they putt from. After a quick trip to the sporting goods store to find a putting cup, and some time with a Google Doc, we’re all set!

To start, I created a Google Slides presentation with 6 games.  Each game has 6 numbers for students to put in order:

During the game, all students in the class had about 2 minutes to place the numbers in order.  They, we randomly drew our “contestant”, who came to the board to fill in the 6 boxes on the board.

order

Next, we went through the numbers from left to right, and determined how far the contestant had gone in successful ordering.

puttOn the floor, 6 lines were taped.  Line 1 was on the other side of the room, and the lines were closer and closer to the hole. If a student had 4 numbers correctly ordered, they were allowed to putt from line 4.  Two students were able to order all of the numbers and tried their putt from about 2 feet away.

Those who made their putts earned candy to share with their group.  In about 20 minutes, we got through 4 games – not bad for ending a Friday on a fun note.

But be careful! My last “contestant” – one of my less cooperative students and a sometimes hot-head – was able to putt from line 6 with the help of his group.  After missing the first putt, I reminded him that the game is really Hole in One – OR TWO, and had a second chance. Lining up the putt…he took it easy…and missed again.  This is when he raised the putter up and, for a brief second, it looked like the putter could end up flying in my direction.

“Sean, just pick up the ball and put it in the hole….here’s some candy…”

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)!