1st Grade Notice/Wonder

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Yesterday, I had the chance to teach the 1st grade lesson I planned here. It was so much fun and SUCH a learning experience for me! After all of the conversation in the comments and on Twitter, I decided to start with the open, one sentence Notice/Wonder. Only having 45 minutes and this being the students first time doing a N/W, I decided not to begin with a number talk/routine (which I usually always do).

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The students started on the carpet, I put up the sentence, read it and asked, “What do you notice and wonder about this sentence?” Just then a student exclaims that he just noticed that “Notice” was not, “Not Ice.” At that moment, I began thinking maybe my question had them looking at the physical pieces of the sentence/words so I quickly rephrased, “I would love to hear what you notice and wonder about what is happening in the sentence.” They used their Number Talk signals, thumbs up when they had a notice or wonder and then used their fingers to indicate more than one. I was so impressed by all of their thoughts, but I did realize that is it hard to end their wonderings! The amazing thing was how all of their wonderings really could turn this sentence into a story in their ELA class because they were all really important details they could add to it. Here was how the board ended.

I asked which wonder we could work on together today in class and there was a unanimous vote for “How many kids are on the bus?” however there were a few that suggested, “How many student can the bus hold?” because “math is counting things and we could count the seats.” I starred the wonder “How many kids are on the bus?” and told them next time they get on their bus I would love to hear how many seats and students they found are there. We discussed whether they know how many students were on the bus by reading our sentence and they said no, they only know that there were 3 stops. I asked, what they would want to know and they wanted to know how many kids were at the stops. I wrote that at the top.

When I told them they would get to choose how many students were at each stop, they were so excited! I gave them a paper with the sentence at the top, let them choose a partner and sent them on their way.

As I walked around and asked students why they chose the numbers they did, I quickly wondered how much I should have helped organize their work for them. I found so many with numbers everywhere and it was hard to see where their bus stop numbers were, let alone their total. Should I have put Bus Stop 1____, Bus Stop 2____, and Bus Stop 3_______ to have a clearer picture while also modeling for students how we can make our math work more clear? Quite a few looked like this…

There were so many interesting papers, so I love WordPress’ new tiling feature for pictures to make it look less cumbersome!

Top Row, Pic 1: This student had 24 as two stops. When I asked him how many stops we noticed in the beginning, I got a “Oh gosh” and he wrote Bus stop 1, 2, and3. He then stuck with the 24 and when I came back he had 8, 8, and 8. I didn’t see this until after class so I am curious how he arrived at that answer. I also realized that these 1st graders move fast and it is SO easy to miss the cool things they do so quickly!!

Top Row, Pic 2: They said 3 and 22 were easy to add and then they just chose another small number. The interesting thing here that I need to find out more about is the 5×6 with the one box shaded in. I loved the commutativity showing up here!

Top Row, Pic 3: This was so interesting because I had never thought that a student would first think of how many students were on the bus and divide it up from there. They thought 30 students would fit on a bus so they made the stops fit that information. (They saw the error on the last one during the share).

Bottom Row, Pic 1: This student said that because there were 3 stops, there were 3 students at each one and ended with 9.

Bottom Row, Pic 2 &3: This student wanted big numbers so his first response, after he insisted on re-writing the sentence, was 1,000 and 1,000,000 and 4. Then on the back of his paper he wanted 6 stops and chose 6 new numbers. This led to some great conversation during the share.

This student figured that if there were 1 or 2 at a seat then there would be 55 students on the bus. I love all of this work so much! Then when I asked her about the students at each stop, she said, 30, 20 and 5.

We shared as a group back on the carpet and I tried to capture why they chose the numbers they did:

I then gave them the original problem and asked them to solve it individually. After seeing them work on this problem, I think there are so many interesting conversations that could happen Monday morning!

This is where I had so many questions as to how we get the younger students to make their thinking more visible. I found so much of it happens on fingers, 100s chart, and number line on their desk that I was getting an equation on the paper. It is great when I am sitting there asking, but that cannot obviously happen when they are done so quickly and there a bunch of them! Is this something that comes with practice? I did find that once I asked them if they could explain to me on the paper how they solved it, they did a great job. My next question is, would taking the 100s charts and number lines off the desks help push students to look for friendlier numbers? I found the majority of them went left to right, counting on instead of using the 6 and 4 first. This is something that I think a structured share out on Monday could bring to light for those who never thought of it.

Here are a sampling of the papers I look forward to chatting with the teacher, Lisa, about on Monday. We can chat about how we can structure this share out.

