# Counting Collections Extension

Today in 1st grade, we did a counting collections activity. One thing I am thinking about as I see this activity happening in K-2 classrooms, is the extension. While choosing questions for students who are struggling is difficult, choosing questions for those who are finished quickly, and correctly, I find just as difficult. What questions can we ask those students who organize, count and can explain their count perfectly?

I have been toying around with this idea for a bit. I have thought about asking them to combine collections, ask how many more they would need to get to another number or mentally adding tens and hundreds to their count.

I know I have seen tons of people on Twitter using counting collections and would love to hear of other ways we could extend this activity in the comments!

# Obsessed With Counting Collections

If you have seen my recent Twitter feed and blog posts, you can probably tell I am currently obsessed with Counting Collections! Because of this obsession, during our recent K-2 Learning Lab I made it the focus of our conversation. This was our first chance to talk across grade levels during a Lab and to hear the variation in ways we could incorporate counting in each was so interesting! Based on this lab, yesterday, I had the chance to participate in both a 2nd grade and Kindergarten counting collection activity and while there were so many similarities, I left each thinking about two very different ideas!

2nd Grade: Naming A Leftover

Based on our Learning Lab discussions three 2nd grade teachers had the amazing idea to combine their classes for a counting activity. While it was a great way to give students the opportunity to work with students from other classrooms, it also offered the teachers a chance to observe and talk to one another about what they were seeing while the activity was in progress. I was so excited when they sent me their idea and invitation to join in on the fun! I have never seen so much math in an elementary gymnasium before!

There was a lot of the anticipated counting by 2’s, 5’s, 10’s and a bit of sorting:

And while this is so interesting to see students begin to combine their groups to make it easier to count in the end, there were three groups counting base 10 rods that particularly caught my attention:

1st Group (who I missed taking a picture of): Counted each rod as 1 and put them in groups of 10.

2nd Group: Counted each rod as 10 because of the 10 cube markings, making the small cube equal to 1. They had a nice mix of 20’s in their containers!

3rd Group: Counted each rod as 10 but had a mix of rods, small cubes and some larger blocks. It was so neat to see them adjust the way they counted based on size…the rod=10, small cube=1 and the large block=5 (because they said it looked like it would be half a rod if they broke it up). After this beginning picture, they arranged the 10 rods to make groups of 100.

Kindergarten: Why Ten Frames?

Every time I am in Kindergarten I leave with so many things to think about! In this case I left the activity thinking about Ten Frames. I am a huge fan of ten frames, so this is not about do we use them or do we not, but more about….Why do we use them? How do we use them? What is their purpose? What understandings come from their use? What misunderstandings or misconceptions can be derived from their use? and Where do these misunderstanding rear their ugly head later?

To start the lesson, groups of students were given a set to count. With a table of tools available to help them organize their count, ten frames were by far the most popular choice. However, not having enough (purposefully) for everyone’s set pushed them to think of other means… which ended up looking like they were on a ten frame as well!

As the teachers and I went around and chatted with groups, we heard and saw students successfully counting by 10’s (on the frames or look-alike frames) and then ones. This is what we hope happens as students work with the ten frames, right? They see that group of 10 made up of 10 ones and then can unitize that to 1 group. It reminds me of Cathy Fosnot’s comment via Marilyn Burns on Joe’s post, which I had huge reflection on after this lesson too!

I was feeling great about the use of ten frames until a first grade teacher and I were listening to one group count their set. I wish I snagged a pic, but I was so stuck trying to figure out what to ask the girls, that I didn’t even think about it. They had arranged 4o counters on 4 ten frames and had one left over, sitting on the table, no ten frame. We asked how she counted and she said, 10, 20, 30, 40, 50…the 1 leftover was counted as a 10. I immediately thought of Joe’s post. Not knowing exactly what to do next, I tried out some things…

• I picked up the one and asked her how many this was, “One” and then pointed and asked how much was on the ten frame, “Ten.” Ok, so can you count for me one more time? Same response.
• I filled an extra ten frame pushed it next to her 4 other full ones and asked her to count: 10, 20, 30, 40, 50. I removed 9, saying “I am going to take some off now,” leaving the one on the ten frame and asked her to count again. Same response.
• I asked her to count by 1’s and she arrived at 41. So I asked if it could be 41 and 50 at the same time. She was thinking about it for a minute but stuck with “that is what I got when I counted.”
• Then I became curious if she had a reason for using the ten frame, I asked. She said it was to put her things on so I began wondering about the usefulness of the 10 frames for her. Was is something, as an object, that represents 10 to her but not able to think about the 10 things that make it up?

