# 2nd Grade Counting,Unitizing, & Combining

The other day, I began writing up my lesson plan for a second grade class I was teaching today. I drafted the lesson, got feedback, revised and ended with this plan, around the 5 Practices, going into the classroom today.

I started the lesson, as I planned, with the students on the carpet like they typically are for a Number Talk. I wrote the sentence “There are 12 people in the park.” on the board and asked them to give me a thumbs up if they could give me a math question I could ask and solve from that statement. A couple students shared after a bit of wait time and I was getting a lot of even/odd talk or questions that involved adding more information to my original sentence. I asked them to turn and talk and one little girl next to me said they could find the number of legs. When I called the group back together I asked her to share her conversation with her partner and after that, hands shot up like crazy. It ended with a board that looked like this…

I asked them if we could think about any of these in the same way? I tried to underline the “same thoughts” in the same color, but they started making connections that is got a bit mixed. A lot of there conversation turned to numbers and so I started a new slide and asked what numbers they thought of when they read those problems and why. I recorded what they were thinking…

I really liked this opening talk (15ish minutes) and really didn’t want to let them go when it was time for their recess break in the middle of math class. So, they lined up and left for 30 minutes.

When they got back, we recapped the numbers and then I gave two groups question #1 and the other two groups question #2. They had individual time to get started and then they worked as a group to share their thinking. Knowing that I was going to be trading seats at groups for them to share their problem with another table, I was walking around looking for varying strategies so I didn’t trade seats and have a whole table who solved it all the same way.

They did a beautiful job working in their original group. I saw students who had different answers for the same problem talking out their strategies and arriving at a common answer. I saw students practicing how they were going to explain it to the new table they visited. I saw students who were stuck working through the problem with their tablemates. I can tell there is such a safe culture established by Lauren, the homeroom teacher. They trade seats, shared their problem and then I had to readjust my plans.

At this point, I wanted the tables talking about what was the same and/or different about the two problems but I was running out of time. In order to pick up with that conversation tomorrow, I decided to have them come to the carpet and I chose two papers (of the same problem) that had the same answer but different strategies. I asked the students privately if they would want to share and they were both excited so I put them both under the document camera and had them explain their work. I thought they was similar enough for students to easily see they both drew the figures out but as I walked around I heard the 1st student counting each one by ones and the 2nd student counting by twos after he wrote the equation. I had them explain their work and asked the class to think about what was the same and what was different and we discussed it. Here are the two I chose:

They pointed out all of the similar things such as feet, people, two’s (but were counted differently), and the same answer. The difference was the equation which was an important thing to come up. I saw quite a few students with the correct answer but incorrect equation. A lot arrived at 22 by counting by wrote 7+2 as their equation so that was an important thing that a student pointed out.

I only had 5 minutes left, so I decided to collect their papers and pick up with the sequencing and connections tomorrow. Which I kind of love because it gives me time to be more thoughtful about how they should share them and also time to talk to their teacher about what I saw today.

So, from my previous plan, I am picking up here:

Practice 4: Sequencing

In the share, after each group has presented to the other groups, we will come to the carpet for a share. The sharing will be sequenced in the way I discussed in the Selecting part, asking students during each student work sample how it is similar and different than the ones we previously shared.

Practice 5: Connecting

The connecting I see happening through my questioning as we share strategies. I am still working on writing this part out and looking for the connections that can be made, aside from the picture to number representation connections.

The connections I would love to see students making throughout the work and sharing, is how we can combine equal groups. For example I would like the student who is drawing ones and counting them all to move to seeing those ones grouped as a 2 or a 5 depending on the context. I would love the student who is seeing the five 1’s as one group of 5 to now see that if they have 2 of them it will make a 10 and if we have 4 of them we would have 20 and really start looking at different ways to combine those groups.

The problem I am seeing in this plan is the differences in the two problems. As I sit here with the papers all over the table, I am struggling to make a sequence involving both problems. So, do I sequence a set for each problem and give each 1/2 of the class time to talk about the similarities and differences? or just choose one problem and go with that?

For problem 1, I like this sequence in moving from counting by 1’s to grouping them and then to the finding half of 34.

