Percents and Number Lines

So I was monitoring this conversation on Twitter the other day:

Can these be placed on a number line? What say you, #MTBoS?? #mathchat pic.twitter.com/tkuQC2Br7P

— Nat Banting (@NatBanting) January 20, 2016

I was going to join in but realized that my thoughts required WAY more than 140 characters.  There were so many great points made by members of the #MTBoS (as always).  My initial thought was "of course a percent is a number, so it can be placed on a number line", but then I began to doubt myself following some points made by @letsplaymath.
It took a while but I think I clarified it all for myself.  The question is can we put a percent on a number line.  I'm sticking with my initial YES.  Here's why: When we create a closed number line (not an open number line used by @Mr_Harris_Math to teach arithmetic strategies), we MUST put at least 2 numbers on the number line.  This inherently defines a unit on the number line.  This unit is, in a sense, the interval.  It is the distance covered by 1.

All the other numbers on the number line are defined by this unit.  The number 42 is 42 of these units lined up away from zero.  A percent is a special type of fraction where the denominator is 100.  Putting 85% on a number line means taking that unit distance, dividing it into 100 equal sub-units and traveling 85 of these sub-units away from zero.  It is both a number and a ratio.  When we have a closed number line, the unit distance (interval if you will) is inherent to the line as soon as two numbers are placed on it.  Any ratio or percent you'd like to plot on this line are then defined in terms of this unit.  A percent just defines that we will be traveling in increments that are 1/100th of the defined unit.

There was an argument that we can't put 80% on the number line because it is relative to another number.  Is it 80% of 20?  Is it 80% of 1?  When we place 80% on the number line, we are placing it relative to 1 unit on the number line.  This is the key to the argument that we CAN indeed put percentages (and any other ratio) on the number line.  It is implied that we will be placing them on the number line relative to the distance of 1 unit.

Intro to the Coordinate Plane

I always had a handful of students that continually reverse coordinates or plot points incorrectly because they forget that a negative sign indicates going down or left.  So I designed a lesson that I was hoping would help students understand the importance of conventions in the idea of an ordered pair.

I told students that they would be on a mini-treasure hunt.  We blindfolded a student and sent them out of the room.  The rest of the class was charged with giving the blindfolded student directions to the "treasure".  For my class, I used a dollar bill and put it in the ceiling which conveniently had square drop tiles.  The class had to give all of the directions to the blindfolded student BEFORE that student was allowed to move.  

They brainstormed and came up with a plan to tell the student to walk 5 steps forward and 3 steps right.  Immediately they began to see some issues with the directions.  The blindfolded student took much smaller steps than the student that had measured the 5 and 3 steps.  The student was way off.  We had a discussion to decide where things may have gone off track.  After some more brainstorming, students decided to use the tiles on the floor as a way to track distance.  This required me to change the blindfold for a "You're only allowed to look straight down" direction to the treasure seeker.  This process allowed students to understand the need for a consistent interval when plotting on the coordinate plane.

We repeated a few times with success (I kept the dollar bill though!).  After two successes, we did the same, but I went into the hall and had the student come in through another door.  The directions the class gave were assuming we were going to use the same door as the last few times.  The class was upset with me because I was being "unfair", but it opened the discussion that if we were to give directions ahead of time, we needed to know WHERE to start.  Again leading to the understanding of why we always plot points starting at the origin!

The last twist was that I put a challenge to the students to give directions in as few words as possible.  They quickly reduced directions to something like 2 left, 5 forward.  Then I said, no words, all numbers.  Having familiarity with a number line, students were able to arrive at a negative for one direction and a positive for the opposite, but it took some serious prodding for them to get to the point where they understood that they needed to come to a consensus about WHICH number would come first.  This was exactly what I was looking for in terms of their coming to grips with the coordinate plane.  

Equation Golf

We were talking about PD experiences the other day and it inevitably turned to "What was the best PD you've ever been to?"  Having never attended a TMC (and unable to go this summer...sad face) I reflected upon one of the more useful district supported PD days.  We met a group of math teachers from another high school in the district.  Prior to the meeting we were asked to write out a description of our best.  It could be a lesson, a technique, an explanation.  I honestly don't remember what I brought to the table, but it wasn't nearly as good as the stuff I got out of the day.

Below are quick descriptions of two review games that have been great of great use in the classroom for a tweak on an "quality, basic" problem set.

Equation Golf:  Have a problem set of 15 problems that you want the whole class to do?  Give students a goal, or a par.  You want to get all of the correct answers to the problem set by calling on 18 people or less.  (Numbers used are purely for illustrative purposes, you can have 20 problems and call on 26 people, just make the goal reasonably attainable).  Have students work on the problem set for the desired amount of time.  Individually, partners, groups, your call.  At the end of the time, tell students that they must pay attention to play golf.  Pick a student at random (Popsicle sticks, random # generator) and that student picks whichever problem they would like to answer.  You verify or reject the answer.  New person.  Each person counts as a "stroke".  Goal is to make par.

This helps work on listening skills as well.  I always tell the students that they must pay attention to which answers have been given AND marked as correct.  The students must announce the problem number first, then give their answer.  This way, if a correct answer has been given for #3, and a student says "I'd like to answer #3, the answer is..." I can cut them off, let them know that that question has been answered correctly already and that counts as a stroke.

In the past, I had given an entire class grade for this activity.  Usually a relatively small grade that didn't really affect the overall grade in the class, but enough to keep them interested.  As I tried to move towards more standards based grading, I might make the change to allowing a student that answers correctly to use that as evidence of learning for a particular objective.  This would have the added bonus of helping students to identify their own areas of need, work on that with students in the class during the class work portion, and trying to answer more difficult questions based on their individual needs.

Bluff:  Take a similar problem set as Equation Golf but divide the class into two teams.  Give one team a problem to complete (the other team should simultaneously complete the problem).  On Team 1, after a certain time, ask anyone that feels that they have the correct answer to stand up.  Someone from Team 2 picks a Team 1 member to answer.  If it is correct, Team 1 gets a point for each person standing.  If it is incorrect, Team 2 can steal the points.  Then the next question goes to Team 2.

This game can be a bit touchy so you have to have a responsible group of kids that will be willing to put themselves out there.  Class culture is extremely important to this game.  I have seen times where students will continually pick on one person, so it is up to the teacher to ensure that the game stays positive.  In cases like this, the teacher doing it was awesome (disclaimer, it was not me...though I am pretty awesome!) and when this situation arose, he turned it and made the kid a champ when he kept getting the questions correct.

A bit of strategy goes into the game as well.  A team cannot call on the same person 2 turns in a row, so the last person called should always stand.  I don't share this with kids, but it is fun to watch them figure it out.  And a student can stand even if they do not think they have the right answer in an effort to increase the point value for their team (hence the name Bluff).