I left that class thinking about how complex unitizing is. We hope students are able to count 10 things, know those 10 things are still there even when we start calling a unit, 1 ten, and then combine those units but still know there are 10 in each one of them. WOW, that is a lot! However, they can easily appear successful in counting by 10’s, which is one of the many reasons Counting Collections are so powerful. They bring to light the misunderstandings or missing pieces in students’ thinking.

I then start to think of recent conversations I have had with 4th grade teachers about students who are struggling with multiplying a number by multiples of ten and wonder if this is where we can “catch” those misunderstandings and confusions before they compound?

What to do next with this class? Erin, the teacher, and I quickly discussed this as she was busy transitioning between classes. We were thinking about displaying an amount, lets say 23, with two full ten frames with 3 extra. Say to the class, “Here are two sets, do they look the same? How can  you tell? Two groups counted this amount two different ways.  One group counted it 10, 20, 21, 22, 23 and the other group counted it 10, 20, 30, 40, 50. Can it be both? If so, how? If not, which one is it?”

Would love any other thoughts. I am heading back to re-read all of the comments on Joe’s post to gain more insight, but I would love your thoughts too!

# 1st Grade: 1st Day of Subtraction

This week, I planned and taught with Kala in her her 1st Grade classroom! It was the students first day of subtraction in the official “take away context type of work”sense, so it was really exciting to see what they would do with it. During our planning, Kala discussed ways in which she anticipated students would solve the problems and we thought about the framework below that is also presented in the Investigation’s Teacher Notes for that unit.

Since it was their first day with subtraction, we wanted to be sure to capture their strategies as clearly as possible to help in planning future lessons. To do this, we designed a quick table with the following headings that we planned on jotting notes about the students’ work on as the lesson progressed.

The lesson was supposed to open with giving students the following problem:

Max had 9 toy cars. His friend Rosa came over to play with him. Max gave 3 of the cars to Rosa to play with. How many cars did Max have then?

We decided to make this a Notice/Wonder to really allow students to think about what is happening in the context, over focusing on the numbers. So, instead, Kala posed this (we changed the names from above on accident):

David had some toy cars. One day his friend Max came over to play with him and David gave Max some of his cars.

Right away, they began to exclaim, “That was a short story!” As they shared, Kala recorded their noticings and wonderings. The very first notice was about not knowing how many David gave to Max because it said “some.” On the very last notice, the words subtract and minus came out. She couldn’t quite pronounce subtraction so she went with “minus” for her notice. She said David was a minus because he gave some, but Max was a plus because he got some. I loved that way of thinking and had not expected that at all! We then said we were going to answer some of their wonders….David had 9 and he gave Max 3. Then, like a Number Talk, they gave a thumbs up when they had an answer and we shared as a class. Kala and I just continually asked, how did you decide on the 9? Why did you do that with the 3? Where was that in the story? We decided not to write these strategies down because we didn’t want to influence their work during Roll and Record. We wanted to see where they were in their own thinking.

I then put up the game board for Roll and Record and we talked about how this was the same and different than the Roll and Record they have been playing with addition. They did notice that this board started at 1 instead of 2 and ended at 11 instead of 12. This will be something I would love to have them thinking about more in later lessons…why is this board this way? We played a few practice rounds of rolling a number cube, taking away the number we rolled on the second die, sharing how we subtracted the two numbers and recording it on our sheet.

They then went back to their seats and played with a partner. They had cubes, number lines, and 100 boards available if they chose to use them. These are a few pics I could snap before the game boards were erased.

As they played, Kala and I both walked around with our sheets and recorded the strategies we saw happening around the room. I was really amazed at the thinking in so many ways! I could tell they were comfortable using a variety of tools and by the way they could explain their thinking, it was obvious they were very used to doing that as well. This was my completed sheet at the end of the period. There were a couple of students I missed and a couple were absent, but between Kala and I, we had a really great picture of where the students were in their thinking. I was really surprised I only saw a few directly modeling the subtraction. The arrows were from me just starting to write their thinking and forgetting to put it in the appropriate column!

After they finished playing, we decided to give them the original problem context with two different numbers (8 and 5) to see if they could create a written response of how they solved the problem as well as they were able to verbally explain to us. We did this because Kala had mentioned during our planning that she often sees them resort to drawing and crossing out when asked to show their work even when she knows the student has a different strategy in mind. I completely felt that same thing in 5th grade as well!

They did not disappoint in their journals! I am wondering if, in us walking around and asking them to verbalize their thinking, it helped them have a clearer picture of what they did? Just a hunch. We also asked them to “show your thinking” instead of “show your work” because we think that has something to do with the direct modeling at times too.

