# Adding & Subtracting: Tools and Representations

There is always a lot of talk about students using an algorithm, process or set of rules, for addition and subtraction. Whether talking about “any algorithm” or “the algorithm,” I am certain, in most cases, people are talking about a process that is absent of tools such as a 100 charts, number lines or base ten blocks. But, what happens when we see the tool becoming an algorithm in and of itself? Can moving left or right on a number line, making jumps of 10s and 1s, writing problems vertically, or jumping rows and columns become an algorithm where students lose sight of the numbers themselves because the process is one more thing to learn?

This was the exact conversation I had yesterday with two 3rd grade teachers as I was leaving school. The students had been playing a game called Capture 5 and struggled making various jumps on the 100 chart. The teacher, understandably, was concerned students were confused about adding and subtracting tens and ones. After more conversation, we began to wonder if the students saw the 100 chart as a set of rules to follow in order to add and subtract instead of a place to look for patterns and structure as we add and subtract. Were they getting caught up in the left, right, up, down movement and losing sight of what was actually happening to the number?

As I thought more about it last night, I wondered about other tools and representations  students learn that could easily turn themselves into an unhelpful set of procedures. I also wondered how often we make connections between these tools and representations explicit. Like, when is one helpful over another? How are they the same? How are they different?

I emailed the teacher my notes (below) and we decided we would try this out this morning.

If you can’t interpret from my notes, the plan was to have each student in a group using a different tool or representation as I called out a series of operations to carry out. After the series of addition and subtraction, they compared their answers and discussed any differences. They rotated seats after each series so they had a chance to try out each of the roles.

We came back together to discuss their favorite one. The recording is below…what do I have against writing horizontally, really?? I found this entire conversation SO incredibly interesting!

• They found the base ten blocks to be “low stress” because they were easy to count, move and trade, but did agree that bigger numbers would be really hard with them because there would be too many.
• They really did not like adding and subtracting on the number line with multiple jumps. It got messy.
• They liked mental math because there was nothing (tool) to distract them and they could focus but they didn’t like that you couldn’t check your answer.
• The 100 chart’s only perk was they didn’t have to write the numbers in, they were already there for them.
• I really loved that they mentioned the equations were they only way they could track their work. So if someone in their group messed up, the equation person was the only one that could help them retrace their steps easily.

I am not sure what I learned today, I am still thinking a lot about this. I know that I loved having them compare the tools and representations and that the teacher felt much better about their ability to add 10s and 1s. I feel like there are so many other cools things to do here, but my brain is fried today so that will have to wait!:)

# Fraction & Decimal Number Lines

Recently, our 3rd, 4th, and 5th grade teachers had the opportunity to chat math for 2 hours during a Learning Lab held on a professional development day. It was the first time we had done a vertical lab and it felt like perfect timing as 3rd and 4th grade would soon be starting their fraction unit and 5th would be entering their decimal unit. Prior to the meeting, we read the NCTM article, “Identify Fractions and Decimals on a Number Line” by Meghan Shaughnessy, so we started the meeting discussing ideas in the article. We then jumped into playing around with clothesline number lines and double number lines, discussing what they could look like at each grade level based on where students are in the fractional thinking.

I have co-taught number line lessons in both 5th grade and Kindergarten this year, but both were a bit different in not only number, but organization. In 5th grade we used one clothesline with the whole class, while in Kindergarten we used tape on the floor and students worked in small groups. Leigh, 5th grade teacher, was interested in trying the small group number lines on the floor. As we planned the lesson, the one thing we thought would be difficult about having small groups is getting around to each group to hear their conversations, especially when we were planning cards purposefully to address misconceptions and misunderstandings. However, knowing we would have the two of us circulating, as well as two 3rd grade teachers who wanted to see the lesson (yeah!), we knew we had plenty of eyes and ears around the room to hear the math conversations.

During the lesson, in groups of three, students placed 24 cards on their number line. There were two sets of cards, so after placing all of their cards, each group visited a number line with a different set of cards to discuss. Instead of boring you with all of the number choices we made, here are a few of the choices in cards and the reason(s) we chose them:

1/3 and .3: Students often think these two are equivalent so before the decimal unit we were curious to see how they were thinking around that idea and how they used what they knew about fractions or percents to reason about it.