Lisa, through number talks and investigations, has been working a lot on having students noticing number patterns leading to generalizations. It was neat to see this work of adjusting addends and keeping the same sum showing up here too. It seemed to show up most after they had their answer and were playing around with the numbers, which I love!!

I am happy to have started with the open notice/wonder because I learned so much about how they think about problems and I think the opportunity to choose their own numbers got them thinking about the context over solving for an answer to an addition problem. I am, however, extremely curious how it would have changed the work if I had given students the problem with the 13 given and the other two missing? Would I have seen more about how they choose numbers to make the 13 easier to add a third number? I am hoping to get into another 1st grade classroom to try this out with another teacher but I would love it if any other 1st grade teachers would there would love to try it out and report back!!

I am so looking forward to Jamie’s post on this because her student work looked amazing on Twitter yesterday!!

~Kristin

Yeah, Jamie’s post is up! Check it out here! Cannot wait for our Google Hang Out tomorrow to chat all about it!

1st Grade Story Problems

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Tomorrow I go into a 1st grade classroom to teach a lesson on addition and subtraction story problems. This Investigations lesson for the day centers on students solving these 6 problems, however I am looking to change it up a bit.

While reading my CGI book, Children’s Mathematics, to learn more about the trajectory in which students solve these types of problems, I found this diagram really helpful and interesting….

I went into this planning thinking I was going to be looking for how students combined numbers in the context of the diagram above. From there, I was planning to have students do a structured share of their strategies, comparing and contrasting along the way. However, as I got ideas from Jamie (@JamieDunc3) on Twitter, I started to think how much more I would learn about their thinking in talking about their noticings, wonderings, and number choices. My goal has now changed to looking at not only their strategies for combining but how they choose numbers in which they will have to combine.

So…I took the second question, removed the actual question and made it a notice/wonder:

Assuming the wonder of how many students were on the bus arose, I would see how students combined the numbers. Would they look for friendly combinations? Would they count all? Model it? Count on? or any combination of those?

Then, I thought I could keep the 13 and leave the other two numbers blank to see what numbers they chose.

Did they pick a combination that was easy to add to 13, like 5 and 5? or would they keep adding onto the 13? how would they add with the 13, would they choose 7 to make 20 and then another 1 digit number? would they choose all 2-digit numbers to challenge themselves?

But then, I thought what could happen if I took all three numbers out?

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For some reason, without the numbers it seems more “wordy” to me. I don’t know why that is? So THEN, I went to this last option….

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I really love this one, although, I must admit, I feel a bit out of control of the course of the lesson in choosing this one over the others. But, I think that is what makes it such a beautiful choice. After taking noticings and wonderings, I am thinking of having the students work in pairs to create their own story and solution for one of the wonderings.

In creating their stories, I am concerned that students will choose numbers such as 0 at two stops and 1 at the third and I won’t be able to get a picture of how they combine numbers, however I will have a possible picture of their number comfort level. If they do this and finish quickly, I will be ready with the second choice above to see how they deal with now having the 13 in the problem.

In their journals I will ask them to tell me why they chose the numbers they did for the problem.

I am still thinking about this, so please feel free to leave suggestions and comments! Thanks to Simon, Fran, Graham, and Bryan for their thoughts on Twitter, always appreciated!

~Kristin

2nd Grade Lesson Follow Up

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Over the past few days I have been blogging about my 2nd grade planning, teaching and reflecting. Today was the follow up lesson…

This morning, I popped in the classroom to chat with Lauren, the classroom teacher, to catch her up on the lesson and chat for a bit about where we could go from here. I was so excited to hear she had read the post yesterday so we could pick up our conversation with the student work! We looked through the student work samples and struggled with how to structure the share…. should she just focus on one question or look at both? We decided to start with showing two student work examples (and having the students explain how they solved it) from the same question that were different in how they represented and grouped their numbers to count, but ended with the same answer. After the class shared what things were the same and different about their way they solved it, we were planning on sending them off with a group who did the same question as they did and look for all the similarities and differences. In this, we were hoping students would start to notice how others grouped their numbers and move some of the students who are still drawing every picture and counting by one’s forward in their thinking.

I had to leave to go meet with another teacher, but after Lauren opened the lesson like we had planned and took her students to recess, (that is that weird split in their schedule that we usually hate but worked out well here) she came down to talk really quick. It seems that the class could not express similarities or differences beyond the “sameness” of their answers or representations they had drawn. And while, in the moment, I thought this was such a difference between 2nd and 5th graders. However, as I reflected later on, I recalled this same thing happening with my 5th graders last year!

