Category Archives: Kindergarten

Both Addends Unknown in Kindergarten

A few weeks ago, I planned a lesson with one of our Kindergarten teachers, Linda. The beginning planning stages and readings behind this lesson are described in this previous post. Based on the work she had been doing with dot image number talks, she was seeing students combining addends to arrive at a sum and also writing equations to match their thinking. After reading the NCTM article, she was curious to see how students would think about the addends when they weren’t right in front of them, as they were with the dots. Also, since we have been reading Connecting Arithmetic to Algebra recently, a lot of our work has been thinking about how students make conjectures and prove claims. This would also be before their Investigations activity called Toss the Chips so I was really interested to see if the movement of bunnies gave a different visual for students than flipping the chips over to different colors?

We posed the image of the bunny cage and 6 bunnies (in my previous post) and did a quick noticing. After noticing the bunny ears and explaining how we need to take really good care of them, the class noticed some really important things around the math: there were 6 bunnies (we counted to be sure), the cage had an inside and outside part, and there was a door for them to go inside and outside.

We explained that with their partner, they were going to see how many different ways these bunnies could be in the cage. Then I asked,”Since we don’t have the bunnies here with us, what could we use instead to help us?” After a suggestion of building a bunny, a couple students suggested the connecting cubes they had on the shelf, perfect. Each pair took their cubes, paper, and pencil and went to work. We purposely didn’t tell them or model how to show or organize their work because we were curious to see how they would do it on their own.

Things I noticed as I circulated:

There were so many amazing ways students organized their information!
A lot of partners started with 3 and 3.
The commutative property was not showing itself at all, so possibly they saw 4 and 2 the same as 2 and 4?
Many partners were moving the cubes as if they were the bunnies to start a really cool pattern but stopping when they got to 6 and 0.

Here are some pieces of work that I thought highlighted my noticings:

This one looked like they were going in a particular order but then jumped to 3 and 3. I loved the labeling on both the drawing and the list! So clear!

This group’s work is not quite as organized as the one above, but they definitely were showing a movement of a bunny. They believed they were finished at 6 and 0, as many did, which leads me to believe the commutative property feels like the same combination to them.

Again, had a pattern going and then jumped back to 3 and 3 after we asked them if they had all of the possible ways.

After walking around the room while they were working, we brought the students back to the carpet to talk about what they did. We started with 3 and 3 and I recorded it every single way I saw it being recorded. This is where I shifted from my initial goal of eliciting all of the combinations to the question,

How are these representations the same or different?

From here it was really nice to hear that “and” is the same as the adding because, “We had 3 and 3 more, so it is 6.” A lot of students easily connected the first and second examples above and we talked about how we could tell which bunnies were inside/outside. I didn’t ask which one was clearer to understand, for fear of making the students feel like their work wasn’t amazing, however the partners of the second example quickly said they could have written in and out over their numbers and it would have been the same. We ended by listing all of the combinations and the commutative property did come out from one group so we ended with them thinking about whether 6 and 0 was the same as 0 and 6.

Things I am left wondering…

Should we have worked with an even number? As I walked around and began looking for any use of the commutative property, I began to wonder how I would have answered questions about 3 and 3. Technically it is the same exact equation, however in context, it would have been 3 different bunnies, so it is not. Would I have written 3+3 twice? I think I would have if the argument came up, but is that something to support in future work?
From here, the students move to Toss The Chips. Do the red and yellow chips reinforce this work or without the context is it different? I know mathematically it can elicit the same discussions, but does the movement of the bunny (as something that moves itself) better support the conjectures of adjusting the addends? Does it not make a difference?
I am so curious to see if the sharing of their organization structures transfer to their Toss the Chips activity. I would like to see them play the game without the table at first to see what they do with it!
How does this thinking change with you play around with more than two addends? (I was so curious about this one that I planned an activity with a K teacher on just this question…that is my next post).

