Tag Archives: 3rd Grade

The Beginning of Arrays in 3rd Grade

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: CGICarpenter, 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.

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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…

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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.

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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:

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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:

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

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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?

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

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

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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:)

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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!

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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

3rd Grade Dot Image

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!

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The start of my planning….

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My new thoughts on these images and responses…

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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

3rd Grade Dot Image Number Talk

Since the 3rd graders are entering their multiplication unit, I find it the perfect time for some dot images!! I used the image below as a quick image in which I ask them to think about how many dots they saw and how they saw them. Quick images are so great for pushing students to visualize the dots and move beyond counting by ones and twos. I flashed the image for about 3 seconds, gave students time to think, and then gave them one more quick look at the image to check and/or revise their thinking.

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They all saw 20, however the way they the 20 varied a lot and the conversation was amazing from there! Here is how our board ended up…

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Recording is something that I am always working on, making truly representative of the students’ thinking. The first thinking was adding groups of four to get 16 and then the additional middle 4 to arrive at 20. The second was skip counting, so I asked the student to do that for me and how they knew to stop at 20. He said he knew it was 5 groups of 4 so he needed to stop after 5 fours. Then I wrote under that “5 groups of 4.” From there a student jumped on that and said that was the same as 5 x 4, because they were talking about that the day before in class.

Then, the thing I was hoping happened, happened. A student said she did 4 x 5 because that was easier. I wrote it down and, of course, ask if that is the same thing? We began on open discussion and they agreed it was the same answer but the picture is not the same. I asked how it changes and a student told me to move a dot to the middle of each of the outside fours to make fives. I drew the arrow and then one student said that is like division, 4 ÷ 4 because you are splitting that 4 between the 4 groups. I let that sit for those not ready for that yet.

The last strategy was counting by twos so I had him skip count for me and recorded that. I asked if we had an equation to match that thinking and got 10 x 2. At that point, I was ready for them to do some algebraic reasoning.

So I wrote 5 x 4 = 10 x 2 and asked them if that was true or false. They unanimously agreed yes so I asked them how they could prove that and to write what they noticed and/or wondered about it. Here are their whiteboard work:

This one showed the 5×4=4×5 to me but I loved the notice so much:

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This one was an interesting decomposition of the 4 to show where the two tens are coming from in 2 x 10:

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This was a beautiful notice and wonder on the groups changing and wondering if this is with every multiplication problem….how AWESOME?!:

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This one required a conversation because I couldn’t really understand it. The movement of dots made two groups of five to make the ten they said, but it was more their noticing/wondering that I want to explore more with them: 

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Oh my goodness, how much do I love this mention of al(l)gebra in here and then the notice about the half of 10 is 5 and 2 is half of 4…this could have some potential conjecture-making in future talks:

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This one is incredibly hard to understand and I am not even sure I completely do, but I love how she used one image to “make” the other: 

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This student started with decomposing the four (I know we need to think about that equal sign later) but then moved to talking about ten frames. He said if I put two ten frames on top of one another (one attached under the other) I can see five fours (vertically). Then he said he drew them side by side and he saw 2 tens. HOLY COW! 

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What an amazing conversation with this group! Today I posed them with a few of these noticings and wonderings and asked them to pick one and see if it always worked and why. I didn’t have time to snap pics of their journals but all I can say is 3.5 x 10 came up…so I will have to blog that this weekend!

All of this K-5 work is so exciting and it is so amazing to hear and see all of the great teaching and learning going on around the building!

-Kristin

Dots, Dots, and More Dots…the Planning Stage

About a month ago, Andrew Stadel sent me the following set of dot images and asked for thoughts:

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Of course, being accustomed to doing Quick Images through Investigations, my first thoughts were around what this would look like as a sequence of images. I sent him this reply:

“Are you thinking of these being shown one after the other….like image, discuss how many and how you saw it, next image, discuss how many and how you saw it, next image…etc? Or are you thinking of using them as stand alone dot images? I am not even sure if that impacts my thinking around the purpose, but here are my initial thoughts (but I do want to think about this a bit more…) “

Now, while I am used to Quick Images, they do not have these yellow and red counters that the students use a lot in the younger grades. That made my begin to split my thoughts into how I may use them for 3-5 versus K-2. So, my thoughts to Andrew continued like this….

For my 3rd – 5th I would love to show these in a progression as I could possibly be focusing
on three things: 
  •  How they think about the red vs yellow (the two colors, it screams distributive to me). 
  • If they create an array and subtract out missing, visually move the dots to create a “nicer” image, or if they build in parts. 
  • How the recordings connect…I typically ask “Where is ___ in ___?” For example in the second image a student could see 2 x 4 +1=9 while another could see 4 + 5 =9  so where is 4+ 5 in 2 x 4 + 1? Well if we decompose that 5 into 4+1, we have 4+4+1 = 2x 4+1 ….Those conversations are probably my favorite with this stuff!

