Wednesday, January 27, 2016

Percents and Number Lines

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

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.

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