Planning K-5, literally

5 Replies

Tomorrow I have the opportunity to teach a Kindergarten, 2nd and 5th grade class! It is so exciting and interesting to be thinking across all of the grade levels in one day of lesson planning! The most interesting part for me, in thinking through this, is the connections across all of the grades. There is so much potential for conjecture and claim-making supported by their development of proofs.

Background: The 5th and 2nd grade teachers are out at a state math teacher leader meeting so I am teaching instead of the substitute. The kindergarten teacher and I will be teaching it together. I have met with each teacher to chat about where they are within their units and what they have been seeing students do within the current work. I invited teachers both at those grade levels and at other grade levels to pop in if they have the time. I thought it would be great having more people to reflect with after the lessons!

5th Grade: They have just started working with finding a fraction of a fraction using bar models. The initial work is unit fraction of a unit fraction and then moves to non-unit. (My post on that from a couple of years ago on this work, I wish I had done that better, so here is a chance to try something new;) Leigh, the teacher, says they have been really successful in partitioning the bars and arriving at the correct answer. I am thinking about starting with a number routine of either a choral count or a number talk string like 1/2 of 12 = __ of 24… As far as the lesson, I could continue work with this and have students look at noticings after and explore them deeper.They have done these noticings with whole number times a fraction or mixed number, so this could be a revisiting of similarities or differences. OR I could do this cornbread task as a formative assessment as the next piece they will move into is an area model. It may be really helpful for Leigh to see how they are thinking about this before they jump into the work. This is my least planned because I keep bouncing all around with ideas.

2nd Grade: They have been working with even and odd numbers and counting by groups of 2’s, 5’s, and 10’s. All of this work is within contexts of break a group of students into equal teams or everyone having a partner. Tara, the classroom teacher, said the students are really great at determining whether a number is odd or even, however when asked how many would be on each team, a lot of students struggle. They are great if they know a related double fact, however if they don’t they resort to “passing out” by tallies or drawing the picture and physically dividing the number of things in half. For example, if they do not know 11+11 is 22, then finding the number of on each team become passing out 22 things into two groups to find 11. While they are successful in this, Tara and I were wondering why they do not say 10+10=20 and 1+1=2 so 11+11=22. They are able to add 11 and 11 but unable to decompose it as fluently.

In thinking about this, I am inclined to want to connect that addition to halving. I am thinking a counting collection would be fabulous for this. Give students a collection of things to count. Share how we counted them because I am positive they will not count them by 1’s given a large set. We can share as a whole group, record ways in which we counted and determined if our number was even or odd. Then, put the collection back together, switch with a partnering team and then split the collection into two groups. The share would be, “Could you make two equal groups?” “Was your number even or odd? How did you know?” Record strategies. Ask for noticings/wonderings about how they counted and how they divided into two groups.

Kindergarten: The students in this class have been doing a lot of work with ten frames, dot images, counting jars, etc and having students counting and adding to compose a number. They have just begun working on decomposition of number so I immediately thought about the mice activity in Thinking Mathematically. Linda, the teacher, and I planned to do this activity with the students. In preparation, we read NCTM TCM’s article by Zachary M. Champagne, Robert Schoen, and Claire M. Riddell, Variations in Both Addends Unknown Problems. We are going to use 6 bunnies and see how students show all of the ways the bunnies can be inside and outside in a pen. Instead of just giving a context, I was imagining that the students may need a visual of the rabbit pen so I created this image to launch with a quick notice and wonder:

Screen Shot 2015-12-06 at 9.51.28 AM.png

We will then let the students work on finding the different ways in partners and then come back for a whole group share and record the ideas on the board. We are really looking to see two things….1-how they organize their information and 2- the strategies they use. The students will do a notice/wonder about the recorded information. If there is time it would be great to see if students, when given a different number, would apply any of the strategies and/or organizational tools shared.

Going for a run to think through this a bit more! Would love any thoughts/suggestions, as always!

~Kristin

Kindergarten Number Lines…The Lesson

7 Replies

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

7 Replies

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!!

