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 allow 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 4th grade students trying to place 4/5, 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 three 6th grade classrooms using the clothesline to encourage students to develop strategies on how to solve equations and reason about the value of expressions. We addressed many of the 6.EE standards. However, this particular lesson pathway is appropriate for 6th-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 6th grade teachers. This lesson was taught to 3 classrooms of 6th grade students that had a variety of learning needs. All classes had covered the proportional reasoning standards. One class had not yet started the unit on expressions and equations while the other two were in different places in the expressions and equations chapter. All three 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 60-70 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):
- What do they remember about proportional reasoning strategies?
- How well does the clothesline keep students engaged?
- How does the clothesline help us understand how students are making meaning of equations? What misconceptions do students have?
- How best to scaffold this learning experience and sequence problems to maximize success for all students?
Learning objectives (for students):
- You will use number lines to show your thinking about the values of expressions.
- You will communicate your reasoning to each other.
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 (tents) 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. Be careful though! Each clothesline increases not just student engagement, but also increases more thinking (chaos?) 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 (part one):
(Quick Note: Because students had not seen the clothesline manipulative before, we had a very low-floor and gradual progression through this lesson. The first step was to introduce a single number line and have students plot numbers to get familiar with spacing and placing numbers. The second step was to introduce the idea of a second number line using a familiar concept: proportional reasoning. The third step that comprised a bulk of the lesson was using double number lines to reason about the value of expressions and solving equations. If you and your students are more familiar with single and double number lines, you may be able to start on a higher floor. As a result of this three-step lesson preparation, this write-up is longer than my usual write-ups.)
Today, we’re going to use a number line as a to tool to solve some problems. Number lines can a be a useful tool for some because math gets pretty abstract and symbolic moving forward. It’s hard to see what’s going on. Number lines can help keep things visual and tangible.
But before we start, I want to make sure that we’re good to go on how to use our number lines.
Can I have a student come up here and place 0?
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.
Any place on the number line is a valid place for 0 at this point. There are no wrong answers, but some answers are more useful than others. Most likely your student will put 0 in the middle of the number line or to the left side.
To the student(s) at the board: Why did you choose to put zero where you put it?
The middle space allows to graph integers. But if we’re only doing whole numbers (or positive integers), the left side of the number line may be better.
To the class: Does anyone think that there might be another useful place to put zero?
Discuss as necessary.
Next, hold up number tents with “2” and “-2” on it.
I’d like you to tell me where 2 and the opposite of 2 should go on your number line. If you’re working at your seats, please draw your answer on a number line using your dry erase boards.
We intentionally stressed “opposite of 2” language instead of “negative 2” because we wanted students to think in terms of spacing. We monitored the room checking to see that students recognized the need for equal spacing.
After 1-2 minutes of work time, return to the group at the clothesline at the board and ask them to explain their reasoning.
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.
We then brought the class to attention and then asked the students at the board:
How did you decide where to put 2 and the opposite of 2?
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 0 the middle” and “There needs to be equal space on each side of 0” 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.
What other groups put 0 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 2 and the opposite of 2 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.
We expanded their thinking by moving 2 and -2 to different places, but still maintaining equal spacing.
Would this answer be just as accurate? Yes.
Now I would like you to tell me where 3 should go. Hand a number tent with a 3 to the group at the board. Monitor the room quickly. Check to see that students are spacing correctly. How are they figuring out how much space to put between the 2 and the 3?
We checked with the group at the board and had them explain their reasoning. Then we brought the class to attention and discussed.
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 (usually where the 4 should go). 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.
We then asked for another group to come to the board to try a similar problem. We wanted them to reinforce their thinking and skills.
This time, tell me where 0, 1, 1/2 and 6/2 should go. Show your work on your whiteboards at your seats while the group works on the clothesline on the board.
Hear are two samples we saw:
Yup. They struggled.
We made a choice not to get bogged down here. We unpacked student thinking quickly and used the language about placing and spacing. Teachers made note of which students might need further review about rational numbers on the number line.
Into (part two):
After introducing the clothesline as a manipulative for a number line, we introduced the idea of a second number line. The second number allows for more discussion around the relationship between quantities. Most importantly, the story becomes more interesting when we introduce a second number line because there’s more to talk about. Double number lines can be used to solve proportional reasoning problems and other algebraic equations.