We knew sending them off, as we had planned, would be ending with the same level of conversation, so we started brainstorming! How could we get them to look at HOW the strategies varied in process and not just look at the numbers and same answer? It is such a difficult thing to relate to the thinking of others, and it probably doesn’t help that the work and drawings were all over the place! Excuse the sloppy mess, but we decided to take strategies we had seen in the students work and frame it as work that Lauren and I had done in solving the problem and ask them to connect those! Then we added in a third teacher’s thinking from across the hall. You can vaguely see the pink counting by ones, the next by 2’s and then 4’s and 2’s, hoping students would think about how the numbers formed different-sized groups. I took a pic of this, texted it to Lauren so she could have it back in her room and she had to run to pick them up from recess! FullSizeRender 21.jpg

After my afternoon meeting, I popped back in to see what happened! She said they did so amazing seeing the similarities and differences when it was written out like we had done. For the first time we were asking them to do this work while looking at student work, I probably should have thought about how to better structure the “look” of the work.

Lauren said after the share she asked them to find the number of eyes in the 7 people, 2 dog problem thinking about what they had just shared. i just got to catch a few of their work samples, but it was already so amazing to see the movement from drawing everything to different groupings. This is where I wonder if students just need to have the permission to now “show their work” in ways that are not pictures? I am not sure but something I am thinking about. Here were a few samples of the new problem, I can’t wait to chat with Lauren about it more tomorrow!

~Kristin

2nd Grade Counting,Unitizing, & Combining

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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:

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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.

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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.

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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:

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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!

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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

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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

Practice 5: Connecting

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

The Beginning of Arrays in 3rd Grade

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Today, I was lucky enough to be asked to teach a third grade math class because the teacher was going to be out. Since I have never taught the Arranging Chairs activity in third grade, I was excited when two of the other third grade teachers, Jen and Devon, wanted to plan with me yesterday. Before meeting I read up in Children’s Mathematics: CGI, Carpenter, Fennema, Franke, Levi, Empson to think more about this idea of equal groups, meets arrays, meets area model builds. Here is one piece I found that connects them in a very nice way.

We decided to change the Ten-Minute activity from a time activity to a dot image number talk. We thought since the students have been doing so many dot images involving equal groups, that it would be interesting to see how they thought about one image with a missing piece. We were curious if students would use any structure of an array to think about how many dots were in the picture. The board ended like this…

For the most part, students either added rows (so they were seeing the array structure) or looked at the symmetry of the picture. They had so many more strategies they wanted to share, but for times sake, I did a quick turn and talk so they could share their ideas with someone before they left the carpet. Because I heard a student talk about filling in the middle, I asked him to describe to the group what he and his partner talked about. He said, “You could fill in the missing dots and then do 4,8,12,16 minus 2.” I heard the word array thrown around so I asked them to tell what they knew about arrays. A few students built upon one another’s definition ending with something with rows and columns.

Next, we introduced the activity on the carpet right after the Number Talk. You have 12 chairs to arrange in straight rows for an audience to watch a class play. You want to arrange the chairs so that there will be the same number in every row with no chairs left over. How many arrangements can you make? They talked to a neighbor and I took one example, a 6×2, and constructed it on the board. We talked about what that would look like on the grid paper. The grid here felt like a very natural way to move students between arrays as equal groups to rectangular arrays. They went back to their table, with cubes, and worked on making as many arrangements as they could. We shared them as a group.

They talked about the commutativity in the rotation of the arrays. We discussed the fact that since we were talking about seating arrangements in this activity, we would consider them two different ways to arrange the seats. This is where I saw the arrays as such a beautiful way of visualizing commutativity in a much different way than they previously had discussed in rearranging number or groups and group sizes.

Next, each group was given a number to create as many arrays as they could, cut and paste them on a piece of construction paper. Choosing the numbers for each group was something we spent a lot of time in during our planning. We wanted to be sure that noticings around sets of numbers such as primes, composites, evens, odds, and squares would surface, as well as relationships between different sets of numbers, we tried to be really thoughtful around this. We came up with a first set of numbers and then decided on a second number to give that same group if they finished early. So, this list is first number/second number (although we knew not all would get to the second one).

11 / 27 – Prime number and then an odd that wasn’t prime

25 / 5 – Odd square number and then relationship to a multiple they did of that number.

16 / 8 – Even square number and then halving on dimension

9 / 18 – Odd composite and square and then double a dimension

24 / 12 – Even number and then half a dimension (we didn’t think they would get to this one because 24 has quite a few to cut out:)

18 / 36 – Even number to compare with another group and then double a factor (36 could also relate to other groups numbers in various ways)

15 / 30 – Odd composite and then double a factor. We didn’t think they would finish 30.

13 / 14 – Prime number and then how adding one more chair changes what you can make.