Kindergarten Dot Image Number Talk

Number Routine PD: What Do I Know About…

2 Replies

My colleague Nancy and I facilitated a K-2 afternoon professional development session yesterday afternoon. The 2.5 hour session was with a wonderful group of teachers from across our state who we are fortunate to work with several times over the course of the school year. Our major focus over the course of this school year centers around connecting arithmetic to algebra based on a book by Virginia Bastable, et al, that I blogged about here: https://mathmindsblog.wordpress.com/2014/11/20/articulating-claims-in-math/ I thought blogging about this experience would be helpful for any of our teachers who could not attend and for any others who facilitate PD.

I find planning for professional development is much like planning for the classroom. Many of the same questions arise:

What content will be engaging and relevant? (especially being an afternoon session when everyone is winding down on a Friday)

What is the trajectory of the content?

Where are they? Where are they heading?

What questions or prompts will encourage conversation?

When are points for table conversation? Whole Group conversation?

How will be know where they are in terms of the content when they leave us?

How will we follow up?

After much planning, videoing, and organizing this was the flow of the afternoon:

We opened by getting into grade level groups to discuss the homework from last month, doing a group planned Number Talk with their students. They used this form to plan together and brought back recording sheets of their work to discuss these two questions:

With the number talk being planned by the group, I felt a sense of ownership over the results in the classroom and, really, who doesn’t like talking about all of the wonderful things our students say during a number talk?

We continued with a quick recap of last month’s session on the book, “Connecting Arithmetic to Algebra” to plant the seed for our routine of focus that day, What Do You Know About….?

Now into the really fun stuff! Working with a Kindergarten teacher in my school (@jennleachteach) who is also a part elementary pd group, we planned and videoed a math routine called “What Do You Know About 15?” in Jenn’s class.

We mixed the grade level PD groups up at this point so there was a range of K-2 teachers (and a few math coaches) in each group. They got a blank planning sheet to brainstorm what they think the planning would look like for this routine in a Kindergarten classroom in January. It was great conversation, with the Kindergarten teachers being the experts at each table. I thought this was such an interesting dynamic since we often tend to pose a mathematical idea and ask what previous understandings K-2 need to build to get there, however, with this opportunity, it was starting in the opposite direction and really focusing on what Kindergartners know at this point of the school year.

After they predicted what our planning sheet would look like, Nancy brought 6 teachers up to act as students in a fishbowl enactment of the Number Routine. The other teachers in the room were observers focusing on two particular aspects of the talk, what you notice about the teacher recording and what you notice the “students” noticing. Importance of recording was a previous topic in an earlier pd, so we wanted to be sure that resurfaced. Nancy did the routine with the teachers and we came back as a group to discuss the observations of our focus questions. Our discussion also touched on the use of the talk moves she used to clarify and illustrate student thinking.

We then watched Jenn’s Kindergarten class do the same exact Number Routine, focusing now on the follow up piece of the planning sheet. What did they notice the students noticing? I wish I had permissions from everyone because Jen did a beautiful job in facilitating the talk and her students said some amazing things. We also took a look at the planning sheet that Jenn, Nancy and I had done for this routine. Here is the planning sheet and anchor chart that arose from the talk:

As a group we discussed what they noticed the students noticing that could lead to future “claims” in their classroom. Teachers noticed such things as, “We can count by fives to get to 15” “It is three fives” (of course I am thinking about groups of and multiplication right there!) “A teen number is a group of ten and some more” “Looking at equality with related equations” and “The 1 means one ten”

Jenn then gave students “random” journal entries to see how students were thinking about the numbers after the talk. To differentiate, we decided to give students 12, 19, or 21 depending on where we thought their entry level was into this thinking. After students completed the journals they chatted with someone who had a different number, to talk about their ideas.” Here are the student samples our PD group looked at and discussed:

We ended with Virginia’s conclusion slide about Connecting Arithmetic to Algebra and our homework for the group:

We also gave an Exit sheet to help us in future planning. We got some very useful information as to where the teachers feel they are. I am very excited to hear about everyone’s journey back in their classrooms next month!