Kindergarten Number Lines…The Lesson

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Two days ago, I planned this kindergarten lesson with Nicole and we taught it today! It was so much fun and I just have to say, I have such an admiration for Kindergarten teachers..that hour was tiring!

The Number Talk was a sequence of two dot images, both showing 7. It always amazes me to see the students counting, explaining their counting and writing equations so beautifully this early. In both images we heard counting by ones, counting by “2’s and 1 more,” and saw students count by ones and twos in different orders, solidifying the concept that the order in which we count does not change the total dots in the image. There was such a wonderful culture in place where students were open to agree, disagree, share answers (right or wrong) and all of this was shown to be valued by Nicole.

Next, came our number line adventure. Nicole had strips of painters tape around the room and sent each group of 4 to their assigned tape. As Nicole handed every group the first card, we (Jenn Leach, another Kindergarten teacher, Nicole, and I) walked around to ask students why they placed the card where they did. In keeping with the plan, the number order and observations were like this:

1 – Every group except one placed it on the far left. It was interesting to me that each group put the card under the blue tape, not on it.
10 – This was a great one to watch. One group put it at the very end of the tape, others “counted out” from 1 to ten to approximate where it would go, and some just put it in the middle without much of an evident strategy. When we asked the groups that placed it in the middle, they said they needed to leave room for other numbers. I asked what numbers would go over there and they said, “big ones, like 100.”
0 – They all shifted the 1 card to the right and replaced it with the 0. I saw one group have a group member place it at to the very left of the blue tape, just before the blue tape actually started and a group member said, “That would be a number if you put it there, but zero is a number” as he moved it under the very beginning of the tape. So cool.
3 – This is where some serious shifting happened. I didn’t get to see all groups do their moving, but as I walked around, I did see the 3 very close to the 1 and all of the tens that were at the end of the line, moved down. It seems their spacing strategy had taken over.
9 – All of them attached it to the left side of 10.

Before we gave them 5, where we really wanted to see how they dealt with the half, Jen, Nicole and I convened quickly to figure out how we were going to see that. We thought the ten at the end would be much easier to see their thinking about 1/2 so we decided to tell the students that 10 was going to be their biggest number to see if that changed their line. We got a couple, “Ohs” and slides of the 10 and 9 to the very right end.

5 – Most went back to counting spots but I did catch a couple groups looking at spacing. One group was using the 1 card to decide on the spot for a 5 while another group said they knew 5 and 5 was ten but was having a hard time using that to place the card.

Because we were running long on this part, we gave them the rest of the cards to place, finalize and tape down. This is what a few of them looked like (the others were all like the third pic):

The above group worked from the right.

Loved the extra space before the 0 here!

This was by far the most popular line!

Then we had students walk around to other lines and talk about similarities and differences to their line. It was great to see the group who started on the right notice that the other groups started on, “that end” while the spacing was a huge topic of conversation. One little girl, whose group had placed all of the cards touching, said she knew why they spaced them out….”They took a breath. Like one, take a breath, two, take a breath, three, take a breath…” I had never thought about how the visual could impact the way we think about timing in our counting! The closer they are the quicker we count, the more spread out, the slower we count. Loved it!

We regrouped on the carpet and talked briefly about what they noticed….

All of the groups went started at 0 and went to 10.
They all went in order, “Not, one and then four and then three and then two…”
Some were spread out far.
Some had the cards squished together.

All really important ideas! Next we went to our big clothesline to play around. Nicole placed the zero all the way to the left and I placed the ten all of the way to the right and said, for this part, we are going to say the zero and ten cannot move. Each pair of students (each from a different original group) got a card to talk about for a minute and then we called them up in the same order as the individual activity to place the cards. It started off all shoved to the left until one little girl went to place her number and started spacing them all out so it “looked better in her brain.” We asked the others what they thought about that. Some said, “it looks right” (says a lot about how equal intervals are visually appealing and seem instinctual for some) while others said they need to all be “at that end” (attached to the zero). We never reinforced one was better than the others but more that there are many ways we could think about this. I have video, but here is a pic of a piece of the final line…

Then, because we are just so curious to hear about connections they make, Nicole asked if they saw anything the same about the ten frames they have been using in class and the number line. A few students said both had ten and one little girl said it was like 5 and 5. Then, it was pretty awesome…she went up to show it was 5 and 5 and started counting at the zero card, so zero was 1, the one card was 2 and so on so needless to say when she ended her second 5 she was at 9. She said, huh? Loved it! Another student raised her hand and said it was because she counted too many, she started at zero and there is no zero on a ten frame.

it was SOOO much fun and I feel so lucky to get to see and hear all of this amazing math conversations across these K-5 classrooms.