For question 2, I see this sequence from pictures to grouping them by people and dogs, the third shows the 8 composed but broken apart on the number line and the paper before it, and the last one starting at 14.

I collected their papers and asked them, in their journals, think about how many people and dogs there could be in the park if I just told them there were 28 legs. I thought that after their share tomorrow of this problem it would lead them into a nice problem from which some great patterns could arise. Here were a few I grabbed before I left:

and this last one was getting at some really great stuff as she got stuck at 9 people and couldn’t figure out the number of dogs. I asked her to write what she was telling me!

Looking back, I would have probably chosen just one problem to work with to make it more manageable in sequencing and making connections during the share. Having two problems was nice as far as having them explain it to others, so I like that, but I am wondering if we did #1 through this process and then split for questions for #2 and #3.

I look forward to hearing how it goes tomorrow!

~kristin

# 2nd Grade Collaborative Planning Using the 5 Practices

This Tuesday, I am teaching a 2nd grade lesson for a teacher who will be out that day. I offered to this for all of the teachers if my schedule permitted. I thought it was a great way for me to learn more about each grade level, possibly plan and teach it with other grade level teachers for that lesson, and it saves having to use a “sub plan” lesson which we all know either leaves us with more papers to grade or even worse, having to redo when we return. After doing this same type of thing for a 3rd grade classroom last week, and getting great suggestions in the comments after the lesson, I thought this time I would try throwing it out there before I taught it. I would love to see how this lesson could take shape with the input ahead of time!

Lately, I have seen a lot of tweets regarding using the 5 Practices when planning. Now, while I don’t use them to the extent the book lays out for every lesson (because, you know, time), I do always have them playing in the back of my mind when I plan. So, I am going to plan here, one piece at a time, using the 5 Practices. I will pose my questions where I have stopped and look forward to feedback in the comments!

Here is a little background information…

The Investigations Unit Summary:

I see the CCSS highlighted most in this lesson:

Up to this point the students have been doing a lot of addition/subtraction story problems and sharing of strategies, counting by equal groups, and working with evens/odds. In their work with evens/odds they have been deciding if numbers can form two equal teams or if they allow each person to have a partner. As of a week ago, this was the class noticings around even/odd:

The lesson I am planning is structured as a workshop in which one piece calls for the students to individually solve the following pages, however I am thinking I want to turn these two pages into the lesson because I think they could lead to some amazing thinking!

Practice 0: Mathematical Goal

[Planning 1]Students use equal groups when thinking about a context. I am not sure if this is too broad, but there is so much here. What I would really love to see is students moving beyond drawing each one out and counting by 1’s but I am also so interested to see how multiplication and division show their beginnings here!

[Final Plan] After a conversation with a colleague, my goal for the lesson is for students to begin unitizing the equal groups when combining the groups. I also have this subgoal of proportional reasoning when thinking about people/eyes or dogs/legs.

Practice 1: Anticipating

[1st Planning Thought] Before moving on here, I need to decide whether to focus on both pages or just focus my planning on one or two problems. Although they all involve equal groups, I am wondering if focusing on a particular one brings out more conversations and connections between the ways in which we can count? I am leaning to #4, but I it would be helpful for me to also see how they think about 1-3 before thinking about the share of #4. OR, do I leave 4 for the next day after gathering info and sharing strategies together for 1-3?

[2nd/Final Planning] I am thinking now that I am going to launch with a simple sentence of “There are 12 people” and ask student what problems we could solve based on that sentence. Talk about ears, eyes, fingers, legs…etc and then how we could represent our work. I am thinking to not actually DO the math but write the ways as a reference back at their seats. For example, “Draw pictures, Use numbers, Use cubes, Write equations, Use words, Use tables…etc” In planning with another 2nd grade teacher today, we saw that “show your work” at the top pushed some students back to pictures when they were not necessary.

After this, I am going to have 1/2 of the class working (in groups) on problem 1 and the other half on 2. Before they jump right into group work, however, I will ask them to take individual think time to get into the problem. After the groups have arrived at an answer, I will  have a couple students swap seats and explain to the new table how they arrived at their answer. They will then discuss what was the same and different about their problems and ways they solved their problems. After they share among tables, I will bring them to the carpet for a group discussion about these similarities and differences.

