For the past few months, I’ve been using a “clothesline” as an instructional tool in lesson inquiries to help teachers find ways to engage their students in:

- thinking about whole numbers and fractions on a number line,
- reasoning proportionally,
- wondering about the value of expressions,
- and solving equations.

The clothesline is a simple low-tech visual and effective manipulative for fostering student engagement around ordering numbers on a number line.

Clotheslines allows teachers to:

- use student arguments and reasoning to structure classroom discourse,
- expose student misconceptions and use them to promote student thinking,
- and help students attend to precision.

Just watch this short video of students trying to place 4/4, 1/5, and 5/5 on the number line. What do you notice about student thinking? What misconceptions do you see?

Rich discussion opportunities, right? Did you notice the girl on the left trying to reason out the spacing? What misconception did the boy on the right show?

Clothesline math activities are fun for teachers and students! I encourage you to try them out for yourself. To help guide your thinking, I’m writing up what I’ve learned from my experiences using the clothesline as the backbone of some lesson inquiries I’ve conducted.

This write-up is about my experiences in 4th grade classrooms using the clothesline to encourage students to develop strategies on how to plot and compare values of fractions on a number line (4.NF.1, 4.NF.2). However, this particular lesson pathway is appropriate for 4th-9th grade students depending on their learning needs.

(Note: This work would not be possible if not for the amazing brains and noble efforts of Chris Shore, Andrew Stadel, and Dan Luevanos. Both Chris and Andrew have created video tutorials about how to set up and use the clothesline in the classroom. Andrew’s short introductory video can be found here and is a great place to start. Chris has a whole clothesline math website and offers a more in depth video here. Chris’s video is worth watching for the Hawaiian shirt alone!)

## Some quick background:

I conducted a lesson inquiry using the clothesline with a team of 4th grade teachers. This lesson was taught to 2 classrooms of 4th grade students that had a variety of learning needs. Both lessons were unique experiences with their own twists and turns and choices. The lesson pathway below shows the main flow of the learning experience and took about 80 minutes.

You’ll notices some “clothesline teaching tips” throughout this lesson. They are things we found very useful to pay attention to in the planning and teaching process of this lesson.

## Our wonderings (as teachers):

- Does the clothesline help students see that fractions are like whole numbers in that they have places (addresses) on the number line?
- How precise are students? How granular do they get with placing and spacing?
- What misconceptions do students have about fractions as numbers on a number line?
- How best to scaffold this learning experience and sequence problems to maximize success for all students?

## Learning objectives (for students):

- You will use reasoning skills to order numbers on a number line.
- You will communicate your reasoning to each other.

## Set-up:

We had one clothesline taped on the front board. Students were in pairs/groups and had access to dry-erase boards at their seats to draw their own number lines and show their thinking.

You need to make “tents” with the numbers that you will use and hang on the clothesline.

Clothesline Teaching Tip #1: Both Andrew and Chris have some excellent PDFs on their website to use as “tents” for numbers to hang on the clothesline. I find that it’s often easier and faster to take a stack of about 5 sheets of regular paper (colored and/or white), fold them down the middle (hot dog style, not taco style), and cut the folded papers in to 4 equal parts. This makes 20 slips of paper that you can use for numbers. Make sure to use a dark marker to write your numbers so that students in the back of the room can read them.

Clothesline Teaching Tip #2: Students are more engaged and interested when their group is plotting numbers using the clothesline compared to when they are working at their seats. Consider setting up a second clothesline in the back of the room so you can have two groups working on the same problem using a clothesline to promote more engagement. I’ve been in some classrooms where we’ve set up 5 or 6 clotheslines. (See the update at the end of this lesson.) Be careful though! Each clothesline increases not just student engagement, but also increases more chaos and thinking that needs to be facilitated by the teacher (not to mention you’ll have to make lots and lots of sets of number tents unless you have your students do that as they go along). If this is your first lesson using the clothesline, I strongly suggest just having one at the front of the room. You can add more as you become more familiar with how to navigate clotheslines activities.

## Into:

**Today, we are going to order numbers on a number line. We’re really interested in how you communicate you’re thinking with each other.**

**If you notice, I’ve got this string set up here. This string acts as a number line that we can use to organize numbers.**

**I need a group to volunteer to come up here and help me out. **

Hand student volunteers 0, 3, and 6 while you proceed with the next prompt.