0.3, 0.33, 0.333, 1/3: The 1/3 and .333 were there to think about equivalency, while the others were there to think about what is the same in each and how much more each decimal has to make it larger. Which you can see caused some confusion here:

2.01 and 2.08: We were curious about the distance they put between these two cards.

1 6/9 and 1.6: We wanted to see how students compared fractions and decimals when they can’t easily convert 6/9 into a decimal. Then, if they began thinking 6/9 and 6/10, how would they decide on the larger fraction and then how much distance do they put between them?

The group below practically had them on top of one another:

While this group had a bit of a space between them:

2.8 and 2.80: Tenths and hundredths equivalency. They all seemed to handle this with ease.

.005 and 1/100: Curious to see the placement in relation to the other numbers. This 1/100 is close to 0 but I wonder about it in relation to the .2. Definitely a conversation worth having!

2.8 and 2 7/8: To see how they compared the 7/8 to the 8/10.

After they visited other number lines, they had a chance to meet with that group and discuss card placements they agreed with and placements they did not. Groups then made adjustments accordingly…

Here was a group’s completed number line and my first stab at panoramic on my phone!

The journal entry we left them with was, “Which cards were the most difficult to place on the number line? Why?”  Many were just as we suspected.

The conversation as I walked back over to the other building with the 3rd grade teachers was, what does this look like in 3rd grade? Could we use array images to place on the line instead of the fractions? Could the pictures include over 1 whole? What whole numbers would we use? Do we play with equivalent pictures with different partitioning? Being mindful of the students’ second grade fraction exposure, below, we are planning on trying out something very soon! I am thinking the cards like these on Illustrative, with the pictures but no fraction names at this point.

CCSS.MATH.CONTENT.2.G.A.2
Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.
CCSS.MATH.CONTENT.2.G.A.3
Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.
…and then could the journal could be, “Can you name any of the fractions on your number line? How do you know?”…or something like that!

# Kindergarten Number Lines…The Lesson

Two days ago, I planned this kindergarten lesson with Nicole and we taught it today! It was so much fun and I just have to say, I have such an admiration for Kindergarten teachers..that hour was tiring!

The Number Talk was a sequence of two dot images, both showing 7. It always amazes me to see the students counting, explaining their counting and writing equations so beautifully this early. In both images we heard counting by ones, counting by “2’s and 1 more,” and saw students count by ones and twos in different orders, solidifying the concept that the order in which we count does not change the total dots in the image. There was such a wonderful culture in place where students were open to agree, disagree, share answers (right or wrong) and all of this was shown to be valued by Nicole.

Next, came our number line adventure. Nicole had strips of painters tape around the room and sent each group of 4 to their assigned tape. As Nicole handed every group the first card, we (Jenn Leach, another Kindergarten teacher, Nicole, and I) walked around to ask students why they placed the card where they did. In keeping with the plan, the number order and observations were like this:

• 1 – Every group except one placed it on the far left. It was interesting to me that each group put the card under the blue tape, not on it.
• 10 – This was a great one to watch. One group put it at the very end of the tape, others “counted out” from 1 to ten to approximate where it would go, and some just put it in the middle without much of an evident strategy. When we asked the groups that placed it in the middle, they said they needed to leave room for other numbers. I asked what numbers would go over there and they said, “big ones, like 100.”
• 0 – They all shifted the 1 card to the right and replaced it with the 0. I saw one group have a group member place it at to the very left of the blue tape, just before the blue tape actually started and a group member said, “That would be a number if you put it there, but zero is a number” as he moved it under the very beginning of the tape. So cool.
• 3 – This is where some serious shifting happened. I didn’t get to see all groups do their moving, but as I walked around, I did see the 3 very close to the 1 and all of the tens that were at the end of the line, moved down. It seems their spacing strategy had taken over.
• 9 – All of them attached it to the left side of 10.