Practice 2: Monitoring

During the work at their seats, I will be walking around, and asking questions when necessary to generate conversation (I don’t know this class as well as I would my own so I do not know what to expect as far as conversation) and looking at strategies.  Questions: How did you arrive at your answer? Does everyone at the table agree ? Where do you see [the ears, people, eyes, fingers] in your work? Is there an equation to match your work?

Again, after discussing this with a colleague, I will not only be monitoring student understandings but also monitoring for which students to switch and share. I would not want students with the same strategies to switch and not have anything to build upon so this is a great opportunity to structure a better situation for conversation.

Practice 3: Selecting

I will choose papers based on a variety of strategies that build along a trajectory. I would like to see students who drew out the problem by 1’s, 2’s, 4, 5’s or 10’s, then others who used one group to represent the 2’s,4’s, 5’s, or 10’s (unitizing), then students who used equations or number operations w/o the pictures.

Practice 4: Sequencing

In the share, after each group has presented to the other groups, we will come to the carpet for a share. The sharing will be sequenced in the way I discussed in the Selecting part, asking students during each student work sample how it is similar and different than the ones we previously shared.

Practice 5: Connecting

The connecting I see happening through my questioning as we share strategies. I am still working on writing this part out and looking for the connections that can be made, aside from the picture to number representation connections.

The connections I would love to see students making throughout the work and sharing, is how we can combine equal groups. For example I would like the student who is drawing ones and counting them all to move to seeing those ones grouped as a 2 or a 5 depending on the context. I would love the student who is seeing the five 1’s as one group of 5 to now see that if they have 2 of them it will make a 10 and if we have 4 of them we would have 20 and really start looking at different ways to combine those groups.

For the journal, I will give them the scenario that there are people and dogs in the park and 28 legs, how many of each could there be? This will offer multiple solutions (Thanks Simon) and allow for them to see some great patterns the following day!

I will let you know how it goes!

Follow up Post #1

Follow up Post #2

~Kristin

# Even or Odd…So Much To Think About!

At the end of the day, Lauren, a 2nd grade teacher and I started chatting about her upcoming lesson on even and odd numbers. I have done a lot of thinking about even and odds in 5th grade when we entered decimals, however I can honestly say I have not thought about it much more than a number being able to be broken into two equal parts or it can’t because there is 1 left over.

Enter this student activity page from Investigations…

Now, I know Investigations is so purposeful with how they structure their pages so I was immediately curious about the set up of this page and wondering why thinking about even and odd in these two ways was so significant. My mind went right to the foundation for the commutative property. For example, with 10, will each have a partner? yes, 5 pairs or 5 groups of 2 or 5 x 2. Can we make two equal teams? yes, 5 on each team or 2 groups of 5 or 2 x 5. I would like to extend this sheet to include a space for the expressions: 2+2+2+2+2 = 10 = 5+5.

Then of course I tweeted it and got this great stuff from Tracy:

So much great stuff to think about and I absolutely cannot wait to see how these students deal with conjectures and generalizations! I would love any more thoughts on this work because I am sure there is more great stuff in here!

-Kristin

# My Week In The 2nd & 3rd Grade Math Classroom

While I am loving my new role as the school math specialist, I am definitely finding that my blogging has taken a bit of a slide. I have come to realize that my main inspirations for blogging is having a class every day in which I am thinking things through with and the student work that is the result. Working in various classrooms around the building does not offer that consistent look at student work, but I am SO excited to see so many teachers in my building using student math journals! I think they are finally starting to get used to me snapping pics of all of that great student work at the end of class!

This week, I had the chance to plan and teach with second and fifth grade teachers and do number talks in 3rd, 4th and 5th grade classrooms! Ahhhh…finally student talk and work that gets me excited to learn and inspires me to blog!:)

Our second grade begins the year with Unit 3 of Investigations which centers around addition, subtraction and the number system. What the teachers and I realized, during the lesson we planned, was that, while the students did an amazing job adding and were finished fairly quickly, they all used primarily one strategy and if they did use a second one, they did see it as different.