K-2 I am still really learning a lot about and full disclaimer, in my purposes with them,  I typically lean toward making connections to 10 (100 for 2nd) and comparisons. If my purpose was to see how they see the dots, recreating the image, and counting this progression would be perfect…especially that last one!! 

However, if my purpose was to have them compare (more or less) and then creating a proof, I
would  have the second image to build upon the first….like maybe add a yellow on the top an
bottom of the first image…so the first one they say, “I saw 4 (of course we ask how they saw
that four) then 2 and 2 and 2.” We ask how could we record that? 4 + 2+2+2 = 10. First flash of
the second image, “Is it more or less than the first? How do you know?” Second flash of the
image, “how many, how did you know? did you know it before I flashed it the second time?” I
would imagine most would do 10 + 2 very quickly and know it before the second flash. Could
be cool to ask how we could use 14 counters in the next image and have them design the
14th.”

After chatting with Elham, Graham, and Andrew, it was interesting to see the different ways we each looked at the images. (Joe Schwartz conveyed his thoughts to Graham, so I was able to hear those as well) There were distinct differences of when the color of the dot mattered to each of us and when it didn’t as well as a difference of how we arranged the dots to make them easier to count.

These were the things that jumped out at me when I counted each one…

Image 1: Color of the dots mattered. I saw red and yellow, 4+5=9. The arrangement made no difference to me.

Image 2: Color was irrelevant to me. I squished it together to make is a 3 x4 array with one missing. Arrangement mattered here and I built up to the total.

Image 3: Again, color irrelevant to me. I saw the array and subtracted out the missing parts.

Image 4: I didn’t know what to do with but the colors played an important way in which I saw the total. I needed to have those reds to easily see how many missing dots I had to subtract out from my total. So in this one arrangement and color both mattered.

Now, in planning to use this with a third grade class who have not officially started their unit on multiplication and arrays, I was curious most about how they would approach the 3rd and 4th image. Because I wanted to push them to be thinking about combining without having to count by ones, I decided to do them as quick images where I flashed the image for about 3-5 seconds and then covered it back up. I did that twice before taking any answers. In the 3rd image, I wanted to see if the colors of the counters made any difference to them or the arrangement was more important. How did they see the dots and how did they combine and then talk about the way they combined?

The 4th image, I will be honest, I didn’t know what to do with it at first. I knew I couldn’t spend the entire class period with it up there because it was a part of a number talk that I wanted to take about 15 minutes. I had to think about what I really wanted to see the students thinking about when looking at the image. Four things came to mind…

  1. Could they come up with an estimate after one flash of the image or two?
  2. What did they look for when given two flashes of the image. Were they counting rows and columns like the work they would soon be doing in the array work of the multiplication unit?
  3. What did they look for on the second flash? Were they looking for the missing pieces first or second?
  4. Could the students be metacognitive to think about what they were doing each time the image flashed and understand how they counted each time?

I had the chance to go into the classroom, do the Quick Images and film it! Because of time and length of this post already, I am going to leave you with this planning stage and post what I saw tomorrow!

In the meantime, you can play around with what you think 3rd graders would do with these images OR suggest other ways we could use them at various grade levels!

To be continued….

3rd Grade Multiplication Talking Points

This week I had the chance to work with a third grade teacher, Andrea! Her class is just about to begin their unit on multiplication and division so she wanted to do Talking Points to see what they knew, and were thinking about, in relation to these operations. During our planning we discussed the ways in which this Investigations unit engages students in these ideas, misconceptions students typically have, bounced around ideas, and played with the wording of the points. Being my first talking points activity with third graders, I was so excited to see how students engaged in the activity. I have found that even during Number Talks, the younger students are very eager to share their own ideas, but listening to others is difficult.

In looking for how students “saw” multiplication and thought about operation relationships, we designed these Talking Points..

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Andrea introduced the activity and we did the first talking point as a practice round in which we stopped the groups after each of the rounds to point out the important aspects. We pointed out things like,”I liked how Bobby was unsure and explained why,” and “I liked how everyone was listening to Becky when she was talking,” and “I thought it was great proof when Lily drew something really quick to support her thoughts.” Then we let them go and walked around to listen as well!