Before we gave them 5, where we really wanted to see how they dealt with the half, Jen, Nicole and I convened quickly to figure out how we were going to see that. We thought the ten at the end would be much easier to see their thinking about 1/2 so we decided to tell the students that 10 was going to be their biggest number to see if that changed their line. We got a couple, “Ohs” and slides of the 10 and 9 to the very right end.

• 5 – Most went back to counting spots but I did catch a couple groups looking at spacing. One group was using the 1 card to decide on the spot for a 5 while another group said they knew 5 and 5 was ten but was having a hard time using that to place the card.

Because we were running long on this part, we gave them the rest of the cards to place, finalize and tape down. This is what a few of them looked like (the others were all like the third pic):

The above group worked from the right.

Loved the extra space before the 0 here!

This was by far the most popular line!

Then we had students walk around to other lines and talk about similarities and differences to their line. It was great to see the group who started on the right notice that the other groups started on, “that end” while the spacing was a huge topic of conversation. One little girl, whose group had placed all of the cards touching, said she knew why they spaced them out….”They took a breath. Like one, take a breath, two, take a breath, three, take a breath…” I had never thought about how the visual could impact the way we think about timing in our counting! The closer they are the quicker we count, the more spread out, the slower we count. Loved it!

We regrouped on the carpet and talked briefly about what they noticed….

• All of the groups went started at 0 and went to 10.
• They all went in order, “Not, one and then four and then three and then two…”
• Some were spread out far.
• Some had the cards squished together.

All really important ideas! Next we went to our big clothesline to play around. Nicole placed the zero all the way to the left and I placed the ten all of the way to the right and said, for this part, we are going to say the zero and ten cannot move. Each pair of students (each from a different original group) got a card to talk about for a minute and then we called them up in the same order as the individual activity to place the cards. It started off all shoved to the left until one little girl went to place her number and started spacing them all out so it “looked better in her brain.” We asked the others what they thought about that. Some said, “it looks right” (says a lot about how equal intervals are visually appealing and seem instinctual for some) while others said they need to all be “at that end” (attached to the zero). We never reinforced one was better than the others but more that there are many ways we could think about this. I have video, but here is a pic of a piece of the final line…

Then, because we are just so curious to hear about connections they make, Nicole asked if they saw anything the same about the ten frames they have been using in class and the number line. A few students said both had ten and one little girl said it was like 5 and 5. Then, it was pretty awesome…she went up to show it was 5 and 5 and started counting at the zero card, so zero was 1, the one card was 2 and so on so needless to say when she ended her second 5 she was at 9. She said, huh? Loved it! Another student raised her hand and said it was because she counted too many, she started at zero and there is no zero on a ten frame.

it was SOOO much fun and I feel so lucky to get to see and hear all of this amazing math conversations across these K-5 classrooms.

The harder part, or at least what I am grappling with right now, is where to go from here. When it is a lesson within Investigations, I find it quite easy to pick up and move on but since this one is something we did outside of the curriculum, it requires a different plan. I am not quite sure where to go with this, but I have a couple thoughts (and would love others)…

• I wonder if students could think about when the number line would make sense to have all of the cards closer together. Like if a lesson was adding to 20 and 20 was on the end now, what would happen?
• Could we think about measuring things that are really short versus things that are really long? That feels like choosing the appropriate unit of measure to me.
• Could we just leave it up and see if students reference it? and maybe refine the distance between each number?
• Could we find some children’s lit that are around measurement and reference the line?
• Could we put some painters tape in the hallway and see how they interact with it? Could they think about walking every tile line versus the feel of two tiles each time?
• Could they model addition on there? Like in connection to maybe their dot image number talks?

So much to think about and I don’t know if any of these ideas are right or wrong or even age appropriate, but I am loving learning this stuff!! I am just so thankful to have such unbelievable colleagues who love to play around with these ideas with me!

~Kristin

# Kindergarten Number Lines

Today I had a great day of planning with a kindergarten and 1st grade teacher for lessons we are teaching together on Thursday and WOW, has it been such a learning experience for me! The best part is, we have a whole day to get feedback from anyone who would like to offer it before we try this all out!

