Fraction Fest, 2013

Today, we started our discussion about Fractions!  Part of this unit is allowing the kids to "prove" their thinking in terms of what a fraction is.  We want kids to use mathematical language to discuss math with each other and to be able to not only look at their own thinking in a critical way, but also the thinking of others.  This is a huge step in a young person's development.

Today, we explore breaking different shapes up into fractional parts.  On purpose, Tom and I did not answer questions about how to do this.  We left it up to the kids to figure it out.  Sometimes, they were stumped on how to break their shape into equal parts.  Here is how the lesson developed.

First, the students worked with their math partners to trace classroom shapes.  They had to pick a square, a rectangle or a circle to be their "whole."  They would use the same "whole" to do each of the activities.

Their task today was a simple one.  Separate each of their "wholes" into each of these fractional parts.  We started with the halves first....which all the students complaining how easy it was!!

But as we moved to the thirds, things got a little interesting....

The kids noticed right away that something was amiss....although, they weren't quite sure why.  Could both of these circles be correct?  There was some waffling back and forth....in the end, the kids were undecided.  As many kids who thought that the circle on the right was wrong, that same number of kids thought it was a perfectly legit way to divide a circle into thirds. 

Again, Tom and I didn't answer any questions, we wanted the kids to construct their own understandings.....which means they have to first make mistakes in order to build up their knowledge.

We kept going, dividing our shapes into 4ths, 5ths, 6ths and 8ths.  The kids all talked about how much harder it got the smaller the sections became.

At this point, the kids still felt like this was an accurate fraction.  They decided that the middle sections were skinnier and taller, and the 2 end pieces were shorter, but wider....so, it all evened out....each section was an equal part.....but there were still some kids who weren't buying it.

Then we got to the 6ths....and this showed up on our chart.  Using their previous thinking. I asked if this section was divided into 6 equal parts.....The kids had a spirited discussion about since these sections all have the same height....are the outside rounded sections equal to the inside sections.....again, no real agreement....but more kids were questioning.  They were beginning to doubt that this kind of fraction could work, but they weren't sure why.

When we got the breaking shapes up into 8ths, all the circle groups decided to split their circle up in the same way.  Kids who were previously drawing straight lines decided it was easier to start with a center point and work outwards (they called it the pie graph way).  In the end, they said it was easier, and it felt like a more PRECISE way to break it apart.  

Some of our square wholes got into the action as well.

 

 

At the end of our session, were left were several fraction decisions, and one question.....

1.  Fractions have to be equal sized parts of a whole.
2.  The larger the fraction, the smaller each section of that fraction.
3.  Some fractions are easier to make than others.....1/2, 1/4, 1/8

But one main question is left.........Does equal parts mean equal shapes?  Can a "whole" be divided into equal parts that look different? 

Check back tomorrow when we try to answer that question.