Students were asked to look at an area model and 3 different numerical representations (2 1/3, 7/3, and 7/9) given by different students to represent the area model. They were asked which student they agreed with and why. They then shared their reasoning with partners and with the class as a whole. To structure students’ responses and provide access to all, students were asked to use the sentence frame: I agree with ___ because.
MICHELLE KIOUS: In yesterday's lesson, there were a couple, were a little bit confusing, so I want you to take a look at this question and decide which of these Students you agree with, and then you need to explain why. So make sure you're using complete sentences when you're explaining. Yes, you can. Um, Cornelius asked a good question, and he asked if you can agree with more than one person, and I will say yes. And we're about ready to share. How many people need a little more time? So I'm just going to give you one more minute. We're about ready to share with partners. So I need everybody, making sure that they're at their desk. They're ready to share with a partner. Cornelius, you need to be back at your desk. It's okay. So I'm going to have you share with your partner, and I want you to share your answer. I'll tell who goes first, odd or even, who you agree with, and why. And I know there are a couple of people that agreed with more than one person. So when you turn to your partner, I would like the odd person to start and share yours, and then the even person can tell. So go ahead.
STUDENT: I agree with Teri because there are two shapes shaded, and there is one of the thirds, so that makes two and one-third.
STUDENT: Uh, I agree with Juan because there are three equal parts in each pie and seven are shaded.
STUDENT: Two wholes shaded at, in and one-third of the circle is also...and also shaded. I also agree with Juan because one whole shaded equals three and three times two equals six, plus the one-third shaded equals seven, seven-ninth.
STUDENT: I agree with Teri because she wrote two and one-third, and if one pie is fully colored, then it is one whole. I also agree with Juan because the denominator is three, and there is three pieces in one whole. And if you count all the pieces, then it is seven. It is also an improper fraction.
STUDENT: I agree with Teri because I saw two is shaded altogether and that will equal two, and I saw one-third that was shaded in, and that will equal two and one-third.
MICHELLE KIOUS: Okay, if you agreed with Jasmine, can you give me a thumbs up? And if you agreed with Teri, can you give me a thumbs up? So I would like someone who agreed with Juan to raise their hand and tell me why they agreed with Juan. And now Abigail, why did you say thirds?
STUDENT: Because, um, each...each, um, circle is cut into three pieces.
MICHELLE KIOUS: Yeah, and so that we said that these were pies. So each pie is cut into three pieces. And when you're talking about the denominator, what does that three mean in the denominator? Three what? Three, three pieces to what?
STUDENT: Um, three pieces and a whole.
MICHELLE KIOUS: Three pieces and a whole. And can you come and point out the nine? Where do you see the nine?
STUDENT: Um, there's one, two, three ...
MICHELLE KIOUS: Louder.
STUDENT: There's one, two, three, four, five, six, seven, eight, nine pieces.
MICHELLE KIOUS: Thumbs up if you can see the nine that she pointed out. Nara, what's similar about these two fractions? What do you see that's the same with these two fractions? What did they both have? Just by looking at the fractions, what do you see that's the same? Hands down. They both have the seven, and we counted those seven. Janelle counted those seven. So the difference is the denominator. Abigail, what did you see a denominator represented?
STUDENT: Um, how many pieces in a whole were.
MICHELLE KIOUS: How many pieces are in a whole? So thinking about a whole pie and deciding if you agree with Jasmine or with Juan. So I want you to talk to your partner again. Do you think there are three pieces in a whole pie or nine pieces make up a whole pie? So turn to your partner. Even start.
STUDENT: Um, I think Jasmine is right because there are three parts in a whole.
STUDENT: I think ...
STUDENT: They are the parts.
STUDENT: I think Teri is right because there's two wholes and one-third of pie.
MICHELLE KIOUS: And you're going to put thumbs up if you think three pieces make up a whole pie in this diagram. Okay? And, uh, now thumbs up if you think that nine pieces make up a whole pie. So now I need somebody with Teri. Who thought Teri was correct? Who thought Teri was correct? Um, Cornelius?
STUDENT: Um, I believe that Teri is correct because there are two full circles shaded in, and then this. So that's two wholes and then they only, um, shaded in one-third of the other ones, so that's what represents the one-third of the fractions.
MICHELLE KIOUS: Thumbs up if you see the two whole pies in the diagram. Near your body. Okay. Uh, Cornelius, can you explain the one-third? What does that mean?
STUDENT: Um, the one-third means that one, only one out of three pieces are shaded in.
MICHELLE KIOUS: Thumbs up if you see that one out of three shaded in. Near your body. Okay, thank you. Um, how many people think that there might be more than one right answer? Raise your hand if you think there might be more than one right answer to this. Hands down. Um, who heard Abigail's word that she used to show more than one fraction representing the same diagram? Maria, what word did you hear her use?
STUDENT: Improper fraction.
MICHELLE KIOUS: She called this an improper fraction. What would we call this? It's not an improper. Michael?
STUDENT: A mixed fraction.
MICHELLE KIOUS: Mm, mixed number. Yeah. We do have a fraction and a whole number, so it's a mixed number. Um, and then Abigail used a word that, um, indicated that these two representations were the same. Did anybody hear that word that Abigail used when she was explaining? Abigail, I know you heard your word. Kyle, did you hear the word?
STUDENT: Um, equivalent?
MICHELLE KIOUS: Yeah, so these two would be equivalent. Okay? So, we're going to do a little bit more work with fractions today, and we're going to start out with our white board.
I chose to start the lesson with this warm-up activity to address misconceptions from the first day of the card sort. Many students were confused with an area model that represented more than one whole. For this activity, I wanted to emphasize the concept of the denominator representing the number of equal parts in a whole. I also wanted students to use the word “equivalent” to talk about two fractions that represent the same quantity. I asked students questions to address the misconceptions without just giving students the information. Students had to justify their own reasoning. This connects to CCMP3.