Monday, February 11, 2013

Fractions Day 8: Have Your Cake and Eat it Too?

fraction misconceptions
Well. . . another day in the fraction trenches for the fourth graders in the Studio--that's for sure!  We had a late start today for icy roads, so I thought to myself, "Let's do a quick warm up discussion then get back into those fractions of sets that we struggled with on Friday."

Mmmmhmmmm . . . quick.  Riiiiiight.

Here's how today's discussion unfolded.  Check out the prompt--see the square divided into pieces?  The prompt simply asked them to explain what fraction is shaded--and to explain their thinking.
critiquing reasoning
The good news--students immediately got to work and were confident in their answers.  
The bad news--MORE MISCONCEPTIONS!
fractions

Here is one student's work . . . she added a dotted line and explained that if you draw a line through it you get 8 pieces--and one is shaded so you get 1/8.  Smiles. 

but....
critiquing reasoning
Then I moved to the next table and saw this one . . . 3 1/2.  Three "whole" sticks and 1/2 of another one.  Sigh.  Deep breathing.  So I finished my rounds and went to the easel (Smartboard on the fritz today) and wrote out the most common answers that I had seen--see photo below.  I chose to NOT put the 3 1/2 up there . . . I was hoping the discussion would help this student see the error in her thinking--but I will be watching her closely--and also the person next to her who had the right answer and then CHANGED it to copy hers.  Sigh.  Two steps forward, one step back.
fraction instruction
So I asked students to vote on which answers are correct.  See the number of votes in the photo below.  Interesting, wouldn't you say?
fractions

I thought this was intriguing (NOTE:  I had 21 students present--so students DID vote for more than one in some instances--always allowed . . . and very "Smarter Balanced" friendly) so I wrote the following question underneath our information.  I asked the students to turn and talk with one another to see what they thought.  Lots of heated chatter and hand motions followed!
teaching fractions
So we voted again.  I asked how many of them thought that the number sentence was true.  NOTE:  I still had 21 students present.  I got 100% agreement--that NEVER happens in my room!
teaching fractions

So I called their attention back to the easel and said something like, "I notice that each and every one of you feel that 1/8 is the same as 1/2 of 1/4.  Let's vote again about which of these answers you feel represent the shaded part of the square.  Look. At. The. Results.  Are you as puzzled as I am?  I gave them some additional talking time, and a few more crossed over . . . but not everyone.
fraction lessons
So I tried this approach.  I drew a new picture with NO lines.  I explained it was my son's birthday cake--and I asked how much he ate.  After some brief discussion, they came up with the idea that it was 1/8 of the cake (except, mind you, of the ONE student who in the above photos refused to accept that we were talking fractions because the pieces were not equal--he refused to admit that this was 1/8 as well.  Tenacious little bugger!)
So to make things interesting, I explained that I really precut the cake like this. . . and I asked if he ate the same amount or a different amount.  I got some tentative "same" responses.  Some said "No, this time he ate 2/16."  Others weren't sure.  Others said "So 1/8 is the same as 2/16?"  
I think I see where things are headed next . . .So much for my easy warm up!  I'll be putting this one on the back burner to get a little more confidence going with fractions of sets and number lines--then it will definitely be time to tackle equivalent fractions!  It's time! Thanks for stopping by--hope everyone had a safe Monday! 

***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!  



6 comments:

  1. I'm loving your fraction posts! We just got done with them a couple weeks ago but it's still insightful to read what your kids are doing!
    Jessica
    ideas by jivey

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  2. Fun! We'll start doing fractions for the 2nd time pretty soon this term. I'm going to borrow some of these ideas :) Hey - check your email. I think some are going missing - I sent you an alternative address this morning.

    Lynn

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  3. Please tell me your journal prompts are a product in your TpT store?! I LOVE the thinking and discussion they generate!!

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    1. Dawn--thank you so much. I actually AM working on a product that will hopefully capture this unit and will make it available in my store. I hope other people find it useful as well! :)

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    2. Hooray! I can't wait! This type of thinking, writing, and discussion are so valuable in math classrooms. I am looking for ways to incorporate it into mine.

      I love reading your posts, since I know you focus on thinking with your students. Kudos!

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  4. What do you do when you have one student who everyone trusts to follow (i.e. they always know the correct answer, so everyone just goes along with what they say)? I find exploration math difficult when students refuse to think for themselves. Thoughts?

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