# 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!!

# Isn’t Math Really Just How You Look At It?

Over the past year and a half, I have attended numerous CCSS trainings, read the standards and examined the CCSS learning trajectories. It is evident there is an emphasis placed on understanding of the properties of operations in the elementary grades. I don’t know about anyone else, but I remember it being taught to me as a lesson: Commutative Property is a+b=b+a… and such. No meaning behind it, simply some symbols, that if you could memorize and recite each, you were considered successful (as far as grades were concerned) in math class.

Fast forward to my second year as a K-5 math specialist. Having taught nothing below 5th grade in my previous 15 years in education, I am slowly wrapping my head around the depth of conceptual knowledge in grades K-1.  I always knew K-1 was very “hands-on” but I have to admit, I really did not understand the complexity and beauty in the way kindergarteners “see” math until this year.

The other day I did a number talk with a class of kindergarten students. I displayed various dot images with anywhere from  5-10 dots arranged in different patterns. My goal was to have students subitizing the dot patterns and writing addition equations to match the groupings.

I flashed the first dot image on the smartboard for @ 2 seconds and the students wrote the number of dots they saw on their dry erase board. Students shared their answer with a partner and showed me their boards. I put the image back up and asked how they saw (visualized) the dots.  We talked about different groupings, circled the dots for each, and practiced writing a couple equations together.

Feeling confident about the goals i had set for the number talk, i began to rethink them a bit after the following image:

Students quickly shared the answer of seven and then I asked, “How did you see the dots?”

The first student said,”I saw 2, 1 ,1,1, 2.” I had him circle the dots the way he saw them on the SMARTBoard and asked the students to write an equation for that grouping. Many successfully wrote a version (with some backwards 2s) of 2 + 1 +1+ 2+1=7. As I was looking around, I noticed one little girl had written all of the possible ways to arrange the 2s and 1s in the equation on her dry erase board. I realized at that moment, THIS is the commutative property in action! We shared all of the equations and I wrote them on the Smartboard.  I posed the wondering to the class: How can these equations look different but still have the same answer? They talked to their neighbor and the common response was because no dots left the picture…not exactly what I was looking for, but good answer.  I thought maybe it was too many numbers in the equation to see the commutative property or i just asked the question wrong, so i continued.

I asked for another way they saw it. Tons of thumbs went up (this is our sign for having a strategy) and the next student came to the board and circled 5 and 2. She knew it was a five, she explained because of a dice and she just knew two (there was the subitizing i wanted, but at this point we were going deeper). I asked students to write an equation for that grouping. They shared with their partner and we recorded 2+5=7 and 5+2=7. I was excited because two students had already written both equations on their boards before the share out. Now I posed the same type of question, worded differently, “What do you notice about the two equations we just wrote?”

I got responses like:
“The have the same numbers”
“Seven is at the end”