The harder part, or at least what I am grappling with right now, is where to go from here. When it is a lesson within Investigations, I find it quite easy to pick up and move on but since this one is something we did outside of the curriculum, it requires a different plan. I am not quite sure where to go with this, but I have a couple thoughts (and would love others)…

I wonder if students could think about when the number line would make sense to have all of the cards closer together. Like if a lesson was adding to 20 and 20 was on the end now, what would happen?
Could we think about measuring things that are really short versus things that are really long? That feels like choosing the appropriate unit of measure to me.
Could we just leave it up and see if students reference it? and maybe refine the distance between each number?
Could we find some children’s lit that are around measurement and reference the line?
Could we put some painters tape in the hallway and see how they interact with it? Could they think about walking every tile line versus the feel of two tiles each time?
Could they model addition on there? Like in connection to maybe their dot image number talks?

So much to think about and I don’t know if any of these ideas are right or wrong or even age appropriate, but I am loving learning this stuff!! I am just so thankful to have such unbelievable colleagues who love to play around with these ideas with me!

~Kristin

Even or Odd…So Much To Think About!

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At the end of the day, Lauren, a 2nd grade teacher and I started chatting about her upcoming lesson on even and odd numbers. I have done a lot of thinking about even and odds in 5th grade when we entered decimals, however I can honestly say I have not thought about it much more than a number being able to be broken into two equal parts or it can’t because there is 1 left over.

Enter this student activity page from Investigations…

Now, I know Investigations is so purposeful with how they structure their pages so I was immediately curious about the set up of this page and wondering why thinking about even and odd in these two ways was so significant. My mind went right to the foundation for the commutative property. For example, with 10, will each have a partner? yes, 5 pairs or 5 groups of 2 or 5 x 2. Can we make two equal teams? yes, 5 on each team or 2 groups of 5 or 2 x 5. I would like to extend this sheet to include a space for the expressions: 2+2+2+2+2 = 10 = 5+5.

Then of course I tweeted it and got this great stuff from Tracy:

So much great stuff to think about and I absolutely cannot wait to see how these students deal with conjectures and generalizations! I would love any more thoughts on this work because I am sure there is more great stuff in here!

-Kristin

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1st Grade Dot Addition

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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

Kindergarten Number Lines

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Today I had a great day of planning with a kindergarten and 1st grade teacher for lessons we are teaching together on Thursday and WOW, has it been such a learning experience for me! The best part is, we have a whole day to get feedback from anyone who would like to offer it before we try this all out!

In Kindergarten, the students have been counting collections, counting dot images in various ways and since I have been obsessed with the clothesline lately, we thought this could be the perfect mash-up! When I read the counting and cardinality learning progressions, however, I did not see anything in there about number lines in Kindergarten but I did find this in the measurement progression:

“Even when students seem to understand length in such activities, they may not conserve length. That is, they may believe that if one of two sticks of equal lengths is vertical, it is then longer than the other, horizontal, stick. Or, they may believe that a string, when bent or curved, is now shorter (due to its endpoints being closer to each other). Both informal and structured experiences, including demonstrations and discussions, can clarify how length is maintained, or conserved, in such situations. For example, teachers and students might rotate shapes to see its sides in different orientations. As with number, learning and using language such as “It looks longer, but it really isn’t longer” is helpful. Students who have these competencies can engage in experiences that lay the groundwork for later learning. Many can begin to learn to compare the lengths of two objects using a third object, order lengths, and connect number to length. For example, informal experiences such as making a road “10 blocks long” help students build a foundation for measuring length in the elementary grades.”

