# Clothesline Math as Professional Development

“Here are K-1 teachers at Calvert School in Baltimore creating a number line with different representations for numbers 0-10. We had words, digits, ten-frames, tallies, geometric shapes (number of corners), dot patterns, double dice pips, and so much more.”       — Beth Curran

Beth is sharing one of many of experiences that is bearing great fruit … Clothesline Math as professional development for teachers. In her example above, teachers are learning how to use Clothesline as a tool while simulatneously learning about various representations of numbers. In other words, teachers are not just learning about the methdodology involved in teaching with Clothesline Math; they are learning about math pedagogy in general as they enhance their own mathematcial content knowledge. Beth and I are not the only ones discovering this triple triered benefit.

There are those who are strengthening their own number sense on an open number line,

as well as those who were learning where number sense activities fit in the overall curriculum,

and how Clothesline Math fits into a repertoire of number sense tools for the classroom.

Best practices indeed!

# Single or Multiple Clotheslines?

My first experience with Clothesline Math was on a single line, so most of my early lessons were exclusive single number lines. I now make use of the single, double or triple clothesline depending on the nature of the math topic involved.

I must admit, though, I was slow to warm up to multiple clotheslines, mostly because I did not see the advantage of it. I thought, why put up a double clothesline when you can simply pin the values together?

However, when I saw the Yogurt Shop example by Kristen Bennett and Andrew Stadel, my eyes were opened to the instructional potential of multiple lines. The top line in the picture below represents the price of the yogurt serving; the bottom line represents the number of ounces in the serving. The example shows that a 5-ounce serving costs \$2. It begs students to determine the cost of yogurt per ounce.

So how do I decide when a double or even triple clothesline is superior to a single? The criteria is simple: When the units being compared are the same or there is one function being analyzed, go single. When the units, the mathematical functions or the representations are different… go double or triple. In other words, multiple units, relationships or representations need multiple lines.

In our yogurt example above, it would not make sense to pin the ‘5’ to the ‘2,’ so a double line set-up is almost demanded. The units are different, so the two lines also help develop the concept of rate. Another terrific example of using a double clothesline to compare different units is in the conversion between degrees and radians.

Often times, it is not units that are different, but the representations as when comparing fraction, decimals and percentages. In this case it would be appropriate to pin say ‘0.5’ to ‘50%,” but it is more powerful, and less confusing, to have the three representations on three separate lines.

Multiple clotheslines are also effective when comparing different data sets in statistics or different functions in algebra.

For some topics, the multiple clotheslines are demanded for the sake of clarity and concept development. For many topics, it is a nice option, but the single may still work, as in the following example on operations with fractions.

As with all things in the classroom, your professional judgment will be needed to determine the best instructional move. Now that your eyes have seen what I see, you will now be capable to branch out and effectively customize your Clothesline lessons to the mathematical concept of the day.

# Unit Circle on the Clothesline

I recently had a teacher inspire me to take the unit circle (converting back and forth between degrees and radians) onto a double Clothesline.

“I am teaching the Unit Circle for the first time, and I think doing a double clothesline would be so awesome for the Unit Circle! We just taught how to convert from degrees to radians, and vice versa, and I think it would be really powerful to give each student a tent with a degree, have them convert to radians, and place their tent on the clothesline from 0 to 2pi. At the same time, give other kids a tent with a radian and have them convert it to degrees, then place their tent on the clothesline directly below the radians clothesline, to see the parallel of radians and degrees. Are there tents already made?”

— Christina Diaz (Depweg), Montclair High School, CA

Christina, and all other Clothesline Math enthusiasts, the tents for the Unit Circle are now available on our Functions Page. I did this myself in my Algebra 2 class during our unit on periodic functions (though I placed degrees on the top line).

We started by first calculating the circumference of the unit circle (2Π). After I equated 2Π radians to 360 degrees on the two Clotheslines, I asked for an equivalent radian value for 180 degrees. Easy. Too easy. So I randomly chose a student to pick a group to start our Group Chain. The first group chosen had to chose and place the appropriate radian measure for any “Multiples of 90 degrees” that have not been placed, yet. We tied both Π/2 and 3Π/2 back to their place on the unit circle to how 90 degrees partitions the circle into quarters. Since one-fourth of two pi is Π/2, every 90° interval is equivalent to another half pi. The next group was responsible for multiples of 45°, the next multiples of 60°, and then finally multiples of 30°, until both lines were complete with common benchmark measures. Many students pointed out before I could that each multiple “counted up by the same fraction,” either by halves, fourths, thirds or sixths (SMP 8). Another student also pointed out that all the fractional coefficients between Π and 2Π where greater than but less than 2 (SMP 7).

# Welcome to the Clothesline

Welcome to the Clothesline, one of the greatest tools for teaching number sense.

The dynamic nature of the Clothesline makes it more cognitively challenging, yet more engaging and easily facilitated, than a traditional, static number line. It’s manipulability strongly reveals and teaches both proportional reasoning and understanding of variables (it’s two greatest strengths).