# 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…

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

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

# Growth Patter Number Talk….3rd Times a Charm

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…

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

# Inspired Thoughts on Number Talks

During the majority of the professional development planning I have been doing this summer, I feel like one of the common threads is Number Talks. After each conversation, more and more questions start spinning in my head….questions about how often to implement, questions about teacher recording, and most importantly, questions about their purpose.

I wasn’t inspired to write them all down until I read @gfletchy’s post: http://gfletchy.com/2014/07/22/on-you-marks-get-set-number-talks/.  BTW *If you do not follow his blog, you most definitely should, great stuff*

1 – Through the math conversations, it fosters a safe, collaborative culture in my classroom.

2- Their conversations embody the Mathematical Practices in my eyes. Their use of structure of the number system, creating viable arguments, critiquing the reasoning of others and repeated reasoning is always music to my ears.

3 – I struggle with purpose…is the purpose a particular strategy? That is how Parrish’s book frames it. There is a string, centered around a certain strategy. Not that other strategies do not emerge, but the numbers are such that they lend themselves to a particular path. So, my conclusion is this – When doing a number talk string, I  am not pushing a certain strategy, but instead, encouraging the students to truly think about the numbers before simply “computing.” I do want students to think that if they are multiplying 39 x 45, to think about 40 and taking a group away rather than breaking both numbers to get 4 partial products. In thinking about the numbers more deeply, they call on their conceptual understandings to develop fluency.

4 – Is the purpose to generate connections between strategies? I do think there is a benefit to putting up one problem and recording all of the strategies to make connections between them. I use that as a formative assessment as to what my students know and also to identify misunderstandings/misconceptions that emerge.

5 – Fawn’s blog has sparked an interest to branch into more visual patterns to switch it up a bit. What that would look like in my 5th grade class, is something I need to work through but I think the algebraic reasoning behind them would be intriguing.

6 – I Completely agree with Graham, they must be a daily routine, they build computational fluency (based in conceptual understanding) and must not just happen on Fridays! Also, it is important for students to use their Number Talk reasonings in other math work.

7 – Teacher recording is something I am still trying to improve upon daily. Recording their thinking is harder than one would think! Also, I find WHAT I write can change the direction of the talk itself.

I am a huge proponent of Number Talks and would love to see our elementary work with them to start to move into the middle/high school classrooms!

-Kristin

# Number Talks – Fractions

Through doing Number Talks with students K-5, I started to realize that one thing I look for students to use in our whole number computation discussions is using known or derived facts to come to a solution. I feel like the problems I have been using are crafted to use the answers from previous problems to reason about the ending problem.

In the younger grades, I would like to see students using the double known fact of 7+7=14 to know 7+8=15. I want them using 23 + 20=43 to get 23 +19 = 42. I don’t want them treating every problem as if they have to “start from scratch” adding all or adding on.

An example in the upper elementary:

18 x 2

18 x 20

18 x 19

This progression leads them to use a known or derived fact (18 x 20) in order to solve 18 x 19. To build efficiency, I don’t want them to the treat the final problem in the progression as a “brand new” problem in order to reason about an answer.

Along these lines of thinking, as I observed students working the other day, I realized that students weren’t using this same use of known/derived facts when working with fractions. For example, a student was adding  3/4 +  7/8. He used 6/8 as an equivalent of 3/4, added that to 7/8 and ended with an answer of 13/8. Don’t get me wrong, I loves his use of equivalency and I am a fan of improper fractions, however I started wondering to myself if it would have been more efficient (or show that he actually thought about the fractions themselves) if he used a fact he may have known such as 3/4 + 3/4=1 1/2 to then add an 1/8 on to get 1 5/8? Or used 3/4 + 1 = 1  3/4 and then took away an 1/8? Is that the flexibility I want them using with fractions like I do with whole numbers?

I thought I would try a Number Talk the following day to see….

1/2 + 1/2

Thumbs went up and they laughed with a lot of “this is too easy”s going around.

1/2 + 1/4

Majority reasoned that 1/2 was the same as 2/4 and added that to 1/4 to get 3/4. Some said they “just knew it because they could picture it in their head” I asked if anyone used what they knew about the first problem to help them with the second problem? Hands went right up and I got an answer that I wish I was recording. It was to the effect of,”I know a 1/4 is half of 1/2 so the answer would be a 1/4 less than 1.”

1/2 + 3/4

Thumbs went up and I got a variety here. Some used 2/4 + 3/4 to get 5/4 while others decomposed the 3/4 to 1/2 + 1/4, added 1/2 + 1/2=1 and added the 1/4 to get 1  1/4.

3/4 + 3/4

Got some grumbles on this one, because it was “too easy” – 6/4…Duh! The class shook their hands in agreement and they were ready to move on to something harder.  I noticed that when the denominators are same, they don’t really “think” about the fractions too much. I waited….finally a student said, “It is just a 1/4 more than the previous problem so it is 1  1/2″ and another said each 3/4 is 1/4 more than a 1/2 so if you know 1/2 + 1/2 = 1 then you add 1/2 because 1/4 + 1/4 = 1/2.” I had to record that reasoning for the class bc it was hard for many to visualize.

3/4 + 5/8

Huge variety on this one and I thoroughly enjoyed it! From 6/8 + 5/8 = 11/8 to decomposing to combine 3/4 and 2/8 to get the whole and then 3 more 1/8s = 1 3/8.  There were many more students who used problems we had previously done.

What I learned (and questions I still have) from this little experiment:

– Students LOVE having the same denominator when combining fractions.

– Do they really “think” about the fractions when the denominators are the same? Can they reason if that answer makes sense if they are just finding equivalents and adding.

– Students can be flexible with fractions if you push them to be.

– Subtraction will be an interesting one to try out next.

– I would much prefer if I remembered to use the word “sum” instead of “answer”…. I tell myself all of the time, but in the moment I always forget.

– Using known or derived fact and compensation are invaluable for students when working with both whole numbers,  fractions and decimals.

– Are there mathematical concepts that present themselves later in Middle School or High School in which known and derived facts would be useful?

Happy Thanksgiving,

Kristin

# Reversing the Number Talk

I am a huge fan of number talks and use Sherry Parrish’s book at least two to three times a week to conduct a number talk with my students.  Sometimes I pose just one problem for students to solve mentally and discuss strategies while making connections between them or I do a string of problems targeting a specific strategy.  Recently, I have been focusing on partial products and using friendly numbers as strategies to multiply. I noticed that as the string went along, they wanted to try and predict what the final problem (or “the hard problem” as my students would say) in the string would be. I started taking a few predictions each time and the conversation was really intriguing to me.

For example, the other day, the string was:

5 x 10

5 x 50

10 x 50

15 x 50

15 x 49

As they predicted the final problem, they actually made a more difficult prediction than the ending problem, 15 x 49. They predicted problems such as 15 x 47, 30 x 51 and 15 x 52.  Their reasonings were targeting the strategy of using friendly numbers without me having to outwardly say it.

So I thought it would be interesting (and fun) to go in the opposite direction and give them the last problem of a string  to see if they could develop the string of three problems that would come before it.  I gave them “36 x 19” and they ran with it.  Here are some ideas i captured from the journals:

This is a great formative assessment for me to see their thought process through our multiplication problems. Definitely adding it to my list of favorite activities!

~Kristin

Number Talks by Sherry Parrish: http://store.mathsolutions.com/product-info.php?Number-Talks-pid270.html