Research has shown that students’ overall IQ, family income, and family level of education do not predict future algebra success as strongly as fraction understanding (Siegler, et al., 2012). So 3.NF.A (Develop Understanding of Fractions as Numbers) is a critical cluster heading within the standards, not only in relation to students’ understanding of the concept of fractions, but also to their future success in math.
There are big implications for students in grades 3 through 5 as they begin to make sense of the expanded number system beyond whole numbers. This is where the number line comes in. The number line is specifically called out in the standards: 3.NF.A.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.
As we began planning for students in Fraction Lab (see more about Fraction lab here) we thought about the best way to help them develop the understanding of fractions as numbers. We wanted to specifically focus on:
3.NF.A.2.A Represent 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. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line, and
3.NF.2.B Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
As part of a pre-assessment, the students we were going to be teaching in Fraction Lab were asked to locate “1” on two different number lines. Almost all students simply put the number 1 at the end of both number lines, without attention to the fractional numbers that were already on the number lines. We decided to use a yarn number line routine each day to help build students’ understanding of fractions in relation to other numbers. (Read more about the pre-assessment and our decision to also use a choral count routine here).
What is a Yarn Number Line Routine?
A yarn number line routine is an instructional routine that can be enacted in the classroom in a short amount of time. The teacher simply creates a number line by hanging yarn across a wall or board. Each day the routine is enacted, students place numbers (integers, whole numbers, fractions, decimals, etc.) on the yarn number line, attending to quantities that are already placed on the number line, as well as placement based on magnitude of the numbers. Students then justify the placement of the numbers. This routine develops mathematical concepts of number sense, place value, number magnitude, and number comparing and ordering.
Dropping into a Classroom
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?
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.
As the yarn number line continued, students were asked to place 2/2 on the number line.
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.
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?
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.
Planning Yarn Number Line
To plan a yarn number line routine, there are several things to consider.
1. Identify a learning goal. Begin by thinking about what mathematical ideas you want to surface by using the yarn number line. Do you want students to work with whole numbers, integers, fractions, decimals, a combination of those? Also, think about what mathematical ideas you want students engaged in. During the yarn number line routine in Fraction Lab, we wanted students thinking about fractions greater than 1 and their placement on the number line in relationship to unit fractions. This helped us with the next planning steps.
2. Determine numbers students will place on the yarn number line, and in which order, based on knowledge of students’ mathematical needs and your instructional goal(s). The powerful thing about a yarn number line routine is that once there is a starting number, students then must make the next placements based on the number that is already on the line. In some cases, students may need to move numbers around to make the segments on the number line work for the magnitude of the numbers they are placing. When planning for a yarn number line routine, consider how the numbers you ask students to place on the line–and the order in which you give the numbers–will bring out the mathematical ideas you are aiming for.
For example, in Fraction Lab, students were struggling attending to magnitude and being precise when placing numbers on the number line, attending to length units on the number line. We chose to have students place “0” and then “3.” Next, we showed both “1”and “2” at the same time, and let them know they would be placing both numbers next. This decision was made to help students think about placement of 1 and 2, attending to splitting the number line into three equal-length units. Although some students may naturally think about partitioning 0 and 3 into three equal segments, the student we were working with had struggled with this and, when given another number, always tended to place that number in the middle of what was already on the number line. By giving students 1 and 2 together, we were asking them to attend more precisely to the partitioning.
3. Think about questions you will ask and the structure you will use to elicit student reasoning. It is important to think about how you will ask students to explain their reasoning. In some cases, we ask students to take a minute of private think time to determine where they will place a number and why they will place it there. This gives all students access to the reasoning. We also sometimes ask students to turn and talk with a partner, which prompts lots of pointing and excited talking. We also ask students individually to come place the number on the yarn number line, sometimes asking that student to explain, and sometimes asking others why they think the student placed the number in the location the student put it. The goal is to plan for engagement and for all students to have an opportunity to reason about the numbers. Pre-planning questions based on the instructional goal is important for eliciting the thinking of the students that will lead them to the mathematical ideas.
4. Plan for misplacement. It is important to also plan for misplacements and how those will be handled. While it is hard to anticipate all the possible ways students may misplace the numbers, it is important to pre-think about misconceptions that may come up and how they will be addressed. Some misconceptions can be addressed with guiding questions, but some misconceptions may need additional scaffolds, such as connecting to another concrete representation. It can be hard to react in the moment when you’re caught off-guard with a misconception, so thinking about possible misconceptions before enacting the yarn number line routine can help you give better feedback.
We have created a planning template that’s helped us in planning yarn number line routines.
Enacting Yarn Number Line
When launching a yarn number line, it is important to get students thinking right away about the choices they will make for placing the numbers, and the norm of providing reasoning and justification. Here is an example of how we launched the yarn number line during Fraction Lab with teacher/adult observers:
Teacher: Ok, so I’m going to do something that we’re going to do a lot of times this week. So we’ve got this line right here, this yarn line right here. Does anyone have an idea of what that can be? A line like that? We’re going to do some math together.
Student: Oh, number line.
Teacher: A number line. We’re going to make a number line, and I’m going to invite you. I have these cards, and the cards can sit on the number line, so does anyone want to help me by putting a card up on the number line. Does anyone volunteer? Diego.
Teacher: And how about if everyone else thinks where might Diego put this on the number line? So how about while Diego’s getting ready, why don’t you tell a partner where do you think he’s going to put this on the number line?
(Diego goes up to front and puts a number at beginning of line).
Diego: At the beginning.
Teacher: At the beginning. Why did you put it there?
Diego: Because it’s 0 to…
Teacher: Oh 0 to 1?
Diego: Oh 1.
Teacher: Hmm ok. Does anybody want to put the next number on? Brianna, do you want to give it a try?
Teacher: Is there a volunteer that will put this on?
Diego: It’s not that hard.
Teacher: Is there an adult that wants to put this on, and tell us? Thank you. Do you want to tell us where you’re going to put it and why?
Adult: Well, it’s an open number line, so I can put it wherever, right? I’m going to put it right there.
Adult: It can be moved, right? Eventually.
Teacher: It can be. So here’s the next one. How about if we do this one a little different? How about if you turn and talk to your partners, where should this be on the number line?
Reflecting on the Experience
As we reflect on our experience using a yarn number line to develop and extend student and teacher learning through the instructional routine, we consider two lenses: a mathematical lens and a pedagogical/sociocultural lens. From a mathematical lens, we think about opportunities for students to be sensemakers, to further understand our number system, and see fractions as numbers and how they are an extension of the whole number system student are already do familiar with.
From a pedagogical and sociocultural lens, we see choral counts leveraging students as contributors to a mathematical community of learners. It is the ideas and contributions of the students that are taken up by the teacher and peers to advance the learning of the group. In Fraction Lab, we emphasized the norms that students make sense of mathematics, share their mathematical ideas (whether they are using words, numbers, pictures, gestures, or tools), and listen to understand some else’s ideas. The yarn number line routine gave us opportunities to reinforce the norms and the opportunity for students to practice the norms.
Try a yarn number line and let us know what you learn. We look forward to continuing the conversation! Find us on Twitter: @jenniebeltro @jody_guarino