# Fractions Day 6: Logic, Pizza, and Critiquing Reasoning

Seriously, we are starting to get into some pretty heavy duty thinking in the Teacher Studio!  I wish I had a video camera to capture some of the amazing moments!  I’m going to try to paint of picture of how things unfolded today.  Sit back–I am still in awe over one of my mathematician’s understandings.  Like–jaw dropping awe. . .  and it takes a lot to impress me!

Today started with a discussion about our homework.  I’m not a huge fan of homework–don’t get me wrong–but I wanted the students to take something home that would show their families the kind of thinking we are doing, but without frustrating them.  See the photo below–I hope you can see the task.

### Critiquing Reasoning and the Standards for Mathematical Practice

In a nutshell, the students needed to identify the shapes that were NOT divided into thirds, then pick 2 of them to defend–to “prove” they were not.  This was a great start to the day . . . students were successful at this but–if you remember–I knew there were LOTS of misconceptions abounding in my room!  So what I did next was hand out sheets of shapes that I had divided in different ways and sent the students off in groups to decide if “TRUE”–the shape WAS divided into equal parts or “FALSE”–the shape was NOT divided into equal parts.  I gave the groups five minutes or so to decide and to come up with their arguments.  Of the 6 groups, 4 came to consensus, 2 did not–so I told them the class would help them decide.

Some of the groups had no problem explaining their thinking:
This group clearly was able to explain that the shape could be divided in half, then each half could be divided again in half to create 4 equal parts.  They were adamant that the DIRECTION of the shape didn’t matter.
What was more fascinating were the two groups that disagreed.  I have to show you their shapes!  Here was the first group . . .
This team was divided evenly (ha–no pun intended!) with two arguing “TRUE” and two arguing “FALSE”.  The “true” team explained that the circle was cut into four equal pieces and then those fourths were cut in half again, creating the 8 parts. The class went wild saying “They aren’t equal parts!  You can’t do it!” (Before you think I have world’s smartest, most on-task class in the world, I should tell you THIS is the point in the lesson where one of my students raised her hand and informed us all that “This shape looks an awful lot like a volleyball.”)
So I asked my mathematicians to try to PROVE it to me . . . I reminded them that just basing it on “looking” can cause problems–we can’t always tell by looking.  I then reminded them that math makes sense (our class saying) and to look for patterns or other ways to explain this.  I sent them back to their teams and here is what one group came up with. Seriously–this is fourth grade!
Their explanation?  If you divide a SQUARE like this, it works . . . half of each fourth is an equal eighth.  But if you make it into a circle . . .
Check the pink post it carefully . . . they have drawn a circle around the eighths.  They used this to PROVE that if you cut the edges off the square to make a circle, you are cutting way more off some pieces than others–so the part left in the circle will not be equal.  CAN YOU BELIEVE?  This is pretty high level thinking, right?  I could hardly control myself when asking the class, “So . . . does their logic make sense?” and the class emphatically agreed.

### Fraction Thinking and Reasoning

So, even though I wanted to end there because I was so proud that I wanted to burst, I had one more group to go–and it was a divided team as well.  Here was their shape.
The class went back and forth on this one for a while . . . really struggling with the “They all look pretty much the same” argument until . . . one of my students–one of those who gets extra help . . . the ones with letters after their names . . . the ones that make our “data” look bad . . . you know the ones–comes up with this.
(Paraphrasing–but not changing his words much)
“Well . . . they must NOT be the same because we need to think about pizza.  You told us that we need to get equal pieces from circles by making slices.  If we pretend this one is a pizza, we can tell the slices all start the same in the middle.  If we turn it into a circle pizza, you can see that there is way more extra left on some pieces than others.”
I wasn’t quite sure if I was hearing him correctly so I asked him if I should redraw the shape and put a circle on it to make it look like a pizza.  He emphatically said yes, so I put the shape on a post it, and drew the circle on it.  He looked at it, then told me which pieces to shade in–and here were the results.
He marched up to the Smartboard and explained as clear as could be to the class. . .  First, he got their agreement that all the circle wedges (pizza slices) were equal and then showed the two blackened sections.  He explained how the two “leftover” pieces were totally different sizes–so the pizza slice plus the extra made pieces that were definitely NOT the same.  You could hear the audible “OHHHHHHHHHHHHHHH!” from the non-believers.  Remember–this is the kid with the letters behind his name.  He doesn’t know his math facts.  He doesn’t write using punctuation.  This is also the child who, when we shared “The Tiger Rising” as a class said, “You know, I think that Sistine is just like Willy May.  They are the only two characters who let their feelings out while the other ones keep them locked inside.”
Yep.  I loved today–not just because my students did some a-mazing thinking . . . but because I was reminded that all students make sense of their world a little differently.  Ask the right questions, give the right context, and they will show what they know.  No standardized test will ever measure what this little guy did today.  I’m glad I was there to be a part of it.
Coming next:  The question of the day:
“Can a shape be both 1/2 and a whole?”
***UPDATE***
This blog post is now a part of my comprehensive fraction unit available by clicking the image below.  Hundreds of teachers have now used it to change the way they teach fractions!