True or False? A Warm-Up That Blows Open Misconceptions About Comparing Fractions

I’ve been filming in Tabitha Eutsler’s classroom all week to capture a progression of student learning.  Every day I have been in her classroom this week there has been moment after moment of excitement as we see the understanding of students unfold through purposeful questioning and student feedback.  

Tabitha is a master facilitator.  She spends lots of time at the beginning of the year (and throughout) building her classroom community because she knows that is what supports students in being able to communicate with each other, disagree, argue, challenge each other and celebrate each other so that every student has not only the opportunity to learn, but the opportunity to belong.  Her students are mathematicians and they know it.

These were the learning goals for each lesson this week:

  • Monday: 
    • Generate equivalent fractions. 
    • Understand two fractions as equivalent if they are the same size.
  • Tuesday: 
    • Identify and generate equivalent fractions.
    • Understand two fractions as equivalent if they are at the same point on a number line.
  • Wednesday: 
    • Represent and compare fractions in a way that makes sense to them.
  • Thursday: 
    • Compare two fractions with the same denominator with the symbols < or >.
  • Friday: 
    • Compare two fractions with the same numerator with the symbols < or >.

Today is Friday and students started the lesson with an instructional routine called True or False?

The first problem posed was:

True or false?

Students were asked to think independently for a minute silently and then Tabitha asked them to share and turn and talk to a partner to see if they agreed or disagree with each other.  She then asked for students to pop their thumbs if they had an answer they were willing to share.  Below are the conversations that followed for each of these tasks:

Student 1: I think it’s false 

Teacher: Why?

S1: Because 2 and 4 are two different numbers and 4 is greater than 2.

T: So you looked at the numbers on the bottom and thought 4 is bigger than 2 so this must be false? Did I get that right?

S1: Uh-huh

T: Okay, thank you for sharing.  Continue. S2 what do you think?

S2: Actually I agree and disagree.  I agree it’s false but actually I would look at the top because it is actually equal, not 4 is greater than 2. 

T: So why do you think it is false then?

S2: Because it says that ½ > than ¼ and actually that is actually equal.

T: So you think it should say ½ = ¼ and they are the same? Ok.  Does anyone have feedback?

S3: I disagree with you because the top are equal, but the bottom numbers are not.  

T: So do you think it is true or false?

S3: False

T: Okay go ahead and give us your reasoning. 

S3: 1 and 1 is equal but the 2 and the 4 aren’t.

S2: I kind of see what you are saying.  The top is equal but the bottom is not. 

T: So are you saying that ½ is the same as ¼ now or you think they aren’t actually the same.

S2: No I don’t think they are the same.  I want to change my answer because of what he said. 

T: I think I’m gonna throw out a different challenge.  How about we get our books out so you have a place to write and I printed some more fractions strips and number lines for you guys.  So how about now you show me what is going on in your brain.  Can you write it down and show me?  So it is asking  ½ > ¼ and we are saying that is false.  So I want you to prove it to me.

Go ahead and talk to your partner.  Is it still false?  Or is it true?

If you want to bring your books up or use the markers you can.  I’m gonna review where we are at. We have 3 of you saying that is false.  You are going to come up and still defend false or if you changed your answers.

S1: I think it’s true because I can see that ½ is bigger than thirds because would you want one of a third or would you want a full half?  The half would be bigger so you would have more.  That’s why I think this is true.

T: Feedback?

S2: I disagree, I think it is false because I drew a number line and on this you said thirds instead of fourths.  Oh yeah fourths. Right here, that would be where ½ happened.  Right here would be ¼. It’s diagonal. 

S1: I agree with how you saying that but you can also split ½ into 1/4s so it would be a smaller piece.  It would be ½ broken up into fourths so you would get a littler piece.  I’ll show you.  We have this whole big piece right here, now you want to get this piece.

T: Can you show him what you mean by pieces using HIS number line?  Can you show him where the pieces are on his number line to compare the sizes?

S1: Yeah. So this is ½ and this is ¼ so this piece is bigger.

S2: Oh, I get it so what you’re saying is this is here.  I agree with S1 now.

T: Everybody we haven’t seen fraction strips yet.  S3 is going to show us fraction strips.  So be looking at your paper.  Does everyone have true written down?

S3: I drew two number lines.  This one is split in halves and this one is spit in fourths.  The half is double the size of the fourth so that’s why I think it is true.

T: Feedback for S3?

S4: I agree because this goes further than this one.  

T: Oh so the half is further than the fourth?

S4: Uh-huh

T: After listening to our family give me a thumbs up or thumbs down is it true or is it false?  Okay.  Thank you.  This is true.  So you might be thinking I kind of see it but we’re in luck because  we have a couple more problems to practice this.

True or False?

T: Are you defending true or false?

S1: I am defending false. As you can see here when you make fourths you get smaller pieces.  When you make thirds you get big pieces.  I labeled them small and big.  

T: Feedback – what do you think about that with the pieces she was talking about?

S2: I agree because I did the same thing but I didn’t notice that.  I didn’t do the number line but I did the picture and it blowed my mind.

T: Thank you so you had a connection that your picture was similar but doing it this way grew your brain even more? Thank you.  Who would like to defend next?

S3: False because if Ms. Eutsler gave me a candy bar would I want the bigger piece or smaller piece? Since it is a candy bar I would want the bigger piece.  So it’s false because that piece is not bigger than that piece.  

T: You shared something with me about the bottom numbers.  Can you share that with the class?

S3: So on the bottom you always want the lesser number because the less number has the bigger pieces than the bigger number. 

T: So if we look back, a yes or no please.  Based on everything that you heard, is this true? It is false.

True or False?

S1: I think it’s true because um if you draw it on a number line you see that 6 is farther away than 8 so the less away that it is is the one that is true.

T: Feedback?

S2: I agree that it is true but you forgot one. These are five pieces and you need to draw one more. 

T: Look at your paper, do you have true based on what our family just told us?  It is true.

Following the warm up, the students worked on a word problem in which two students disagreed about 5/6 or 5/8 being greater. Several students shared and a student ended with this comment:

S: This kind of goes back to our true or false questions that we did earlier where if the numerator stays the same then the denominator determines which is one is greater.

T: And how would you determine with the denominator which one is bigger?

S: When the denominator is smaller.

T: Because the size of the pieces is what?

S: Bigger.

This is the power of a problem based classroom.  The students uncover misconceptions as they begin the task, but as the teacher supports their productive struggle by allowing them time to grapple with new ideas, share their thinking with others, and see solutions from other classmates they begin to make sense of the mathematics and develop reasoning and proof.  This is possible because of the teacher’s understanding and application of the mathematical teaching practices.  Ms. Eutsler poses purposeful questions, supports productive struggle and facilitates discourse so students are positioned as the experts in the room.  When they need support, she provides it in the means of tools, questions, and the purposeful sharing of student work that provide visual representations from which others can compare, contrast and connect to their own. It is the skillful art of teaching.

Problem based classrooms provide students time and space to develop deep understanding of math concepts. They struggle, they perservere. They make mistakes and they correct them. They agree and they disagree. These are the habits of mathematicians. The world doesn’t behave like a math textbook. Students need to know how to make sense of contexts and scenarios by playing with mathematics in a classroom with a teacher who values mistakes and problem solving, holds the belief that every student can do math to high levels, understands that math is about depth not speed, and knows that students will meet the expectations that are set for them.

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