The majority of the students decomposed both numbers and combined the tens and ones like the top two strategies of this student:

When asked to show another way, he quickly did the third strategy. Walking around the room, the teacher and I saw many others thinking in the same way as the third strategy but intricately different.

Thinking in terms of the 5 Practices, we monitored and selected a progression of papers to elicit connections between strategies, however what we found is that as students shared, the others were saying, “I did it the same way, I just broke it apart.” They didn’t see a difference in breaking both numbers or breaking one number or then how they thought about the decomposition and combining of the partial sums. We left class with that spinning in our heads….”It is wonderful they can use a strategy to add, but how do we get them to see the differences in each and think about when one may be more efficient than another?” and for me, being new to second grade math, “How important is it that they do? and Why?” The following class period, which I could not be there due to a meeting, the teacher began creating an anchor chart of strategies as students discussed them and pushed them to see the similarities and differences of each. I am still thinking through the importance of these connections and realizing I have so much to learn!!

In third grade this week, I was asked by a teacher if to come and do an addition number talk with her class. That took no thought, of course I jumped at the chance to chat math with them! I realized both before and after how much easier it was for me to plan for my 5th graders because I knew them and, due to experience, could anticipate fairly well what they would do with problems. I chose a string of addition problems that, while open to any strategies, encouraged the use of friendly numbers. I forget the exact string now, but something like 39 + 43 and 53 + 38. After being in second grade a few days before, it was interesting to see the same decomposition of both numbers to tens and ones and recombining of them. I am beginning to think that is the easiest, most instinctual way for them to do problems because they CAN do it other ways, they just jump right to that first! We did three problems together, and while the use of friendly numbers did emerge, it was definitely not the instinctual choice of the class. I left them with one problem to do “as many ways as they could in their journal (WOOHOO, they have math journals). I went back later to have them explain some of their strategies and take a look at their work.

I was excited to see that while many started with tens/ones, they had a wide variety of thinking around the problem:

Of course there are always a couple that leave you thinking….

In his verbal explanation, this one said he, “Multiplied 35 times 2 because he knew that 30 and 30 made 60 and the two 5’s made 10 so that was 70. Then he added the 14 to get 84.” When he first started talking, I had no idea where he was going and was honestly prepared to hear an incorrect answer at the end. I asked him to write out his thinking and he gave me this great response:

I know we need to be aware of his use the equal sign and make that a point in future number talks, but that thinking is soo interesting. He saw he had two 35’s, one of which within the 49 and then 14 leftover once he used it in his multiplication. Great stuff!

This one I need to hear more about from the student. He said he subtracted from 100 on a number line to end at 84. I asked him why he subtracted and he said he knew he needed to get from 100 to 84. I was confused but in the midst of the class, I didn’t think it was the time to go deeper with this one. I can’t tell if it is connections to things they are working on in class with 100 or something else?

I still have to blog about the 4th and 5th grade fun, but this is getting long already! I will save that for tomorrow!

~Kristin

# Second Grade Number Talk

This was the first week of school and the very first number talk these students had done this year! From the excitement in the room and this poster on the wall, however, you can tell they have done them before…

This string was to see some of the strategies they had used before and how they were thinking about organization, decomposition and notation. I included my reasoning for choosing each one under the image.

Image 1:

I was curious to hear so many things in this first one. I wanted to see if the students saw the numbers in particular ways such as: 4 on top and 3 on bottom, subitize the die 4 to the left then the 3, or 6 and 1 more. After they saw them, how do they combine? Do they “just know” 4+3 or 6 +1, do they count up, do they count all? I was also curious to hear if any students reorganized the dots to fill the five on the top row to create 5 +2. And then do they combine them 5,6,7 or do they know 5 and 2 more is 7 right away?  I was so impressed to hear the students do all of the things I anticipated very quickly and were very comfortable with writing equations, explaining the thinking, expressing where they made a mistake and talking to one another. Yeah K and 1 for building that community, it showed!

