Handout 4.3d [pg. 1 of 3]:
Dropping Into A Classroom
|
This text is excerpted from the Website: Yarn Number Line: A Lever to Develop Fraction Understanding.*
Let’s drop into a classroom to think about the mathematical affordances of a yarn number line. This particular yarn number line discussion happened on day three of Fraction Lab. On days one and two, it became clear that students’ understanding of whole numbers on a number line was limited, especially the concept of how to represent number magnitude by iterating equal-length units with attention to being precise about number placement. After working with those concepts using zero and the whole numbers, 1, 2, and 3, we asked the students to place 1/2 on the number line. The exchange below took place after 1/2 was placed between 1 and 2, but not exactly in the middle.
Teacher: What do you think Diego was thinking when he put it right there?
Anthony: That half and another half makes a whole.
Teacher: That half and another half makes a whole? So I’m wondering about what we talked about, thinking about the jumps we make, and they’re evenly spaced. I’m wondering about that related to a half. Does anybody want to add to it or think about where the half goes… Does ½ and ½ make a whole? What is 1/2 plus 1/2?
Anthony: 2 wholes.
Teacher: 2 wholes or 2…?
Gianna: 2 halves.
Teacher: 2 halves which is how many wholes?
Gianna: 1.
* (2019, April 18). Yarn number line: A lever to develop fraction understanding. Peers and Pedagogy. Retrieved January 31, 2023, from http://achievethecore.org/peersandpedagogy/yarn-number-line-lever-develop-fraction-understanding/
Math | Module 4.3 (Gr 4)
Handout 4.3d:
Dropping Into A Classroom
|
Teacher: One whole. Interesting, so would you say that this right here like if I drew it, so would you say if I had a strip [teacher draws a fraction strip above the yarn number line, between 0 and 1], remember when we did the folding the other day, and I wanted to mark a half, where would that be, I wonder. What do you think?
Student: In the middle.
Teacher: Why would it be in the middle?
Student: Because 1/2 plus 1/2 equals one whole.
Teacher: What were we really careful about when we were partitioning?
Student: Cutting 2 even pieces.
Teacher: Cutting them in really even pieces? So I’m wondering if we should shift that half. Do you want to come do it? Come on up. Do you want to do it with the pen, or do you want to move the…can do the pen. Ok, there you go. [Student writes ½ in the middle of the strip.]
Teacher: So I wonder if it’s important on the number line, just like it’s important with the paper that we fold, to make sure we’re showing that this amount is the same as this amount?
Within this exchange, students are connecting their understanding of partitioning a whole into equal parts (something they experienced concretely using paper folding) with the concept of representing a fraction on the number line. Students are beginning to reason about representing a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts, by connecting that ½ plus ½ equals one whole.
Math | Module 4.3 (Gr 4)
Handout 4.3d:
Dropping Into A Classroom
|
Teacher: So let’s talk about this one. [teacher holds up 2/2 on card] We’ve already said we know what this is. What did we say?
Student: 2 halves.
Teacher: So if this is 1/2 , and this is 1/2 [teacher pointing on number line], and then we talked about… this is 2/2, we can break that apart [teacher draws on board, whole box and decomposing into 1/2 and 1/2 ]. Where would 2/2 go on the number line? Talk with your partner about where you think it would go.
[Students talking]
Teacher: So Camila has an idea, but before she tells us where she thinks it goes, I’m just going ask her to share her thinking with us. What did you think about 2/2?
Camila: 2/2 equals 1 whole.
Teacher: 2/2 equals 1 whole. Do we already have 1 whole on the number line?
Student: Yeah.
This exchange helped students to continue thinking about representing fractions on number lines, and the connection between decomposing fractions into unit fractions with representing a fraction a/b (2/2) on a number line diagram by marking off a lengths 1/b(1/2) from 0. While there was much direction from the teacher in this example, the reasoning of the students became more sophisticated as the next numbers that were presented to place on the number line included 5/2 and 4/2. Extending the number line beyond 1 was an important step for these students who prior to Fraction Lab had all assumed number lines end at 1.
Math | Module 4.3 (Gr 4)