Extras for groups done both: 64, 72, 128. (No one got there)

Thanks to a lovely fire drill in the middle of class, some groups did not get to a second number or if they did, did not get to finish. This is the point where you realize how amazing it is to have more than 1 teachers in the room! Everyone could walk around and listen to their conversations while they worked. We heard everything from frustration/wonderings about prime numbers because they thought there had to be more than one (and the rotation) to excitement when they finally got a second number with more. Here a few of the (close to) final products:

On Monday they will hang them up and walk around to do a notice/wonder about all of the different numbers around the room, but we really wanted them to think about their work today before jumping into comparing others. I also really wanted to capture what they were frustrated by, liked about their number, were thinking about in the moment and were left wondering. So, I asked them to write about what they noticed and wondered about their work today. I expanded on the prompt a bit to avoid, “I notice I could make 4 arrays,” and I said, “You could tell me why you liked your number or didn’t, what you think made your number easy or hard, or what you realized as you were making them.”

There were some beautiful responses that I cannot wait for Andrea (their teacher) to hear on Monday because they were so excited to share!

A nice noticing that could lead to largest perimeter with the same area:

An informative noticing and wonder about commutativity to keep in mind when planning…

Wonderful comparison of why they feel evens are easier than odds, but also great wonderings about “Is that really all you can do?” with prime numbers and why?

I talked to this student and he was using the 12’s for 24 but had trouble articulating it in his journal.

Loved this one wanting a number in the hundreds because it would be more challenging and don’t miss the bottom piece about subtraction!

She was not as much of a fan of the square as I was when I walked up, she said it is, “just the same when we turn it” and I said, “That is an awesome thing!” (I meant her noticing, but I think she thought it was about the square:)

I will leave you with this one that struck me as “We always have more to learn.” I cannot wait to see her working with fractional dimensions in 5th grade!

I cannot wait for the gallery walk and noticings and wonderings from the entire group of numbers. I am also really excited to see this work move into rectangular arrays and seeing students’ strategies around multiplication evolve and how they take this work and form relationships between multiplication and division.

Great day in 3rd grade and I have to say, I think Jen, Devon and I planned really well for this one!

-Kristin

1st Grade Dot Addition and Math Journals

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A couple of weeks ago, I blogged about my planning with a first grade teacher here. After teaching the lesson, the students did an amazing job with the dot images we chose to use. Some students moved the dots to make the dice look the same on both sides of the equal sign while others solved both sides. On the last image they easily decomposed the 4 into the two 2’s to prove both sides were equal so that was something we were hoping to see transfer into the dot image activity.

We walked around, recorded the expressions we saw students writing, and asked students questions about their strategies for choosing cards. As I do with many lessons, in thinking about their strategies beforehand, I referred to the Learning Progressions to see how students progress through algebraic reasoning. If they didn’t know the the addition expression from memory, like 3+3 or 5+5, this clip from the progressions best describes how I was seeing students arrive at the first expression written for each given sum. Because the commutative property was the way most students found the second expression for each sum the day before, this particular day we told the students they had to use different cards than their partner in thinking about writing their expression.

I especially loved this passage in the Progressions about counting on…I had never thought of counting on as seeing the first addend embedded in the total, although it makes complete sense now! I wonder how understanding that could impact the way in which I question students about their thinking when adding?

What we were looking for as we walked around in particular was how students were using either this Level 2 method above or, what the progressions would call it, Level 3:

It is hard to convey all of the conversations we heard, however here are some of the game boards I captured after the finished playing the game. (Some boards were 6,9,10,15 and others were 8,9,12,16)

These partners seemed to think individually about their expressions on the left and right sides of the board. The student on the left appears to use facts they know such as 7+3 to arrive at 4+3+3 (since there were no 7 cards). I love the use of the equal sign between the two columns!

The other two pairs appears to have done the same thing…

The two groups below, I remember talking to because I was so interested in how closely their sides were related. After the student on the left had written their expression, the student on the right either combined or decomposed numbers to write an equivalent expression. I would love to talk to both groups about the sum for 12 because I am curious if they are decomposing and making a “new” number based on what they are “taking from” another number.

After playing the game, we put the equations we saw for each of the sums on the board and asked students what they noticed. Some noticed relationships between the expressions for a given sum while others looked at expressions for various sums. For example, when looking at the expressions for 10 and 15, they noticed that each expression added 5. Then we discussed whether that 5 was always a 5 and students were really comfortable saying that it could be a 2 and 3 or a 4 and 1. They could have shared their noticings for quite a while so we asked them to go back to their journals and describe something they were noticings among any of the equations.