Image 2:

On this one I was curious to hear all of the same things from the first one, but to also hear how they see/think about teen numbers. Do they move the dots to make the 10 and why do they do that? Do they know 8+4 and don’t think about moving the dots? How do they know it is 8 and 4…is it because of 5 and some more or because of the missing boxes to make the 10 or the 5?

Again, all of the things I anticipated came out, however one little girl started explaining how she started by counting the empty boxes so I completely thought it was going to be 20 – 8 =12, however it did not go there.  She did get to 8 empty boxes but then said, “so then I moved two up to make 10…” Ha, not where I saw that going!

Image 3:

Building on what I learned from the first two, I wanted to see if and how they combined 10’s and then added on the extra dots. I didn’t make the 5 a neat row on the bottom because I wanted to see how they organized them. I was excited to see that as soon as I flashed the image the first time, all of their eyes went right to the bottom ten frame. That let me know that once they saw a full ten, they could just keep going and it would be easy to add that on at the end. The students shared their thinking and then I wanted to focus on the 20 + 5 = 25 and 10 + 10 + 5 = 25. Having recently read/reread Connecting Arithmetic to Algebra and Thinking Mathematically, I am really interested in how students in the younger grades build this foundation for algebra. So I told them i was going to write an equation and I wanted them to tell me whether it was true or false and give me a thumbs up or thumbs down on it. I wrote 20 + 5 = 10 + 10 + 5. I was completely anticipating the majority to say false because they are used to seeing one number after the equal sign, so I was SO excited to see more than 75% of the class with their thumbs up. I asked them share why and many students said because the 10 and 10 are the same as the 20 on the other side and the five stayed the same on both sides. Others said because it is 25 on both sides so that is the same. This was such an interesting thing to think about for me…some student look for balance (equal on both sides) while others look to make them look the same on both sides (the 20 is the 10 + 10), a little bit different in my mind.

After the talk, I was SOOOO excited to see that Miss Robertson was starting math journals this year so we came up with a double ten frame (the first one with 9 dots and the second with 7 dots) for the students to explain how they think about the dots? What things to they look for or do to find the number?

Here were some of their responses that I thought we so interesting and leaves me wanting to chat with them about their work!!

I loved so many things about this one. The “10” in a different color makes me feel like that student thinks there is something really special about that 10. Although she numbered them by ones, I don’t think that is how she found the 16, but I would like to chat with her more. I wonder if she wrote 9+7 but then filled in the answer after she moved and solved the 10+6=16?

This was so exciting because it was one of Miss Robertson’s ELL students and look at all of that writing!! While there is no answer, there is the expression, 5+4+4+3 at the top which shows me how he is seeing the dots. He went on explain about a 10, but I did not capture the back of the paper…grrrr… stupid me. I will have to go back to this one!

I was amazed to see so many students write both equations and with such an articulate explanation of the process. I expected, if a student moved a dot, to just see 10+6=16 written. Like this:

But it was interesting the student in the first one wrote both! I am so excited for Miss Robertson to try a number string with them without the ten frames to see what they do with that!

This student showed how they thought about the dots in each ten frame and then at the bottom shows beautifully how he combined 9+7. Under the 7 you can see the decomposition to 6 and 1, how lovely. The bottom thought string needs to be something to think about moving forward as teachers. Making explicit the meaning of the equal sign.

Ok, I am obsessed with this one and I need to talk to this student one more time! I am so curious why this student chose 3’s. Did he see 3’s to start or did he know something about 9 being able to be broken into 3’s? I  could completely see that if the top ten frame looked like 3’s or they were circled like the bottom one and the 3’s to the right were grouped together, however they are circled like he was counting off by 3’s by going down to the next row. Would he have done the same thing if the top ten frame was 8? In my head I am feeling like the student knew that 9 could be three 3’s, thirds, by the way it is circled. I don’t know if that is something students think about at all, so I am so curious. Or do students “see” threes but then circle them in a different way then they saw them?

Now, onto my 1st and 5th grade experiences yesterday….I am not going to be able to keep up with these K-5 blogging ideas this year…so much great stuff!

-Kristin