# How Planning Mistakes Can Lead To Great Student Thinking….

The other day I did this fraction clothesline activity with a 5th grade class and today I had the chance to do it again with another 5th grade teacher, Leigh. It is always so nice to get to have a do-over after having time to reflect and think more about what the students thought about both during and after the activity.

I really thought the conversation was great during the clothesline activity, but it took too long the first time. We noticed that some students began to disengage. To try and improve upon that, Leigh and I decided to give only one card to every pair of students instead of each student having one. However, due to us wanting to keep a few important cards we wanted to hear them talk about, some pairs had two.

I also did not like my placement of 0 being at the very end of the left (when looking at it) end of the string. I moved it out some and talked about the set of numbers that falls on either side of the 0. I felt much better about that this time around!

In the planning of the first clothesline activity, we took fractions from the work the students had been doing with percents and decided on putting 100% in there, completely thinking it would be at 4/4. As the student placed it, however, I started realizing that I never thought about the difference of 100% in terms of the area representations the students had been using versus 100% when talking about distance on a number line. But now, having time to reflect on the card, I thought it would make a great journal entry!

As we neared the end of the card placements, I handed the 100% card to a student and told her it was going to probably cause a lot of discussion but just put it where she thought it went. She said she got it, walked up there and placed it on top of the 2 (the highest number on the line). There were some agree signals going on and some other hands that shot right up to disagree. We talked about it a bit and then we asked them to journal their ending thoughts so we could move on with the rest of the lesson about different sized wholes.

Some thought that 100% was at 4/4 on the number line because it equals 1….

Some thought it was at 4/4, but because of the conversation became a bit unclear…

Some thought it goes on the 2 because it is the biggest number on the number line…

Some related it to different contexts with different wholes…

And one student said it can be anywhere with beautiful adjustments as it moves….

What a great day revisiting my planning mistake!

-Kristin

Today, I had the chance to plan and teach with a 5th grade teacher and it was awesome! Last week, this class had just finished a bunch of 100s grid shading in thinking about fraction/percent equivalencies, so we picked up planning the lesson in Investigations with the fraction/percent equivalent strips. Instead of the 10-minute math activity, we thought it would be really interesting to do the clothesline number line to kick off the class period.

We chose fractions (and one percent I will talk about later) based on the fractions the students had been working with on the grids. We chose fractions based on different comparison strategies that could arise such as:

• Partitioning sections of the line
• Distance to benchmarks
• Equivalent Fractions
• Common Denominator
• Greater than, Less than or equal to a whole or 1/2

We settled upon the following cards:

1/4, 3/4, 4/4, 1/3, 4/3, 5/10, 2/5, 100%, 3/8, 1 5/8, 1 7/8, 4/5, 11/6, 1 6/10, 1/10, 9/8, 12/8, 2

Then, we moved into the fraction cards. We gave each pair of student two cards. In hindsight, for times sake, I would probably only do one card per pair. I gave them one minute to talk about everything they knew about the fractions they had and then we started. I asked for volunteers who thought their card would help us get started and called on a boy with the 1 7/8 card. He went up and stood all of the way to the right and said he couldn’t put his on. I asked why and he said that since the cards were all fractions the line could only go to 1 so his is more than one and can’t go on here. I asked if anyone in the class had a card that may help us out and a student with the 2 card raised her hand. She placed her card all of the way to the right, said “maybe it goes to two” and the other student placed it just to the left of it because, “it is only 1/8 from 2.” Awesome!

We went along with the rest of the cards and so many amazing conversations, agreements and disagreements happened along the way. There are a few things that stand out in my mind as some great reflections on the activity:

1. A student had placed 5/10 halfway between 0 and 1. The next student placed 2/5 just to the left of the 5/10 because, “I know 2 and a half fifths is a half so that means that 2/5 has to be less than 5/10. It is a half of a fifth away.” The NEXT student volunteered and placed 3/8 overlapping just the edge of the 2/5 card on the left. I was expecting percentages to come out, since that was their most recent work with those fractions, however the student said they knew 3/8 was an 1/8 from a half and 2/5 was a 1/10 from a half and an 1/8 and 1/10 are close but an 1/8 is just a little bit further away. Awesome and definitely not what I expected!
2. I wish I had not put the zero so far to the left. Looking back I am wondering if that instills misunderstandings when they begin their work with negative numbers on a number line similar to the original misconception that launched the activity with the 1 7/8.
3. Oh, the 100% card….complete mistake on my part, although it may have been a great mistake to have! In the first class, the student with the 100% card came up and said, “I have 100% and that is 100/100 which is 1” and put it in the appropriate place on the line. Just as she did that, I started thinking how I never really thought about the distinct difference between percent in relation to area (like the grids they had been shading) and 100% when dealing with distance on a number line. No one seemed to notice and since I didn’t know exactly what to ask at that point because I was processing my own thoughts, I waited until another student placed 4/4 on top of it and erased it from my immediate view!
• I stayed for the next class and this time I was prepared for that card and now really looking around to see what students’ reactions were when it was placed. As soon as the student placed it at the 1 location, I heard some side whispers at the tables. I paused and asked what the problem was and they said, “100% is the whole thing.” The next student who volunteered had the 2 card, picked up the 100% card on the way to the right side and put the 2 down and the 100% on top. Lovely and just what I was thinking.

