Category Archives: Number Talks

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:

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

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

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:

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

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

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

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

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

~Kristin

3rd Grade Subtraction Number Talk

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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…

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.

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.

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…

Looking forward to seeing this bunch tomorrow!

-Kristin

Second Grade Number Talk

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

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

Image 1:

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

Image 2:

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

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

Image 3:

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

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

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

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

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

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

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

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

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

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

-Kristin

Making Number Talks Matter – Book Study!

Number Talks vs Number Strings

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When I first saw this image, I have to admit, it didn’t match what I had been calling a Number Talk in my classroom. Having started my journey with Number Talks with Parrish’s book, I saw it as a string of problems with a specific strategy as the focus of the string that got progressively more difficult (which sounds more like a string in reading this slide). As I started creating my own and making variations to them over the past couple of years, I have simply started calling them Math Routines….it was just easier than trying to make things fit into a certain framework in my brain. After going back and forth about terminology, I started looking at these structures more in the sense of purpose than the name and I am finding it very interesting. Please keep in mind as you read, this is completely from my experience with Math Routines in the classroom and I find all of these talks so incredibly valuable!

First comparison: Single problem vs string of problems. In my experience, I think these two designs have a substantial difference in purpose. When I give one problem, I am going for one of two things: a variety of strategies to see where students are in their thinking OR connections/comparisons of multiple strategies. Personally, I like the variety of strategies before and after we have investigated different ideas that could impact their thinking. For both the students and myself, it shows growth and I can use what I find to help me in planning future routines. Connecting and comparing takes a bit longer and, for me, I don’t get as many strategies out because we focus on connecting and comparing only a few for time’s sake.

Second comparison: Difference in having a representation and context. I don’t give a representation or context unless a student brings one up in a explanation. If it comes from them, I go with it, if it doesn’t, I don’t write it up there. If there is a specific representation I am hoping comes up because we have been doing work with it during our math class, but doesn’t, I do have them do a quick journal response. I will ask them something such as, “How was our number talk similar to or different from our work in math class this week?” Then I can read their journals and have some students share the next day about the representation connection they made. I also have had students create contexts after we finish a number talk with a prompt such as, “Write a context that you think fits one of the problems in the string. How does the context change for another problem in the string?” For example if the string has “16 x 4” followed by “16 x 8” I am hoping to see connections between the two problems in the context.

Nothing to compare in the final piece, multiple strategies discussed in both!

In the end, students are talking math with a mathematical purpose so they are win/wins in my book, but I am curious to hear what others think around the purposes of different formats of these routines! Would love to hear other ideas so comment away!

-Kristin

Growth Patter Number Talk….3rd Times a Charm

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Over the past couple of days, with my homeroom, I have tried a few strings of numbers to bring out the different ideas that are important when thinking about growth patterns and finding any term in a sequence, Here and here. Both days brought out many great ideas, conversations, and disagreements, however I couldn’t help but feeling the ideas we talked about in two days, could have been achieved in one and felt a little more connected. I knew it was completely the way I posed the problems, so when my second class came in yesterday, after missing a couple days of math due to testing, I was excited to try and adjust my previous work.

Apologize for the messy board, but I still cannot seem to get a handle on that recording thing…

I started with having a student count by 6’s and wrote that in blue. I stopped them at 4 because I was asking about the 10th term and wanted to see if some would figure our 5th and double. I think that is an interesting thing to think about when the start is different so I wanted it up there. I asked 10th term? 60. Ways to get there? 6 x 10 and, unlike my prediction of doubling 30, one student said 24 x 2 because two group of 4 of them is 8 and then two more 6’s (12) is added to that to make 60. I asked 100th term? 600. 2,000th term? 12,000. I asked them how they were getting those without counting and I got “I did 6 times 100” “I did 6 x 2000” and then one student said you could do any number by multiplying it by 6. I asked how he wanted to write that and I wrote that in green. Another student, who has done Visual Patterns with me in our RTI group, said, “We can also write that as 6 times n = Answer.” I asked them to turn and talk to a neighbor if they thought that meant the same thing. We had all yeses and I had some student prove it. I did the same thing with 8’s and wrote that in orange. They started using “A” for “Answer.”

IMG_0774_2

After that, I asked them to to count by 3’s starting with 6 and stopped them at 15… Asked for the 10th term and got, as expected, 30 and 33. Then the conversation took off with proofs and some really important ideas that was hoping would emerge. I love it when the class is practically divided in half on an answer, we had the 30’s and the 33’s. I asked a 30 to explain how he got the answer and he quickly said 3 x 10=30. I saw a lot of agreement, so I asked for a 33 to share their reasoning. A student said that we “need the beginning number, three, to find out where the tenth one is. 3 x 10 is 30 but then you started three ahead of that so you add 3 to 30.” I wrote that down on the board.