In thinking about this, I tweeted out about number lines in Kindergarten and immediately was reminded by Tracy of her post on this work from last spring! Awesome stuff! I sent the link on to Nicole, the teacher I am planning with, and we were both filled with so many ideas! We were both thinking about relative location on the number line but hadn’t thought more specifically about the equal distances between each number! We also were originally going to do the number line as a whole group, but after reading Tracy’s post we changed our plan to allow for more discovery and exploration of the number line!

Here is the plan….

Students will be in groups of 4. Each group will have a strip of tape on the floor in different areas around the room.
We decided to put the tape the length of 5 tiles to see if any group uses the tiles in thinking about space.
We will hand each group the same card one by one and ask them to decide, as a group, where it should be placed. We went back and forth with this one…we wondered whether we should just let them start placing, but we really were so curious to see their moves and adjustments with each card. We also thought that since they have been ordering numbers lately, the majority would just put each card next to one another on the line.
Now, the order of the cards…this was so much fun to talk about….
- 1 – to see if they place it at the beginning and then the adjustment when 0 comes up.
- 10 – to see if students put it at the end of the line and how they determine the distance from 1
- 0 – to see if students place it to the left of 1 and if they have to move the 1.
- 3 – to see if students but it closer to 1 than 10, how close to 3 they place it, and if they put it less than half.
- 9 – to see if students think about 1 less than 10.
- 5 – THIS IS THE CARD I CANNOT WAIT TO SEE! Since they have been doing ten frames so much, some students are comfortable with 5 and 5 is 10, so do they apply that logic here?
- 7 – to see if students put it right in the middle of 5 and 9.
- 6 – one less than 7 or one more than 5.
- 2 – between 1 and 3.
- 4 – again, one more or one less
- 8 – same
During all of the placing time, we will be listening and recording any important ideas we want to have students talk about when we go to the whole group discussion.

After each group has placed the cards, we will have them do a gallery walk to the other groups’ lines and ask them to talk about what is the same, what is different at each line. We will then gather on the carpet.

We have a clothesline up, much longer than their strips of tape to do the same cards as a whole group. We will give each pair of students one card to talk to each other where they would put it (based on their work in the earlier group work). *Something we did not think of until I just typed this was how we partner the students up…we should match them with a student from a different number line to vary the convo.

We will call the cards up in the same oder they did their group work and ask the pair to explain where they decided to put their card and why. After all the cards are placed, we will ask them what was important to them as we made our number lines and record that for future conversations.

As a future conversation, we thought it would be really cool to see what connections the students make between the number line, ten frame, and dot images they have been working with so much!

Also, if anyone knows of a children’s book that has something moving a distance of 10 or 20 units, I would love to hear about it! Every single book I read dealt with 10 as collections of things, never distance.

-Kristin

Too late to type up the 1st grade one now, but it will be around this Dot Addition game in Investigations: http://www.smusd.org/cms/lib3/CA01000805/Centricity/Domain/198/Dot%20Addition.pdf Will type that one up tomorrow!!

How Planning Mistakes Can Lead To Great Student Thinking….

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The other day I did this fraction clothesline activity with a 5th grade class and today I had the chance to do it again with another 5th grade teacher, Leigh. It is always so nice to get to have a do-over after having time to reflect and think more about what the students thought about both during and after the activity.

I really thought the conversation was great during the clothesline activity, but it took too long the first time. We noticed that some students began to disengage. To try and improve upon that, Leigh and I decided to give only one card to every pair of students instead of each student having one. However, due to us wanting to keep a few important cards we wanted to hear them talk about, some pairs had two.

I also did not like my placement of 0 being at the very end of the left (when looking at it) end of the string. I moved it out some and talked about the set of numbers that falls on either side of the 0. I felt much better about that this time around!

In the planning of the first clothesline activity, we took fractions from the work the students had been doing with percents and decided on putting 100% in there, completely thinking it would be at 4/4. As the student placed it, however, I started realizing that I never thought about the difference of 100% in terms of the area representations the students had been using versus 100% when talking about distance on a number line. But now, having time to reflect on the card, I thought it would make a great journal entry!