Here was the direct model…

We had some interesting counting backs. Some counted back dots, like the die. When doing this some put numbers next to the dots while others just used them as dots to count back on. I was so excited to see a student who had counted back on his fingers write out the process. He even wrote the number he started with and ended at the correct number. <- sometimes I see them count that 8 as part of the counting back process and end at 6.

We had a great variety of tools and models, whether used for direct modeling (like the ten frame and tallies) or used for counting back on the number line. The bottom papers were partners so it was interesting to see them do a jump of +3 versus -3. Just as class was wrapping up they were talking about where their answers were because the partner who subtracted 3, said it looked like her partner’s answer was 8 because he added three and landed at 8. Interesting convo to have later too! Their journals also made us realize that we didn’t use the word difference in a way that students knew the answer was called the difference and not sum (yeah, they know sum though:)! Something for us to think more about next time!

Then we saw some great related facts, like this one. I am assuming the number line above also was thinking this but put it on the number line instead of writing the fact. That is another interesting convo to have in future classes!

This student used a known fact to get the answer and could not have explained it more clearly. I asked her to explain to me what she did and she said, “I knew 8-4=4, if I take one away from the 4 it makes it a 3 and the answer changed to 5.” Wow. Conjectures and claims here we come!

What a great day! From here, the students do “Start With/Get To” cards as their Ten Minute math that will help emerge subtraction as distance as well as reinforce the relationship between addition and subtraction. They will also play Five in a Row which will allow ideas such as 7-5=6-4 emerge when looking at expressions with the same difference. Of course, many context problems follow from here too so it is going to be so fun to watch what they do with this work!

~Kristin

# 1st Grade Notice/Wonder

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

~Kristin

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

# 1st Grade Story Problems

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

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

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

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

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

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

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

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

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

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

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

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

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

~Kristin

# 1st Grade Dot Addition and Math Journals

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

~Kristin

Tomorrow I get to teach with a 1st grade teacher, Lisa! I am super excited! To give a bit of background, the students up to this point have done a lot of dot image number talks. These talks have been a mix of just dots with no particular order and others with subitizable dots. The main focus has been becoming aware of how students are organizing and/or combining the dots. Are they counting all? Counting on? Using known facts? Or using any combination of the three strategies? In their Investigations work, they have been building on these talks using the 100’s chart and number lines to represent the addition and subtraction contexts.

Today in class the students will be learning how to play Dot Addition, so we will be building on that work tomorrow when I join them.

We decided to build on this work and launch the lesson with a string of three dice images. Just to make it a bit interesting, we set the dice equal to each other and ask them how we could prove if it was true or false.

Image 1 – Hear if students recognize that order of the dice doesn’t matter in finding the sum

Image 2 – See if students decompose to form equivalent expressions

Image 3 – See how they talk about decomposition with three addends versus two. Can they be equal with more on one side?

Next we will review the game and show the change in game boards. Instead of finding sums of 6, 8, 10, 12 to 6, 9, 10, 15. Now, here is where I am wondering about what the changes are in student thinking? There is SO much in here! Is it about combining strategies? Is it recording? Is it how they decompose? Is it compensation? Is it the relationships between the addends and sums that students need to start to look at? Holy cow, we had all of these conversations in our planning and we are still not sure we have it right, but here is our plan from here:

• Observe them play on the new game board and take note of how students are find the sums.
• Pull out strategic expressions that we want to highlight in the group share.
• Ask students what they would do if they didn’t have a card they needed. For example, what would happen if there was no 5 and you needed it? What could you do? or Could you have made that expression with more than two cards? How do you know?

After they play, we have two options. If there are a variety of expressions, we will bring them to the carpet to look at a completed game board from my game with Ms. Williams that contains the expressions they have arrived at also. If there is not a variety, we will complete a blank sheet together, gathering all of the expressions they did have and then ask them to turn and talk to see if they could come up with different ways to write these equations with the cards.

If we use our completed sheet, it will look like this:

We were going to ask them to take a few minutes to look at the expressions within each sum and then talk about what they notice. For example, within 6 do they notice that you can either “move a dot” or decompose and the sum stays the same?

If that goes smoothly and we make it this far without running out of time, we will ask them to do the same noticing between different sums. Do they notice that you add three to every expression in 6 to get to 9? Do they notice that somewhere in the 10 expressions there is an extra 1 from the expressions in 9? Do they notice the 5 when moving from a sum of 10 to a sum of 15?

So much to see! I cannot wait! Would love any thoughts and I will be posting the follow up soon!!

-Kristin