It was at this moment when I started to detach myself from the math for a quick second and began seeing how journaling really begins. I found I take it for granted that when I say write in your journal about something, that they understand how we explain our mathematical thinking. I know that writing at various grade levels differs based on so many things such as vocabulary, writing experience, and just how they write words in general. However, one thing I did not think so much about is how students view writing in math. I did not realize until I saw this student showing all of his compensation in numbers by connecting the numbers that were staying the same with lines and showing the number that was “one less” by writing -1 when going from an expression that totals 10 to a sum of 9. He explained it so beautifully but was having trouble communicating that on paper. When he finished talking a girl next to him, asked me, “Can we use words too?” <—- that is when I had an aha! Do students think about writing in math as only communicating numerically? Do we ever explicitly tell them it is ok to write about math in numbers, words, or we can use both numbers and words? I think I have always assumed they knew.

Then I came back later and the very same girl had written all of this wonderful thinking…

This student showed a wonderful connection to what was happening when he went from 6 to 9 and then from 10 to 15:

After they had finished journaling, the students moved to recess, however this student sat for another 20 minutes explaining to me all of the wonderful thoughts he had in his journal. The arrows were movement of numbers that were changing however being able to clearly communicate that in his writing was not something he was able to capture clearly. THIS is the power of writing in math I think…learning to take all of the amazing thoughts and communicate it clearly because the more he talked it out to me, the more arrows he drew, the more he elaborated on his thoughts.

Moving forward from here there is so much to think about for me….in addition to moving students thinking about addition and relating that to subtraction, how do I begin to think more about journaling in math, how does it really start?

For Dot Addition game I am wondering if we could allow some students the option to use subtraction? Make the range of card choices larger to allow for students to play around with that relationship. It is something that I thought about as I looked at the table in the Learning Progressions..

So much to think about each time I leave a classroom!

~Kristin

3rd Grade Dot Image

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The third grade team is planning for a dot image number talk that focuses on this standard:

“Apply properties of operations as strategies to multiply and divide.²Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)”

Before this talk the students have been doing work with equal groups and are moving into array work with the arranging chairs activity in Investigations. They have also been doing dot images with smaller groups and have noticed the commutative property as arranging the same dots into different-sized groups.

These are the three images we are playing around with and anticipating which would would draw out the most interesting strategies based on the properties. We are thinking of having a journal entry afterwards to see if students make any connections between the strategies.

So if you feel like playing around with some dot images and doing some math, I would love anyone’s thoughts on which image you would choose and why!

The start of my planning….

My new thoughts on these images and responses…

After chatting with a few friends yesterday and thinking about which image would elicit the most expressions that could allow students to see some connections between the properties of operations, I am thinking about some changes to the images (in orange).

In image 1, I am wondering if we should split each group of 8 into fours but leave a bigger space between the top four groups and bottom four groups. It may allow students to better see the 4’s and then group them as 8’s and at the same time thinking about “doubling” the top group to get the total because of symmetry. They could then explore ideas like (4 x 4) + (4 x 4) = 4 x 4 x 2 or (4 x 2) x 4 = (4 x 4) x 2 [associative property] or 8 x 4 = (4×4) + (4 x 4) [distributive property] or any fun mix of them. If we leave it as it is, I think it may be hard to move them past 4 x 8, skip counting by 8’s or using 2’s.

In the second image, I love the structure of it but am wondering how students could use that 4 in the middle aside from just adding it on each time? Will we just end up with a lot of expressions with “+4” at the end? I am wondering what would happen if we adding an extra group of four next to it? Would students see the structure of a 5 and double it in some way? (5×4)x2 = 10 x 4 or 5x(2×4)=(5×4)x2 [associative] or 5 x 8 = 10 x 4 [doubling/halving] or 2×4 + 2×4 + 2×4 + 2×4 + 2×4 = 10 x 4

Then what question to pose at the end? Do we ask them to freely choose two expressions and explain how they are equal? or Do we choose the two we want them to compare? Do we have the dot image printed at the top of the page for them to use in their entry?

So much to think about..

~Kristin

Kindergarten Number Lines…The Lesson

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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

Even or Odd…So Much To Think About!

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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

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MathMinds

by Kristin Gray.

1st Grade Notice/Wonder

1st Grade Story Problems

2nd Grade Lesson Follow Up

2nd Grade Counting,Unitizing, & Combining

2nd Grade Collaborative Planning Using the 5 Practices

The Beginning of Arrays in 3rd Grade

1st Grade Dot Addition and Math Journals

3rd Grade Dot Image

Kindergarten Number Lines…The Lesson

Even or Odd…So Much To Think About!