(To get an overview of the problem we did, check out Andrew Stadel’s short video here. It’s really useful and I can’t recommend it enough. He uses some wonderful language to draw learners in to the problem. I insist. Watch it. I’ll wait…….Welcome back! On we go!)
We’ve been comparing a lot of things lately. Sometimes, it helps to have two number lines to show these problems.
We then launched into our own version of Andrew’s Yogurt problem. Because students struggled at the beginning of the lesson, we lowered the floor on Andrew’s task a bit. We also kept a strong story format to the problem.
So you know those ice cream and yogurt places where you get a bowl and you’re allowed to make your own sundae? You get to choose which flavors, how much of each, and all the toppings you want.
I usually act this out too and get in to it. Make sure students are nodding in agreement. Ask them for stores in their neighborhood that may be like this.
So you load up your bowl, and you head to the register, what do they do before you pay? “They weigh it!”
Right! The more stuff in the bowl, the more you need to pay. So I went to one with my family last night. I got 10 oz of dessert.
At this point, I place the unit card “oz” and the number “0” on the left side of the top number line as I’m talking. And then I place “10” near the middle of the top number line. And then on the second number line, I place a “$” card below the “oz” card, another “0”, and a “?” card below the 10. (Again, check out Andrew’s video if this doesn’t make sense!)
How much do you think I paid? Make an estimate.
Quick discussion. And then I share with them that my dessert cost $4. I place the “4” card directly on top of the “?” card.
Do you see how I’ve got the $4 right below the 10 oz? Cards that are stacked vertically like this are equal. So 10 oz equals $4.
This language about equality is important to stress.
Here’s where we shifted to a lower-floor. We did not move directly to the 12.5 oz problem like Andrew does.
My youngest son also got a dessert. But he’s only 6 so he eats less. He ate 5 ounces. Place a “5” between the 0 and 10. How much do you think it cost? Place a “?” below the 5. (I may have missed an opportunity at this moment by placing the cards for them. I could have asked a group to come up and place them, but the lesson was long enough as is. We didn’t want to burn up all their brain fuel.)
Almost all students intuit that the answer is $2. We pressed on students to show visually why the answer was $2.
Convince me that $2 is the correct answer.
We shared visual representations. Not many students found the unit rate and doubled it. Most showed equal spacing and talked about “half of 10 is 5 and half of 4 is 2”.
Y’all seem to agree that 4 ounces equals 2 dollars. Put a “2” where the “?” is. So, the story continues. My wife LOVES frozen yogurt. Hers weighed 12.5 ounces. Place the “12.5” card. How much do you think her’s cost? Place the “?” below “12.5”.
Here, more than half the students really struggled. Like before, we didn’t want to get too bogged down. So we offered an alternative question to work on instead.
The last part of my story is that my oldest son, who’s 16 also came along. He said “Dad! I need $6 for my dessert.” Place the “6” on the bottom number line. I replied, “Jeez son! How much did you get?” Place “?” above the 6 on the top number line.
We then gave students about 5 minutes to work on either/both questions. The second question (16 year old son) helps students reason out another value (15 ounces cost $6). This value then made the first question (wife) more accessible since her dessert was the midpoint between mine and my eldest son.
We then went through and discussed the following problems using the double number line. We placed numbers on the bottom number line because we felt that students needed this structure to experience more success. The fewer the numbers you use, the more thinking students can do.
Our purpose for these problems was to make sure students could visualize equations and use a visual model to solve for variables. We were trying to avoid their crappy textbook which stressed the “subtraction property of equations” without first creating any context for why that property is useful.
What do you think this picture says? x+2=5
Cool. What’s the value of x? Where should you put it? Focus on reasoning.
We had a group work on the board for a few minutes deciding where to place “x”. Then we discussed some reasoning strategies. We were looking for students to connect the words “subtraction” and “addition” with physically moving left and right on the double number lines.
Here’s the solution to this problem. Then we moved forward in a similar way with other problems.
Here are some more examples we used.
We brought closure to the lesson by having students express their reasoning about the following prompt using double number lines. We also encouraged written expressions of reasoning as well.
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.
- 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?
- 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.
As always, feedback and comments are welcome. What inspired you? What opportunities did we miss?
Help us get better together.