A student then said something that made me have a realization, “It shouldn’t change because you are still doing 10 jumps of 3, so it HAS to be 30. 33 is 3 x 11.” In my last class I had a student who kept insisting that the 10th term remain the same no matter where we started and I could not figure out what they were trying to articulate. NOW, I understand. 30 will always be the distance between wherever we start in the sequence and the 10th term, but not the tenth from the true beginning. AH HA!

So, the beginning number was suddenly becoming very important and articulating “10th term from where” was having students agreeing that the 10th term starting from the 6 was going to be 33 but when thinking about a rule for the pattern we needed the true beginning. We were just about to head back to our desks to continue our work when a student (different than the one who had originally said it) said that we could write this one “3 x n + 3 = A” because you have to “add the three you are missing from the beginning to get the answer.” I had them turn, talk and try a few terms out and see what they thought. It was all wrapping up nicely (I was excited about it) when another student said, “You could also write 6 + (3 x n) since you are starting at 6” ….oh goodness, they just don’t ever let it end and I love it:) A disagreement arose that it would have to be “6 + (3 x n -3) because of that extra jump of 3 to start at 6.”

I always hate to say that time got the best of me, but I had missed this group for 2 days of math and I saw this conversation going lonnnnng so I had them write those ideas down in their journal to kick off our class on Monday!

I love when I have the chance to refine ideas that don’t go exactly as I had hoped they would, especially when I know it was completely how I posed the problem or asked the question. After a couple days of talks not connecting as I hoped they would, third time was a charm!

-Kristin

Subtraction Number Talk: My Curiosity Today…

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Subtraction is the one operation that every time it arises in class, throws one more thing for me to think about into the mix. I have two recent posts around decimal subtraction, here and here, and I continue to work with whole number subtraction through number talks.

Today, I only had time for two problems in the Number Talk due to testing 😦 The first problem was 400 – 349. I was most anticipating students would subtract 50 and add one back or add up from the 349 to the 400 (1+50) to arrive at the answer of 51. I was surprised when a student said he “subtracted 100 – 49 to get 51 and knew that would be the same answer because if you added 300 to both numbers it would give you the same problem, so the same answer.” This made me think of a distance model on a number line, but I completely missed that opportunity and moved into the next problem. Seeing what happened next, it may have either made one strategy more clear or completely caused us to miss out on the conversation that followed.

Problem #2: 400 – 274

The student, “M”, on the right subtracted to find the distance between 400 and 274, however did not explain it that way so it left many students wondering how she knew what to subtract. I had a student ask her if that was her second strategy because she seems to have subtracted the answer from the 400.

The student, “C”, on the left solved it the way the majority of the class did, removal in part with some compensation at the end. Before he started explaining, he prefaced with, “I did it pretty much like M.” When he finished, he realized it was not the same and was confused as to where “M” came out with the same answer. He even exclaimed that, ‘I think she got the answer by mistake.”

“M” knew exactly what she did, however, I didn’t let her explain yet because I wanted the rest of the class to think about it a bit more. I told her she would be able to explain it tomorrow after we chat a bit more with it. I had them all end the class with a journal entry (surprising, right?:) I asked them what they understood, saw happening in each, or were not too sure about. It is just the most beautiful thing to read the honesty and reflection in their writings.

Some students could see what was happening…(even though it seems some tables have the vocabulary a little mixed up:)

Some left confused…

Some had a really interesting way of thinking about it…

And then there was “M” who cannot wait to share tomorrow…

Now, the question is, how to approach this tomorrow? I am thinking I would love three groups, one who subtracted in parts, one who found the distance by subtracting back to the minuend, and one group who adjusted the subtrahend and minuend to find the distance between. Have them create a context and representation that shows what they did (still working within the same problem they all have the same answer for) and do a share. I would like the share to go in the exact order of the groups I just listed above. Crossing my fingers I have time to talk some more math with them tomorrow, a silent classroom is probably more torture for me than them 🙂

-Kristin

Fraction Number Talks

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Two days a week we have a Math RTI period built into our school schedule. It is 50 minutes in which students receive additional math support through Marilyn Burns’ Do The Math Program, as well as the use of Number Talks. The groups are smaller than the regular core classes, allowing for more individual time with each student. In 5th grade, we focus heavily on the fraction module and building reasoning within the structure of our number system. When we implemented this structure about four years ago, the majority of the students in the more intensive groups had an extreme aversion to fractions and really just a lack of confidence in their ability to do math. They were just looking for a “way to solve” the problem to get it over with, rather than reasoning and working through a problem.