During the Talking Point round, some things we found really interesting were:

  • How difficult is for them to sit and listen to others without commenting. Not like it is not hard for use as adults though, right? 😉
  • How much students struggled to say why they were unsure. Sometimes it was not knowing what the word division meant, yet they struggled to articulate what it was within that talking point that was confusing them. What a great thing for them to be able to think about!?
  • How they related the dot images they had been doing in class to multiplication and division.
  • How they thought about inverse operations. They said things like, “I don’t know what division is but if I can use subtraction with addition, I probably can use division with multiplication.

We had pulled two of the points that we wanted to discuss, whole group, afterwards, “I can show multiplication as a picture.” and “We can use multiplication problems to solve division problems.” We put them up and just asked them what their table had talking about. The conversation was amazing. Hearing how they thought about multiplication as groups of but 50 x 2 means “fifty two times” while 2 x 50 means “2 fifty times.” We also heard how someone at their table had changed the way they thought about something. And the division conversation was so great and for the students who were unsure because they did not know what division meant, it felt really organic to come out that way… from them, not us.

Of course, we followed with a journal write:) We gave them three choices to write about…

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I was so impressed by the way they wrote about their thinking, by 5th grade, they will be amazing!!

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The student above, during the Talking Points, said that he could show multiplication as a picture because “an equation IS a picture.” It was lovely to see him make the connection to a visual for an equation in his journal.

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I wish the quality of this picture was so much better but her pencil was so light it was hard to see! She does a beautiful writing about how exactly someone at her table changed her mind with such an articulate way of talking about multiplication and division!

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This student above explains perfectly why teaching is so difficult…”…sometimes we have facts about math, we all have a different schema. We were taught differently than other kids.” I am curious to hear more about her feeling about the end piece, “some kids know more then other kids.” Is that ok with her and she understands we all will get there just at different times?

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These last two were two different ways in which students reflected on the dot images they have been doing in relation to multiplication!

What a great class period! I cannot wait to be back in this class to see how students are working with and talking about multiplication and division!

~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!:)

Second Grade:

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:

IMG_0743When 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.

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IMG_0746Thinking 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!!

3rd Grade

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:

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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:

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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?IMG_0762

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

3rd Grade Subtraction Number Talk

So, this year is tough….getting to know students and content across all grade levels is so exciting but always leaves me with so many questions! As much as I use the CCSS as a guide, I go in to every class wondering what students at this grade know, wondering how they talk about it, and wondering how to structure activities to encourage connections. These are all things I took for granted as a 5th grade teacher.

Today I went in and did a subtraction number talk with a 3rd grade teacher. I did a string starting with the problem: 23 – 19 and all of the other problems were subtracting a number with a 9 in the ones place. I thought I could possibly get adding up, removal and/or compensation strategies. For this problem and the following two, I got at least 3 or 4 different answers and a lot of strategies, some correct others not. The most common was subtracting tens (20-10 = 10) and then incorrectly subtracting ones (9-3=6) and arriving at 16 as their answer. Correct or not, I absolutely loved their openness to sharing and looking for errors in their thinking, it was fantastic! Their thinking was definitely not anything I could even begin to really string together because they were really all over the place so all I can focus on now is where to go from here?

The only common thread I saw was the majority of the students were “number pulling and operating” without seeming to think about the numbers first, what was happening or reasonableness. So, my question now is, Is there a type of number talk that would take the focus off of the numbers for a bit and allow students to think about what relationship the pictures have? I don’t know if this makes much sense but I am playing around with these images, but struggling with the wording…

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If I flashed the first one, How many did you see? How did you see them?

Flash the second one, What changed? What is the difference? <—–(I like this one suggested by the awesome 3rd grade teacher) Can you write an equation to represent the change?

I am thinking we could get 20 – 5 = 15 or 15 + 5 = 20.

Next this..same questions.

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Now on this one, 30 – 11 = 19, I think I may bring up the strategy they used today, 30 – 10 = 20 and 1-0 = 1, leaving us with the answer of 21 and see what they think? I can’t tell if that would be helpful or not?? Would love thoughts.

Also, I cannot decide whether to end with a number expression and ask them what the first image looked like and what is different in the second and what the equation would be? Still thinking on this one too.

Trying it out tomorrow and will keep you posted, however I couldn’t sign off without one piece of student work that I loved. I left them today with 36 – 19 in two ways if they could. This student originally got 23 (by the means I described above) but then did the number line and arrived at 17. He went back to the first and realized that 20 and -3 gave him 17, not 23. IMG_0661

When I asked him how he knew it was 17, he said it was like having something 20 feet above the ground and it goes down 3 feet. It has 17 above ground still. I asked him to try and capture that and this is the beautiful piece of work I got…

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Looking forward to seeing this bunch tomorrow!

-Kristin