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.**

### Math Misconceptions

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.

The good news–students immediately got to work and were confident in their answers.

The bad news–MORE MISCONCEPTIONS!

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….

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.

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?

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!

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!

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.

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!

***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!

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