**These students are going to put the numbers 0, 3, and 6 in a place that makes sense to them. At your seats, I want you to work with your partner(s) to draw a number line on your whiteboards and then put 0, 3, and 6 in a place that makes sense to you.**

Clothesline Teaching Tip #3: Have students interact with the clothesline immediately. Give them a low-floor prompt (like this one) that allows them to experience the clothesline at the board, work with one another on their whiteboards at their seats, and internalize the objectives of the lesson.

Give students about 60 seconds to complete this task. Monitor the room to make sure students are following along and how they are deciding to place the numbers. In particular, notice how they are deciding where to place the 3. Then return to the group at the clothesline.

Clothesline Teaching Tip #4: Have students at the board explain their reasoning to you first and have them choose a spokesperson (or two) to explain to the class. This allows students to prepare themselves to figure out what they’re going to say and you can maximize the clarity and usefulness of the discourse.

Bring the class to attention. And then ask the group at the board:** **

**How did you decide where to put the 3?**

Pay attention to the useful language students use. The reasons they give are the same reasons they may choose where to put fractions later in the lesson. “We put it in the middle” and “There needs to be equal space on each side of 3” are two useful replies, but you may hear other valid responses too. Make sure students unpack the reasons why they know that to be true.

In our lesson, we scribed useful words/phrases like “middle” and “equal space/parts” on the board as students said them in discussion.

**What other groups put 3 in the middle for the same reason?**

Make sure groups are making connections between their whiteboard work and the work on the clothesline.

**Can anyone offer a different way to explain where 3 should go?**

This question is useful to invite other groups to share. If you noticed other useful work on whiteboards when you monitored the room, ask for them to share their reasoning as well.

Clothesline Teaching Tip #5: Be open to the unexpected teachable moments. Continue to ponder how students are making meaning.

At this point, I usually give high fives and applause to the group who volunteered and send them back to their seats. I stress that we are applauding their effort, courage, and/or reasoning, not for their answer.

Clothesline Teaching Tip #6: Help reinforce student understanding by asking students to understand an error.

For example, at this point in the lesson, I like to slide the 3 to an incorrect space. And then state: **Let’s say a student put the 3 here. What could we say to this student to help them understand their error? Turn to your group members and discuss. **Then ask for groups to share with the rest of the class.

Clothesline Teaching Tip #7: Andrew recommends using the language “place and then space” when it comes to putting numbers on the number line.

I make sure that this “place then space” language is addressed before the closure of the Into part of this lesson, explicitly putting it out there if need be. And I write the words “place” and “space” on the board as well so that students can use them later in the lesson.

**So what you all are saying is that this student placed 3 in a good spot because it’s between 0 and 6, but they didn’t pay enough attention to the spacing.**

Here’s a video of this moment in a lesson I did. I missed the opportunity to use the question “What could you say to help this student understand their error?” and instead defaulted (ugh!) to a more direct instruction approach. But you might find the video useful. Take a peek!

## Through:

We used the following instructional cycle about 5-6 times to guide student thinking during the middle part of this lesson. We tried to keep each cycle to about 5-8 minutes for each problem we did.

- Write the numbers to be ordered on the board (but don’t write them in order).
- Have a group come up to work on the clothesline on the board.
- Have students work in groups at their tables on their dry erase boards.
- Monitor all groups for student thinking and take mental note of groups who might have something useful to add in step 8.
- Before concluding work time, the group at the clothesline explains to the teacher their thinking.
- Bring everyone to attention. Have the students at the clothesline present their thinking.
- Have groups make connections between their answer and their thinking.
- Have groups share other strategies to place and space numbers.
- (Optional) Students individually record their answers on their personal sheet. Chris has made a handout you can use here.
- High Fives! Repeat the cycle with the next problem with an new group at the board.

During lesson planning, we spent a significant amount of time thinking about a sequence of problems that effectively built on one another. We wanted the problems to be accessible and challenging. We wanted strategies to build logically and started with whole numbers first because we wanted to expose misconceptions early on before diving in to fractions. We did not get through all of these problems in either lesson and had to make adjustments and choices as we went along.

