Clothesline Math

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)

During the discussion of placement on the number line, they referenced percents and decimals, so as that came up we clipped them together to allow them to make comparisons.

After many partner talks, and people coming up to change things they felt were wrong, they came to the picture below…

Note: The 8/15 and the 15/29 are in the wrong order, which led to the discussion below.  I didn’t snap a picture of the correct number line :/

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…NL5

Thankfully as I was walking around I saw that this student (and some others) knew exactly where to place each one.NL6

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.NL7

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.

NL8The 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…..



Comment Codes

Recently on Twitter, in the Principles to Actions (#NCTMp2a) slow math chat, the topic of Comment Codes came up.  The first week of March, Question #3 was, “What are some ways to response to student thinking in your classes?”

I remember reading on Fawn Nguyen’s (@fawnpnguyen) website Finding Ways that she grades using a highlighter.  During the Principles to Actions math chat, Frank McGowan (@frankmcgowa) talked about using comment codes.  Instead of writing the same questions and comments on EVERY SINGLE PAPER, Frank attaches a code to each comment or question. I believe he collaborated with his English Department on this.  Then when the assignment is returned, he gives them a reflection sheet which includes the codes.

So here is how I applied the same idea in my class this weekend as I was grading.  Frank, maybe you can offer your insights as well.

I gave a quiz on proportionality.  Every time I wanted to write down a comment or question, I wrote it one time on a separate sheet of paper and numbered them.  It looked like this:

  1. Where would the decimal be in $4?
  2. Is there a more efficient way?
  3. How could you label the picture?
  4. Why divide these numbers?
  5. If it’s 12 ft 6 inches, how would you represent that as a decimal?
  6. what would it look like if you used ratios?
  7. What properties must proportional relationships have?
  8. No sure I understand your strategy.
  9. How could you use the diagram to help you?
  10. What are you multiplying 4 and 7?
  11. What does the 2.5 represent?
  12. How is this like Growing Rectangles?
  13. Can the same situation be both True and False?
  14. Why multiply by 16 instead of 15?
  15. Does your answer represent seconds, hours or minutes?

These questions were all specific to one quiz.  So next to the question they got wrong, I put the number of the question(s) next to the problem so they would know how to address their mistake, or reflect on their work.



It might be better to label them a., b., c., …etc. as when they are numbered it can be confused with point value.  This took less time and I found it to be way more efficient.  You can also make the comment codes key more of a reflection worksheet in which the students have to interact with the questions you asked them to understand their mistakes.  There are lots of opportunities here for students to develop their learning.

Thanks Frank, Andrew and Robin for chatting about this.


Connections with Proportionality Part 2

I have been spending a lot of time with my students on what it means for two quantities to be in a direct proportion.  Instead of working out of the textbook for the unit, I have done the following tasks:

  • Growing Rectangles
  • Vegetable Garden
  • Skittles
  • Track Meet (I didn’t blog about this one because it was a hot mess and I totally anticipated student responses incorrectly.  Although it’s a great task it won’t be in this sequence next time)
  • MARS Proportion or Not a Proportion

There are already many things I would change about how I implemented these  tasks, especially the Skittles Task (which probably could be an entire unit within itself). After the Skittles task, I gave a quiz.  Most of the kids totally bombed it.  I was so confused.  I felt like the Skittles task was so rich that they should have done better. They lacked the ability to make connections when the context changed. Then when I started grading their Skittles packets (they held on to each piece daily and turned it all in at the end), I realized that even though they were participating in the discussions, they were not writing down other student strategies presented in class.  They didn’t record how the ideas and strategies from class related to their own.  And sadly, the implementation of my new homework (HW) reflections has been mildly successful because they are just not doing their HW.  The kids that do however, have a greater depth of knowledge than those that didn’t take it seriously.  There are many reasons the kids bombed the quiz, and it’s not totally their fault either.  Because of this, I decided to do the MARS activity Proportion or Not a Proportion.

I gave them a pre-test to see how they were thinking about proportional relationships.  I graded them but didn’t assign a score as this was strictly formative. I only wrote questions for each problem they got wrong (yes this took FOREVER).  Then we did the MARS activity, which really helped them work through what it means for quantities to be in direct proportion.  After the activity, I gave them a similar quiz and saw a lot of improvement.

Even with the improvement, the bad quiz scores from before still weighed heavily in my mind because I keep thinking, “Man, I must have done something wrong.”  So on a Friday I gave a short response quiz.  I didn’t want the students to wonder, “Why are we doing these isolated activities?”  I knew they weren’t isolated tasks but did they? So I asked them these questions and requested no less than a 5 sentence paragraph for each one:

  1. How are the activities Growing Rectangles, Vegetable Garden and Skittles all similar?
  2. Does Growing Rectangles represent a proportional relationship? Explain how you know.
  3. Does Vegetable Garden represent a proportional relationship? Explain how you know.

This was also a open note quiz because I wanted them to take out all of their tasks, make generalizations and look for similar structure within the problems.  I was thoroughly please with the majority of the responses.




However, not all of the responses were as good as the ones above.  I’d be lying if I said they were.  I realized that my class that struggles with discourse during class has more trouble connecting the activities.  Their responses went something like these:

“The similarity between the Growing Rectangles, Vegetable Garden and Skittles activities are pretty much they same.  The reason why they’re pretty much the same is because the math that’s involved in it.  You have to find out what goes where and for all of them we made a graph that was also the same as the rest of the activities.”

“The Growing Rectangles, Vegetable Garden and Skittles activity all include multiplication, addition and not proportional relationships. The Vegetable Garden and Growing Rectangles only included (multiplying, addition and areas of ft. or inches).  The Skittles activity included multiplying, addition, doubling, division, number lines.”

“All include graphs, numbers, and answers. Also involve lots of thinking. These activities work without a graph but if you didn’t use a graph, you probably got something wrong.  The activities we did use addition, subtraction, multiplication and division.  Growing Rectangles, Skittle activity and Vegetable Garden all have missing numbers and you have to figure out those numbers by divide, multiply, add, subtract, graph even a number line.”

These are somewhat comical.  They are a great example of writing a whole lot about nothing.  However, they show what they don’t know.  So clearly there is more work to be done on my part.

Next Steps:

I’m really struggling with discourse in one of my classes.  I ask them to partner talk and either they don’t at all, or they talk about something else, even when I’m standing right there!  And even when I ask them a low level question that everyone should have an answer to.  Looks like I need to go back into Principles to Actions for some answers. Do you have advise?

Homework will always be an ongoing battle that I am not willing to fight.  I just know personally that the kids that do spend time on their reflections and do them thoroughly have more depth than the other students and will be more successful later.  I guess I will have to be happy with that.

So what now?  I’m thinking of creating a HW assignment where students actually graph on a coordinate plane some Growing Rectangles.  Then they will see that the rectangles that are proportional to each other fall on the same line, which may clear up some misunderstandings. Then they can make more connections with this new graph to the graphs they did in the Skittles activity.

Until next time…S