As we neared the end of the card placements, I handed the 100% card to a student and told her it was going to probably cause a lot of discussion but just put it where she thought it went. She said she got it, walked up there and placed it on top of the 2 (the highest number on the line). There were some agree signals going on and some other hands that shot right up to disagree. We talked about it a bit and then we asked them to journal their ending thoughts so we could move on with the rest of the lesson about different sized wholes.

Some thought that 100% was at 4/4 on the number line because it equals 1….

Some thought it was at 4/4, but because of the conversation became a bit unclear…

Some thought it goes on the 2 because it is the biggest number on the number line…

Some related it to different contexts with different wholes…

And one student said it can be anywhere with beautiful adjustments as it moves….

What a great day revisiting my planning mistake!

-Kristin

3rd Grade Dot Image Number Talk

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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.

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…

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:

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

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

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:

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:

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:

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!

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

5th Grade Fraction Clothesline

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Today, I had the chance to plan and teach with a 5th grade teacher and it was awesome! Last week, this class had just finished a bunch of 100s grid shading in thinking about fraction/percent equivalencies, so we picked up planning the lesson in Investigations with the fraction/percent equivalent strips. Instead of the 10-minute math activity, we thought it would be really interesting to do the clothesline number line to kick off the class period.

We chose fractions (and one percent I will talk about later) based on the fractions the students had been working with on the grids. We chose fractions based on different comparison strategies that could arise such as:

Partitioning sections of the line
Distance to benchmarks
Equivalent Fractions
Common Denominator
Greater than, Less than or equal to a whole or 1/2

We settled upon the following cards:

1/4, 3/4, 4/4, 1/3, 4/3, 5/10, 2/5, 100%, 3/8, 1 5/8, 1 7/8, 4/5, 11/6, 1 6/10, 1/10, 9/8, 12/8, 2

To start, I put the 0 toward the left of the line (when you are looking at it) and we practiced with a few whole numbers. One student volunteered to be first and I handed her a card with the number 7. As she walked up, looked around, walked up and down the line, looked at me like I was playing some type of trick on her, we immediately had the conversation about how knowing the highest numbered card would be super helpful. She settled on putting it toward the far right side and had a seat. I gave another student the 10 card. He put that at the far right and adjusted the 7 to be “about 2 cards away” from the 10, leaving a really long distance from 0-7 for them to think about. We had some students disagree so we talked about distance and adjusted the cards to be more reflective of distance. Since the conversation of half of the distance to 10 came up, I handed another student the 5 card and he placed it right in the middle. The discussion went back to the 7 and they decided that since 7.5 would be halfway between 5 and 10 that 7 had to be a little bit less than the halfway of 5 and 10.

Then, we moved into the fraction cards. We gave each pair of student two cards. In hindsight, for times sake, I would probably only do one card per pair. I gave them one minute to talk about everything they knew about the fractions they had and then we started. I asked for volunteers who thought their card would help us get started and called on a boy with the 1 7/8 card. He went up and stood all of the way to the right and said he couldn’t put his on. I asked why and he said that since the cards were all fractions the line could only go to 1 so his is more than one and can’t go on here. I asked if anyone in the class had a card that may help us out and a student with the 2 card raised her hand. She placed her card all of the way to the right, said “maybe it goes to two” and the other student placed it just to the left of it because, “it is only 1/8 from 2.” Awesome!

We went along with the rest of the cards and so many amazing conversations, agreements and disagreements happened along the way. There are a few things that stand out in my mind as some great reflections on the activity:

A student had placed 5/10 halfway between 0 and 1. The next student placed 2/5 just to the left of the 5/10 because, “I know 2 and a half fifths is a half so that means that 2/5 has to be less than 5/10. It is a half of a fifth away.” The NEXT student volunteered and placed 3/8 overlapping just the edge of the 2/5 card on the left. I was expecting percentages to come out, since that was their most recent work with those fractions, however the student said they knew 3/8 was an 1/8 from a half and 2/5 was a 1/10 from a half and an 1/8 and 1/10 are close but an 1/8 is just a little bit further away. Awesome and definitely not what I expected!
I wish I had not put the zero so far to the left. Looking back I am wondering if that instills misunderstandings when they begin their work with negative numbers on a number line similar to the original misconception that launched the activity with the 1 7/8.
Oh, the 100% card….complete mistake on my part, although it may have been a great mistake to have! In the first class, the student with the 100% card came up and said, “I have 100% and that is 100/100 which is 1” and put it in the appropriate place on the line. Just as she did that, I started thinking how I never really thought about the distinct difference between percent in relation to area (like the grids they had been shading) and 100% when dealing with distance on a number line. No one seemed to notice and since I didn’t know exactly what to ask at that point because I was processing my own thoughts, I waited until another student placed 4/4 on top of it and erased it from my immediate view!
- I stayed for the next class and this time I was prepared for that card and now really looking around to see what students’ reactions were when it was placed. As soon as the student placed it at the 1 location, I heard some side whispers at the tables. I paused and asked what the problem was and they said, “100% is the whole thing.” The next student who volunteered had the 2 card, picked up the 100% card on the way to the right side and put the 2 down and the 100% on top. Lovely and just what I was thinking.

I have never had students reflect on the difference of talking about percentages with distance versus area because I had never thought about it! It definitely feels like an interesting convo to have and a great mistake that I am glad I made!!

I will be back in another 5th grade class tomorrow and will see what happens…it could make for a great journal writing!

-Kristin

Dots, Dots, and More Dots…the Planning Stage

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

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

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

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

“For my 3rd – 5th I would love to show these in a progression as I could possibly be focusing

on three things:

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

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

However, if my purpose was to have them compare (more or less) and then creating a proof, I

would have the second image to build upon the first….like maybe add a yellow on the top an

bottom of the first image…so the first one they say, “I saw 4 (of course we ask how they saw

that four) then 2 and 2 and 2.” We ask how could we record that? 4 + 2+2+2 = 10. First flash of

the second image, “Is it more or less than the first? How do you know?” Second flash of the

image, “how many, how did you know? did you know it before I flashed it the second time?” I

would imagine most would do 10 + 2 very quickly and know it before the second flash. Could

be cool to ask how we could use 14 counters in the next image and have them design the

14th.”

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

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

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

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

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

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

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

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

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

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

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

To be continued….

3rd Grade Multiplication Talking Points

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

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

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

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

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

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

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

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

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

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

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

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

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

~Kristin

Fraction/Percent Equivalents

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It goes without saying that I miss talking 5th grade math with my students each day. But I am so lucky this year to have a new, wonderful teacher in 5th grade who lets me plan and teach some lessons with her! This lesson was one of her first lessons of Unit 4, Name That Portion.

Since in 4th grade the students do a lot of work with comparing fractions, we designed a Number Talk string in which students were comparing two fractions. We wanted to hear how they talked about the fractions. In the string we used a set with common denominators, common numerators, and one unit from a whole. On each problem we were excited to hear talking about the “size of the piece” being the unit and the numerator telling us how many of those pieces we have. Our 4th grade teachers really do a beautiful job with this work. They also used equivalents to have common denominators to compare and a few used percents, since they had done a some grid work with that they day before.

We started the lesson by asking them how they could shade 1/4 on a 10×10 grid. The majority of the students split the grid in half vertically and then again horizontally and shaded one quadrant. We heard a lot of the “1/4 is half of a half.” As I was walking around, I heard a pair talking about shading a 5×5 in that grid. I saw this as a beautiful connection to the volume unit they just completed in which they were adjusting dimensions and seeing the effect on the volume. I had her explain her strategy and wrote 5 x 5 under the 10 x 10 that was up on the board already and asked how that could get us 1/4 of the whole thing? One student said it looks like it should be half of it because 5 is half of 10, but then one student said since we were taking 1/2 of both it would be a fourth….this is where I hope Leigh (the 5th grade teacher) and I remember to use this when they hit multiplication of fractions!

They then worked in pairs to shade 1/8 and 3/8 and we came back to discuss. We noticed as we walked around that the shading was wonderful on their papers, but when asked to write the fraction and percent, most were blank. I remember this lesson from last year during decimals where the same thing happened. So, we asked them what they thought the fraction was as we got these three answers…

12 r4/100

12 1/2/100

12.5/100

They were not overly comfortable with any of them so we asked them to journal which one “felt right” to them and why…