My first experience with Clothesline Math was at CMC 2015 in Palm Springs where I attended a session with Andrew Stadel (@mr_stadel) where we were solving multi-step equations on a clothesline. I remember hanging in there at first with the basics, but when he got to the multi-step equations, the session moved faster than my brain could process. This only intrigued me. Then I starting following Chris Shore (@MathProjects) who I think created the clothesline. I was in search of some resources on how I might do this in my class. Thankfully my colleague Lisa (@mathgeeksrock) went to a training and received more information on how to implement the clothesline. So naturally I went to her to figure this whole thing out.
It’s the week before spring break so I definitely didn’t want to start a brand new unit. We are approaching a unit on solving and writing equations. I couldn’t think of a better time to introduce the clothesline than right now, and Lisa agreed. I wrote the values from the cards on the whiteboard so the kids could start placing them on their whiteboards (a piece of card stock in a page protector). After about 30 seconds, I passed out the cards and had students go up and place them on the clothesline. Students could make adjustments on their whiteboards as needed. When we agreed on placing and spacing, they copied the final product onto their worksheet which can be found here.
We started off easy with whole numbers to get them used to the number line. Then we moved to fractions and decimals to connect to equivalency. Their second shot at fractions looked something like this…(Their first attempt was with benchmark fractions and they were pretty accurate)
After many partner talks, and people coming up to change things they felt were wrong, they came to the picture below…
The 8/15 and 15/29 created a great discussion/debate on proper placement. The kids were talking about equivalent fractions, the difference between the numerator and denominator, the size of the denominator and what that meant in terms of the size of the pieces and how all of this related to 1/2. There were many partner and table talks that happen with this. Although this may sound like a script between me and one or two students, it was a class discussion:
Me: So how do we know if the 8/15 and the 15/29 are in the right order?
Student: Half of 15 is 7.5 so 8/15 is 0.5 over what the half way would be.
Me: So how does that compare to the 15/29?
Student: Well, half of 29 is 14.5, so 15/29 is also 0.5 over.
Me: So what does that mean?
Student: That both fractions are 0.5 over the half way point.
Me: So how do you know which is bigger then?
Student: What happen if we changed 8/15 to 16/30 by using a giant one?
Me: How does that help?
Student: Well then you can see that half of 30 is 15, and 16 is one over half, and therefore would be larger than the 15/29.
Student: But what if we do the same thing to 15/29?
Student: We got 30/58.
Me: How does that help us?
Student: It doesn’t really. The giant one makes 15/29 equal to 30/58, which is one more than half since half of 58 is 29.
Me: So now what?
The students seemed frustrated at this point because they thought they had an effective method of determining which was larger. The period was coming to a close so I asked them, “How can the denominator help us figure this out?” In a table talk, students realized that fifteenths are larger pieces than twenty-ninths so therefore, the 8/15 would have to be larger than 15/29. I was satisfied with that response, but sad I didn’t have time to talk about this more. The students begged to check with a calculator using division. They felt relief when they found out they were right.
When we moved to variables the next day, they were fine with the variable placement, until I made x equal to 4. I gave out the whole number cards of 2 and 1 which threw them for a loop (see middle picture orange cards to the right). They weren’t connecting the whole numbers to what would be the equivalent x term. They didn’t see a relationship between 1/2x and x. Through some partner and table talks, they eventually figured out the ½x and ¼x were related to the x.
Then I gave them some expressions to place. As you can see below, this was clearly confusing to them…
Thankfully as I was walking around I saw that this student (and some others) knew exactly where to place each one.
While students were talking and collaborating I heard a lot of, “If x were _____ then _____.” They were using the same ideas from placing ½x and ¼x to place the expressions.
The last day we tried 3x + 5 = 5x – 3 and 4x – 3 = x + 6. For the first equation we placed 0, 3x, 5x, 3x + 5 and 5x – 3. They did it right the first time. So then I asked them, what if 3x + 5 was equal to 5x – 3? I clipped the expressions together at this point. This created excitement in the room. We talked about distance on the number line and how we could use that to determine equivalency between other terms. For example, the distance between the 3x and 3x + 5 was 5. The distance between 5x – 3 and 5x was 3. So the total distance between 3x and 5x was 8. But it was also 2x. This told them that 2x was now equivalent to 8 and they could now work backwards to figure out x. A sketch of what the number line would look like is below (from my notes). The other problem looked similar. The students applied the same idea of distance between terms and how we could use whole numbers and variable terms to represent the same distance. When I wrote the actual equation they solved on the board, they were amazed. They didn’t actually feel like they were solving multi-step equations.
The kids really enjoyed this week of clothesline math. It’s like I was doing number talks all week. One of my students said, “I wish we could do this everyday, this is fun,” which is always nice to hear.
Thanks to everyone for the continued inspiration and support as we try new things in our classroom. Now off to Spring Break…..