## Monday, August 24, 2015

### Clothesline

Dear Chris Shore,

Thanks for introducing Tustin Unified to Clothesline during your awesome professional development workshop!
You, Math Projects Journal, and Clothesline rock (David Lee Roth style)!

With all of my math heart,
Andrew

*Check out Chris' post on Clothesline (link coming soon).
**I highly recommend inviting Chris to your district/school for math workshops.

Where do I begin?
I used to have a number line in my old class. But it was static. All of the benchmark numbers were taped to the wall. I used it often, but not often enough.

Flash forward to Chris' workshop last week. He introduced Clothesline using this great quote from Tim McCaffrey:
You better believe my ears perked up when I heard "master number sense maker". Check it out!

When I worked with teachers in Irvine today, my ice breaker was asking the teachers,
"How long does Dyson think it should take you to dry your hands with their machine?"
I've never posted this picture online because it's my favorite ice breaker to do with teachers at conferences and workshops. However, I will use the context to illustrate how I introduced Clothesline to about 150 teachers, coaches, administrators today.

As teachers were discussing, I went around and asked
• two teachers for a guess that's too low (3 and 5 seconds)
• two teachers for a guess that's too high (30 and 40 seconds)
• four teachers for a guess that's just right
I used a big black marker to write the numbers on the red papers for the first four teachers with their "wrong" answers. I then had them place the red papers on the 25 feet long clothesline hanging on the side wall. I had the last four teachers write their "just right" guesses on the green slips.

You'll notice the green slips might be hard to read from a distance. I did this on purpose so that we could get the visual effect (from a distance) of placing their green slips accurately on the clothesline using correct spacing (sorry, no pictures), based on where the red slips were originally placed. I reminded the teachers that this is a dynamic number line. You can move the numbers along the clothesline as you please. Please note that the teachers (in this case) were doing all the work, thinking, critiquing, and adjusting. In other words, students should be doing the same in my class as I help facilitate the conversations.

Throughout the rest of the day, I worked with teachers grades 7-12 during three breakout sessions. Therefore, I made a handful of cards for each breakout session to correspond to numbers or expressions relevant to them and their content area. Here's a sample:

I love when Chris used colored papers to focus on the numbers being placed (or in question). Notice the "benchmark" numbers are on white paper.

THE ROPE
The clothesline is 25 feet long. I think this is plenty long. I went to Home Depot and bought a 100 ft. clothesline. I made three cuts to make four lines of 25 feet. I used a flame to burn the ends of the rope so they stay in tact.

THE CLOTHESPINS
I noticed Chris used these to stack equivalent values together vertically. Brilliant. See above for examples that could stack.

THE NUMBERS (or expressions)
NCTM suggests using 3x5 cards, but then you have to use more clothespins, making the number line more static. Chris suggested using strips of paper. Using strips of paper allows the number line to be way more dynamic, allowing the numbers to slide along the clothesline or making it easy to place the numbers or take them off without the use of clothespins.

VARIABLES
Many teachers loved the idea of using variable expressions. Here's how I determined x for each group. I asked the three teachers in possession of the variable expressions to share how long they had been working in the district. For example, three teachers shared 12, 13, and 15 years. Therefore, 40 was the value of our variable, x.

FUTURE USE
Whenever I give a professional development workshop for teachers from now on, I will be using Clothesline. IT'S AWESOME! It is a master number sense maker. If I happen to be at your district or school doing PD, I'll bring a handful of clotheslines to raffle off (or give away). At the end of my last session today, it was awesome to have two excited calculus teachers be extremely thankful for receiving a clothesline. One walked away saying, "I'm going to use this Wednesday." Their first day of school! Calculus!

Last, but not least, test it out at home if you have the chance. My five-year-old son and I had fun this past weekend. He threw me a few surprises.

Clothesline 1
I just tossed up a few numbers on the clothesline for him to first get acquainted with the idea of numbers on the clothesline. "Move the pieces of paper so they make sense to you."

Clothesline 2
My son caught me off guard when he pursued something he was interested in. 1, 2, 3, 6.
"What?"

Clothesline 3
I wanted to see how my son did with spacing the numbers.
"Show me where four and five go."

Check out Kristin Gray's great post from the other day. I love how much she anticipates student thinking in preparing for a successful Clothesline activity.