Here are the problems and some notes about our rationale and student thinking we were hoping to see. You’ll also see samples of student work we were able to capture. Students had completed much of their fraction curriculum for the year, so we went pretty deep into fraction concepts and skills. Your problems will probably be different because you may have different learning needs. But we hope that it’s helpful for you to see what we did as an example.

- With 0 and 6 on the number line, plot 1, 2, and 4.
- 2 and 4 divide the number line into equal thirds.
- 1 is in the middle of 0 and 2.
- 1 establishes the unit size. There should be 6 of those units between o and 6.
- Here’s a useful mistake we saw and were able to share with the class. This is an example where the numbers are placed in order, but not spaced properly.

- With 0 and 6 still on the number line, plot ½, 1, and 3.
- Have them graph a simple benchmark fraction. Would they use the same “middle” and “equal” spacing language?
- How will they make sense of where to place 1 in between 0 and 3. Would they talk about thirds?

- With 0 and 1 on the number line, plot ½, ¼, ⅓.
- Would they get the order correctly?
- Make sure we contextualize what the denominator means.
- ¼ in the middle of 0 and a ½
- ⅓ closer to ¼ than a ½

- With 0 and 1 on the number line, plot ⅔, ½, ⅓.
- Practice with other benchmark fractions.
- Do they put the thirds where the fourths should be?
- Do they see the symmetry on either side of ½?

- With 0 and 1 on the number line, plot ⅕, ⅘, 5/5.
- Do they see that 5/5 equals 1?
- Is the space between 0 and ⅕ equal to ⅘ and 5/5?
- Is there the right amount of space between ⅕ and ⅘?
- Will any of them add fractions or plot all the fifths?
- Many students drew area models above their number lines as a strategy.
- The video at the top of this post shows students working on this problem.

- With 0 and 1 on the number line, plot 0, 2/2, 3/2, 5/2.
- Connect improper fractions to mixed numbers.
- Will they move the 1 closer to 0 to create the space needed for the other fractions? These students did not, and it led to a productive discussion.

- With 0 and 1 on the number line, plot 2/4, ⅜, 7/12
- With scary numbers, do they keep their cool and see that they can use ½ as a benchmark fraction. ⅜ is less than ½. 7/12 is greater than ½.
- Check for spacing.

- Possible Challenge: With 0 and 1 on the number line, plot 2/6, 3/9, 8/6
- Will they recognize the equivalent fractions?

## Beyond:

To bring closure to the lesson, we asked students to reflect individually, in groups, and as a whole class.

**What strategies have you used today to place and space numbers in order on a number line?**

We made connections to key vocabulary and strategies that were scribed already on the board.

We concluded the lesson by assessing student knowledge with an exit ticket. We asked the to order ¾, ⅛, ½ on a number line. And also asked them which of these fractions was the greatest and to explain their reasoning.

## Reflections, Takeaways, Analysis:

- Students are excited and engaged about the clothesline. They like anything where they can get up and move and talk.
- Be open to the unexpected teachable moments. Continue to ponder how students are making meaning.
- Conversation is strong at the beginning of the lesson and then it wained. As it got harder, kids withdrew. Did we go too hard too fast? How could we tweek it to find a better balance between challenge and success.
- We need to continue to practice having students stay focused especially during repetitive cycles. How could we change the cycle to keep more interest going? Maybe we should do less problems?
- Clotheslines allow students to “groove” more solid understanding into their prior knowledge (and identify misconceptions) about number lines.
- Students have an opportunity to express their mindset, attitude.
- How do we get more kids engaged fully at all times? More number lines? Smaller groups? Clearer roles? Should they do area models as well? We should assign roles?
- Clothesline Teaching Tip #7: Consider assigning a spokesperson role to each student group. Have students take turns being spokesperson. The spokesperson is responsible for sharing the group thinking with the rest of the class.
- It was really telling watching them struggle even with ½. It makes sense about they might struggle. They curriculum they use almost always put ½ on the number line for the students.

## Update:

A teacher in the lesson inquiry wanted to increase student engagement by making more clotheslines in his classroom. Check out the images below to see how he did it!

## An invitation:

As always, feedback and comments are welcome. What inspired you? What opportunities did we miss?

Help us get better together.