I have never had students reflect on the difference of talking about percentages with distance versus area because I had never thought about it! It definitely feels like an interesting convo to have and a great mistake that I am glad I made!!

I will be back in another 5th grade class tomorrow and will see what happens…it could make for a great journal writing!

-Kristin

# The Clothesline Number Line

The other day Andrew tweeted this post about the use of the clothesline in math class…

“Clothesline is the master number sense maker.” says @timsmccaffrey according to @MathProjects. Good call Tim! pic.twitter.com/bVvvJz7CtX

I love the clothesline activity and have used it with students in the context of fractions, decimals and percents. When I first started using it, it would be up with pre-marked benchmarks on it and as we talked about relationships and equivalencies we would add them in relation to the appropriate benchmarks. For example, 0 and 1/2 would be already up on the line and in class the students realize that 1/4 is half of that distance, we would put the 1/4 on the line between the pre-marked 0 and 1/2. It was always a work in progress and it was a nice visual, but static.

After designing a PD with Jody Guarino, I realized the power in having the pieces be constantly movable. So we started with an empty number line and students drew fraction cards to put on the line one at a time. Now, there are two ways to launch this… either the students know the range of the cards they are using or they do not. I prefer the do not, but I see benefits and places within grade levels/units that each would be appropriate.

Let’s say the students did not know the range of the fraction cards (and they can have whole numbers mixed in there too) they were pulling. The first student is almost a blind guess because they do not know where this line starts or ends, but from there, there is constant adjustment to do.  It is a bit hard to describe in words so I hope this makes sense…

Student 1 draws 1/2 and places it in the middle of the clothesline assuming that possibly the line goes from 0-1. We ask why they placed it where they did and hope to hear something like “I think the line goes from 0 to 1 so 1/2 is halfway between those.”

Student 2 draws a 2, places it at the end, but then has to adjust where the 1/2 is placed because now 1 must be in the middle of the clothesline. Makes the necessary moves. We of course as why they adjusted the way they did and how they determined how close to place the fractions to the other cards.

Student 3 draws 3/2, no adjustments are necessary at this point, but the student must estimate where 1 would be and do half of the distance to 2 and place the card OR take the 1/2 distance and replicate that three times. Either is awesome. Of course, we ask all kinds of questions about their placements.

…and play continues, but if a 3 is drawn, then things shift tremendously…the fun part is students don’t know so they cannot have their spot pre-planned and everything is adjustable!

After Andrew’s post, I started thinking about two things…what could this look like with whole numbers and what are the differences between starting with a set on the line to move around vs drawing cards and placing them where they go?

I tried to sketch out a few moves and think about what students could be thinking about as they went. I am sure I have not captured all of the possibilities, so I would love to hear ideas in the comments!

First sketch, numbers 0-6 placed randomly on the line to start (top line)

So, we could be looking for students to place the beginning and end (0 and 6) first and then I would be curious to see if they placed the middle number before dealing with the other numbers (line 2). Or, would students order left to right, least to greatest (line 3)? Would be interested how they think about spacing the cards in this situation, do they touch? Is there the same distance between each? What would they think about if we gave them a card with 10 on it after they finished arranging them on the line?

Now, what if we started with just a zero at the left and didn’t tell students what range of numbers we had?

The first card they could draw would be a 2, they place it in the middle. We asked why they placed it there and maybe hope to hear something like “I think the line goes to 4, so 2 is the middle of 0 and 4.” (Line 1)

Then, let’s say the next student draws a 6. They place it at the end, again assuming it is the highest number and then has to adjust the 2. We ask what their thinking was and hope to hear something like “Since 6 is the end now, 3 must be in the middle and 2 is less than 3.” There could also be some questions around how close the 2 is placed to the middle versus toward the 0. Cool stuff. (line 2).

Next student draws a 10, now the 6 shifts left, just right of the halfway point though because that would be 5. Then the 2 has to shift because the 6 did. Now, interesting questions about all of those moves come out..How did you decide where to place 6? Why did that affect the 2? How does the distance to the 2 relate to the distance to where the 5 would be? to where the 6 is?

Andrew played around with the clothesline with his son here. Great stuff and so many ways to think about how we could use this in our classrooms across all grade levels!

Thanks for the inspiration to blog this Tim, Chris and Andrew!

-Kristin