Clothesline,
930

1. Wow. I love the number sense we can talk about through this post. Teachers and students will all benefit.

1. Right on! Thanks Tammy!

2. We have been using this idea for a few years now. I love the idea of using it with functions (place x on the line and then ask where 2x or x/3 is). We have a set of tent cards that we re use by just putting post-it notes on them (you've probably got a few of those). You can see an image of this on the banner of my blog here http://engaging-math.blogspot.ca/

1. Right on! Thanks for sharing. Have any strategies to share with helping students space the numbers correctly?

2. It's really just practice. We leave the string up all the time and that makes it easy to ask them to "put a fraction between 0 and 2 up" or something like that on the fly and then have that discussion around relative vs absolute position often. The fact that the kids don't have names attached to the numbers means we can point out errors without calling out the students individually.

3. Love this! I've used my smart board to have students move numbers around on a number line, but I think I like this better.

Also, love the idea of raffling off a clothesline after introducing the idea at teacher PD. Totally stealing that idea! ;)

1. Right on. Let me know how it goes!

4. I was curious how well thing would pan out if we did a system of equations (other than equal values) with the clotheslines... I'm still not able to wrap by brain around it. Thoughts?

1. I might need some clarification just to be sure we're talking about the same thing. When you say "system" I think something like y=5x-7 and y=3x+9 and I would think there is some overlap with solving the equation 5x-7-3x+9.
With the system, it's been established there are two different variables and we typically use the coordinate plane where the two axis represent x and y.
However, with the equation with variables on both sides, the focus can be just the unknown value of x. So how does this work with a double number line?
I use the top number line as the placement of concrete numbers. The bottom number line is used for the algebraic expressions and how they relate with their location in regards to the concrete numbers above them.
I'll post some examples soon.

5. Hello! This post was recommended for MTBoS 2015: a collection of people's favorite blog posts of the year. We would like to publish an edited volume of the posts and use the money raised toward a scholarship for TMC. Please let us know by responding via email to tina.cardone1@gmail.com whether or not you grant us permission to include your post. Thank you, Tina and Lani.

1. Thanks for the comment. I appreciate the request and think the TMC scholarship is an awesome idea!
Yes, I grant you permission to include the Clothesline post in your collection.
Let me know if you have any further questions.

6. Thinking of using this when we introduce exponent rules... asking Ss where would you place (x^2)^2 versus x^-4, etc...to see if ,in one variable, students understand, what the rules "do"...
any thoughts on that?

1. Great question. Not knowing what grade level you teach, I'm wondering if it would benefit to students to start with a more concrete and conceptual way of thinking in conjunction with identifying patterns.
For example, where would we place 3^1, 3^2, 3^3, 3^4 and then ask where 3^0, 3^-1, 3^-2, 3^-3, 3^-4 would go...
Maybe once you have formalized some "rules" from a base of 3, maybe introduce x^2 and x^-4.
What do you think?

2. I love that idea! I teach Algebra I. I didn't think about using it to introduce negative exponents but your illustration makes perfect sense.
So without knowing everything about how the clothesline works, would I hang more than one and use the top for base 3 and then use one of the other "lines" for base x???

3. Great question: off the top of my head, I would use one clothesline to hold the expressions (ex. 3^2) and another clothesline to contain the value (9).
I might JUST do 3^0, ^1,2,3 and then remove ^2 and ^3 so that we can expand the range of the number line so it's between 3^0 and 3^1. Now I would start using that space for the negative exponents.
Does that make sense?

4. I get the two lines with 3^2 and 9 above, check.
I lost you a little bit after that... But If I leave 3^2 on the line, have a student add 3^1 and then enquire about 3^0 and get it placed, we could still explore the patterning in the integers (* or / by 3).
But what were you doing by placing 3^0 and 3^1 on the line? The only thing in between those are roots...

5. Thanks for following up, Madelyne. I definitely made a mistake and didn't check my thinking. I'd leave 0 and 3^0 (as 1) up and then start working on the negative exponents.

I'm excited to see what you come up with.

7. I am curious to know what is the first activity/lesson you do with your students to introduce and understand what a clothesline is in class.

I teach 8th grade where they will work often with real numbers and estimating square roots and cube roots and I can see this being a great understanding tool but am not sure how to introduce it.

I've found it helpful to introduce students to the clothesline using carefully selected whole numbers and integers. For example, -8, -3, 0, 2, 8, 10
This way students can get familiar with placing and spacing the numbers while recognizing numerical relationships between the numbers.
Hope that helps.

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