Category Archives: Number Talks

Which One Doesn’t Belong? Place Value

Since the 3rd grade team begins the year with an addition and subtraction unit in Investigations the teachers and I were having a conversation about how students understand place value. While I don’t see teachers using the HTO (hundreds/tens/ones) chart in their classrooms, students still seem to talk about numbers in that sense. For example, when given a 3-digit number such as 148, students are quick to say the number has 4 tens instead of thinking about the tens that are in the 100. I think a lot of this is because of how we as teachers say these things in our classrooms. I know I am guilty of quickly saying something like, “Oh, you looked at the 4 tens and subtracted…” when doing computation number talks, which could lead students to solely see the value of a number by what digit is sitting in a particular place.

We thought it would be interesting to get a vibe of how this new group of 2nd graders talked about numbers since their first unit deals with place in terms of stickers.  A sheet of stickers is 100, a strip of stickers is 10 and then there are the single stickers equal to 1.

I designed a Which One Doesn’t Belong? activity  with four numbers:  45, 148, 76, 40

I posted the numbers, asked students to share which number they thought didn’t belong, and asked them to work in groups to come up with a reason that each could not belong. Below is the final recording of their ideas:

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I loved the random equation for 148 that emerged and the unsureness of what numbers they would hit if they counted by 3’s or 4’s. One student was sure she would say 45 when she counted by 3’s and was sure she would not say 76 or 40, but unsure about the 148. I wrote those at the bottom for them to check out later.

Since the teacher said she was good on time, I kept going. I pulled the 148 and asked how many tens were in that number. I was not surprised to see the majority say 4, but I did have 3 or 4 students say 14. As you can see below a student did mention the HTO chart, with tallies, interesting.

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As students shared, I thought about something Marilyn Burns tweeted a week or so ago…

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So, I asked the students to do their first math journal of the school year (YEAH!):

“For the students who answered 14, what question did you answer?

“For the students who answered 4, what question did you answer?

After the students shared, I revisited the Hundreds, Tens, Ones chart. I put a 14 in the tens column, 8 in the ones column, and asked if that was right. The light bulbs and confusion was great! It was as if I had broken all rules of the HTO chart! Then I put a 1 in the hundreds, 3 in the tens, and they worked out the 18. I look forward to seeing them play around with this some more and wonder if when they go to subtract something 148-92, they can think 14 tens -9 tens is 5 tens.

I had to run out because I was running out of time, but snagged three open journals as I left! (I especially love the “I Heart Math” on the second one!
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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

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

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…

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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:

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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! 

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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!

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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!!

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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? 

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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! 

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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:

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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! 

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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. 

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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

Making Number Talks Matter – Book Study!

I honestly cannot talk about the teaching and learning in my classroom, or across our school, without highlighting Number Talks. I cannot recall the very first time I started Number Talks, but now I cannot imagine my math class without them. My Number Talk journey began with Sherry Parrish’s book and had continued to grow through reading Cathy Humphreys’ and Ruth Parker’s most recent book, Making Number Talks Matter. I implement Number Talks on a regular basis, reflect through writing blog post after blog post and have presented at both NCSM and NCTM around Number Talks. I simply cannot say enough wonderful things about them!

For this reason, it was not surprising when I had the opportunity to meet my fellow Teaching Channel Laureates this summer, that Crystal (@themathdancer) and I struck up an immediate conversation about Number Talks when we began chatting instruction. As an elementary school teacher, I often have middle and high school teachers ask what Number Talks could look like in the secondary classrooms. Would the setting be the same? What would example problems or strings look like? How does the content focus change? How do we get students at that age to engage in these mathematical conversations?…etc. So many questions that I still am trying to wrap my head around.

Fortunately for me, Crystal is a middle school teacher using Number Talks, so our conversations gave me great insight as to how they look and feel in the middle school. We talked extensively about the Number Talk course she had taken with Ruth Parker and shared students’ conversations during Number Talks in our classrooms. It was so exciting to see the connections between our students’ experiences and the path Number Talks can take after my students leave me and enter the middle school.  While I would love the opportunity to plan and observe Crystal’s classroom, and her to visit mine, she lives in Washington state while am on the east coast in Delaware, so our opportunity to collaborate face to face is not a reality.

Enter the wonderful world of technology, amazing resources available through The Teaching Channel, incredible teachers around the world wanting to learn and grow every day, and Ruth and Cathy’s new book Making Number Talks Matter!  Now, Crystal and I do not have to be on this amazing journey alone, but instead we have the opportunity to create an experience in which other educators everywhere can join us!

Together, we have planned and structured a book study unlike any other! Flexible to fit your needs, full of valuable resources, and completely FREE!

Beginning October 5th, each week will be dedicated to one chapter of the book. Conversations will happen on Twitter and Facebook, videos will be posted in our Teaching Channel Teams group, and we will even have a guest appearance by Ruth Parker herself! It is guaranteed to be an amazing learning experience that can only get better with your participation!  

For more information and to register for the book study, follow the links below:

We look forward to learning with you and Getting Better Together!

Check out Crystal’s Number Talks journey here!

-Kristin