In Kindergarten, the students have been counting collections, counting dot images in various ways and since I have been obsessed with the clothesline lately, we thought this could be the perfect mash-up! When I read the counting and cardinality learning progressions, however, I did not see anything in there about number lines in Kindergarten but I did find this in the measurement progression:

“Even when students seem to understand length in such activities, they may not conserve length. That is, they may believe that if one of two sticks of equal lengths is vertical, it is then longer than the other, horizontal, stick. Or, they may believe that a string, when bent or curved, is now shorter (due to its endpoints being closer to each other). Both informal and structured experiences, including demonstrations and discussions, can clarify how length is maintained, or conserved, in such situations. For example, teachers and students might rotate shapes to see its sides in different orientations. As with number, learning and using language such as “It looks longer, but it really isn’t longer” is helpful. Students who have these competencies can engage in experiences that lay the groundwork for later learning. Many can begin to learn to compare the lengths of two objects using a third object, order lengths, and connect number to length. For example, informal experiences such as making a road “10 blocks long” help students build a foundation for measuring length in the elementary grades.”

In thinking about this, I tweeted out about number lines in Kindergarten and immediately was reminded by Tracy of her post on this work from last spring! Awesome stuff! I sent the link on to Nicole, the teacher I am planning with, and we were both filled with so many ideas! We were both thinking about relative location on the number line but hadn’t thought more specifically about the equal distances between each number! We also were originally going to do the number line as a whole group, but after reading Tracy’s post we changed our plan to allow for more discovery and exploration of the number line!

Here is the plan….

• Students will be in groups of 4. Each group will have a strip of tape on the floor in different areas around the room.
• We decided to put the tape the length of 5 tiles to see if any group uses the tiles in thinking about space.
• We will hand each group the same card one by one and ask them to decide, as a group, where it should be placed. We went back and forth with this one…we wondered whether we should just let them start placing, but we really were so curious to see their moves and adjustments with each card. We also thought that since they have been ordering numbers lately, the majority would just put each card next to one another on the line.
• Now, the order of the cards…this was so much fun to talk about….
• 1 – to see if they place it at the beginning and then the adjustment when 0 comes up.
• 10 – to see if students put it at the end of the line and how they determine the distance from 1
• 0 – to see if students place it to the left of 1 and if they have to move the 1.
• 3 – to see if students but it closer to 1 than 10, how close to 3 they place it, and if they put it less than half.
• 9 – to see if students think about 1 less than 10.
• 5 – THIS IS THE CARD I CANNOT WAIT TO SEE! Since they have been doing ten frames so much, some students are comfortable with 5 and 5 is 10, so do they apply that logic here?
• 7 – to see if students put it right in the middle of 5 and 9.
• 6 – one less than 7 or one more than 5.
• 2 – between 1 and 3.
• 4 – again, one more or one less
• 8 – same
• During all of the placing time, we will be listening and recording any important ideas we want to have students talk about when we go to the whole group discussion.

After each group has placed the cards, we will have them do a gallery walk to the other groups’ lines and ask them to talk about what is the same, what is different at each line. We will then gather on the carpet.

We have a clothesline up, much longer than their strips of tape to do the same cards as a whole group. We will give each pair of students one card to talk to each other where they would put it (based on their work in the earlier group work). *Something we did not think of until I just typed this was how we partner the students up…we should match them with a student from a different number line to vary the convo.

We will call the cards up in the same oder they did their group work and ask the pair to explain where they decided to put their card and why. After all the cards are placed, we will ask them what was important to them as we made our number lines and record that for future conversations.

As a future conversation, we thought it would be really cool to see what connections the students make between the number line, ten frame, and dot images they have been working with so much!

Also, if anyone knows of a children’s book that has something moving a distance of 10 or 20 units, I would love to hear about it! Every single book I read dealt with 10 as collections of things, never distance.

-Kristin

Too late to type up the 1st grade one now, but it will be around this Dot Addition game in Investigations: http://www.smusd.org/cms/lib3/CA01000805/Centricity/Domain/198/Dot%20Addition.pdf Will type that one up tomorrow!!

# 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