Sometimes you learn about a teaching tool or technique at just the right time. That’s what Clothesline Math was to me in the past week. I had known about Chris Shore’s work with #ClotheslineMath (think open number line) for a while now, but I only knew the basic number line stuff he had started with. I happened to attend his session at the RSBCMTA #FallforMath conference, learning how many new ways he has taken this concept (Clothesline Math grew up!), and it was just what I need in both of my classes this year (PreCalc and Calculus) at that moment.
I was just about to start exploring the Unit Circle with my students. I know they always struggle with radians and making sense of the fractions. So, I started with a basic fraction Clothesline math for 5-ish minutes at the end of class one day:
It was amazing. This short number line actually took more than 5 minutes, with rich discussion among groups. It was amazing how many strong math students in Pre-Calculus wanted to say 1/2, 1/3 and 1/4 were equally spaced (probably due to the 2,3,4 in the denominators).
The next step was to make a Double Clothesline (!!!). We started making the top one in degrees, going from 0 to 360. Students had to put the other “common angles” on the number line proportionately. Then, we started a second number line below that in radians, going from 0 to 2pi. We talked as a class that 180 degrees would be where pi (or 1pi) would go, and then students had to figure out the rest using fraction reasoning. Here’s what it looked like by the end:
We did not finish because I did not realize how long it would take students. So we came back to this and re-did both number lines again at the start of next class. Worth. It.
In the following days, I have never had students so solid at reasoning through what fraction of pi each of the angles is. The manipulation of the clothesline, the time to really reason through it on their own, and starting with a horizontal number line before moving the fraction reasoning to the circle all contributed to making this a worthwhile use of time. I will never teach the unit circle/radians again without Clothesline Math!
We had just started limits, and I spend the entire first day just having students make tables to see what the y-values are approaching on the table. So if we are talking about the limit as x approaches 4, for instance, students would make a table with an x-column including numbers like 3.8, 3.9, 3.99, 3.99 as well as 4.1, 4.01, 4.001, etc. I really want to emphasize how we are “squeezing” in around a number.
Students do pretty well with these tables. However, many always struggle when we have a problem where we are finding the limit as x approaches 0. Students usually make a table with 0.1, 0.01, 0.001 just fine, but on the other side of 0, they choose -0.9, -0.99, -0.999, etc. They get so into the habit of .001s and .999s with other numbers, they struggle with things reversing in the negative numbers, and being especially unique around 0.
Cue clothesline math! After this first day of tables, we started the next day with a short clothesline math activity. Students put some whole numbers (0,2,3,6) on a number line. Then organized the same fractions that I had used in PreCalc (1/2, 1/3, 1/4). Lastly, I asked them to put the following on a number line:
0, 1, -1, 0.1, 0.5, 0.9, -0.9, -0.4, -0.001
They got it. They just needed the time to refresh their understanding of the number line and strengthen their number sense. I also think some of them just needed permission to draw a number line anytime they want in the future. (This activity helped make drawing a number line seem not-so-juvenile/really important.)
Some quick tips:
- I actually started clothesline math in each class with a “easy” set of numbers: 0,2,3,6. It was a great “easy” way to introduce it and help students understand the idea of scale/proportionality. Plenty of students struggled at first with just these numbers.
- I made the numbers on the clothesline by folding over index cards. Quick and easy.
- Each round (each set of numbers) I had one group come to the front (a different group each round) to put the index card numbers on the clothesline. The other groups were all working on small white boards, drawing a number line and discussing with their group where the numbers went. Thanks to Chris Shore for this pro-tip! (See example below)