The fraction module, through the use of fraction strips, encourages the students to think about the size of fractional pieces, creates a visual for fraction equivalence and looks at the relationships between fractions. Students use these understandings to compare, add and subtract fractions and most importantly build their confidence in their ability to do math. The Number Talks I do with fractions really focus on getting the students to THINK about the fractions before just operating left to right and looking for a common denominator each time. This week I was doing a number talk on adding fractions with my group and put up this problem: 3/4 + 5/10 + 1/8 + 2/16. My thought in choosing the problem was there was some great decomposition and equivalence that could happen.

We usually do these problems mentally, so I don’t typically give them white boards but since I really wanted to see their thinking, I did this time (and I am so glad). Seven students came up with six different answers. It was awesome. I had them lay their boards down and look at them all before they started to explain their strategy. It was all of the great decomposition, equivalence, and addition I was hoping it would be. I especially love 3/4 + 5/10 = 1 1/4 and the bottom left where the student rewrote 5/10 as 4/8 + 1/8 to add to the 6/8.

I started to hear a lot of “Oh”‘s and “They are the same”‘s but the student who got 24/16 thought she was wrong because hers “looked different.” They all agreed the others were equivalent but I asked them to explain to their strategy and discuss the 24/16.

It was such a great discussion and as I was listening to them, I wondered how in the world any teacher could ever want to teach a group of students how to solve problems in only one way when there is such rich conversation in their individual thinking. They loved matching their answer to the others and proving how it was the same. Not to mention the confidence, independence and reassurance in their own math ability when they arrived at the correct answer.

-Kristin

Flexibility, Efficiency or Starting From Scratch?

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I ask myself this question numerous times during the course of school week. During number talks and in class conversations, the students show such amazing thinking and strategies in solving various computation problems. But, just when I think they are constantly thinking about the numbers, their values and sense-making, they seem to start a new problem from scratch without connecting to any of their prior reasonings. Is it flexibility in their thinking, efficiency or seeing each problem as a new one? I was SO glad to see I am not alone when I read Tracy’s tweets yesterday….

The conversation was an interesting one that then seemed to moved into number choice and thinking about what the students were thinking and what we do as teachers from here. We all definitely had a lot more questions than answers, which is always fun to explore!

So, of course I had to test out some of our questions into my number talk today. I had the students do the number talk from their seats so they had their journals readily available. I gave them 36 x 7, asked them to solve mentally and really think about the strategy they were using. I took answers, they all got 252, and I asked them to jot down how they solved it. We shared out and the majority had solved it just as Tracy had mentioned in her tweet, (30×7) + (6 x 7). Then I gave them 36 x 25 to see if, when given a 2-digt x 2-digit, they changed their thinking. I was also interested in the influence of the number choice of 25.

I don’t think it was the two-digit times 2-digit number that changed their approaches, but more so the influence of the 25. A lot went to double/halving because they could get to 50 and 100 and others used the 100 made of four 25s. One student multiplied 40x 25 and subtracted 100 while a few others used the associative property that Tracy had mentioned (4×25) x 9.

The final problem was 39 x 25. Unlike a typical number talk in which I push students to connect to previous responses in route to an answer, I instead asked them to not solve it, but just think about how they would solve the problem. After they had their thumbs up with a strategy, I asked them to complete one of the following prompts: “I used the same strategy I had used before because….” or “I used a different strategy in this problem because…” Here are some of their responses…

My conclusion is: the more students talk about their strategies, reasonings, and choices, the more they think about the numbers and what “makes sense” in the solution pathway. I think some students definitely get into a comfort zone with a strategy that works for them, and that is ok with me, but I definitely want to expose them to other ideas and things to think about. I loved that 25 and 39 influenced their thinking about the way to approach the problems.

I am not sure this answers any questions in our Twitter conversations, but I am always SO incredibly curious to see what the students actually do after anticipating their thoughts. The even better part is, they love sharing what they were thinking without the worry of being wrong. I even had one student who said she changed her strategy for the last problem because she got the one before it wrong after solving it twice. In her words, “It definitely was not working.” 🙂

Hope this gives you something to think about Tracy, Christopher, Sadie, Simon and Kassia!

-Kristin

Kindergarten Dot Image Number Talk

Math Minds

In Math, the Journey IS the Destination.

Category Archives: Number Talks

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

3rd Grade Subtraction Number Talk

Second Grade Number Talk

Making Number Talks Matter – Book Study!

Number Talks vs Number Strings

Growth Patter Number Talk….3rd Times a Charm

Subtraction Number Talk: My Curiosity Today…

Fraction Number Talks

Flexibility, Efficiency or Starting From Scratch?

Kindergarten Dot Image Number Talk