# Dot Image with Halves

Yesterday, I was thinking about some new number routines for the 4th and 5th graders who will soon be in their fraction units. I wanted to find something that both grade levels could engage in so the teachers could try them out and have a common talking point when we met to discuss what we learned about student thinking. I tweeted it out and didn’t want to lose all of the great thoughts so I going to compile them here!

This was the image I created:

Before launching this with the students, I thought the teacher would establish that a full circle was 1 and ask the question we typically use with our dot images, which is

How many dots do you see and how do you see them?

I thought there could be a variety of responses, but I was most anticipating these two responses:

• I know two halves make 1 and so the first row is 3. There are 4 rows so 4 x 3 =12.
• I know two halves make 1, so the first column is 2. There are 6 columns so 6 x 2 =12.

I knew questioning would be important to move from those descriptions to what the equation would look like and imagined we would get to some like this:

• 6 x 1/2 x 4 = 12
• 4 x 1/2 x 6 = 12

Other interesting things I am now thinking about thanks to others on Twitter:

Why am I stuck on dots? Circles makes so much more sense!

Yes, why not leave it completely open,not tell them that circle is 1 and talk about the unit?

It is funny that the same image conjures up different student responses, I always love that! I had not anticipate half of an array. I also love the idea of messing around with quarters here!

Awesome to have them do some scaling of the dots to find how many are there? And then I can see them saying it is half of the array like Michael suggested once it is uncovered!

I like the idea of moving them around however the cutting and tearing thing never goes well in my room!! I do love changing the value of the whole a lot!

I debated this part a lot when I was making it! I thought for the 1st one, a whole number would be best and then move to non!

There is the running list so far! Add away in the comments!

Since the 3rd grade classes are about to begin their multiplication unit, the teachers and I wanted to hear how they talk about equal groups to get a sense of where they are in their thinking. What better way to do that than a dot image? I chose the first image because of the 3’s and “look” of 5’s and the second image because of the 2’s,3’s, and 6’s, all of which students can count by easily.

Image 1 went relatively the same in both classrooms and much like I anticipated. There were two things that stood out to me as a bit different between the class responses:

• There were more incorrect answers shared in the 1st class than the 2nd class.
• In the second class, multiplication came out during the discussion. The “4 groups of 7 and 4 x7 = 28” in the 1st class came out after both images were finished and one student said she knew some multiplication already. She asked to go back to the first image and gave me that.

1st Class

2nd Class

After the first image, I anticipated Image 2 would go much the same, however it was quite different.

1st Class

2nd Class

After the 1st image, I was really surprised at the difference in responses and I have to say it even felt really different. My assumption at this point is that in the 2nd Class one of the early responses was multiplication.

I am left wondering:

• Does that early multiplication response shut down other students who don’t know anything about multiplication yet? While I asked her to explain what she meant when she said 4 x 12, I wonder if that intimidated others?
• How could I have handled that response differently so others felt OK using skip counting or addition to count the dots?
• Can we anticipate that type of reaction from other students when someone starts the discussion with something that may be beyond where the majority of the class is in their thinking?
• Was this even the issue at all? Did the 2nd class just see so many more dots and wanted to avoid adding and skip counting?

The 2nd Class ended with a journal entry after a student remarked, “If we know it is 8 groups of 6, then it is also 6 groups of 8.”

I asked if 8 groups of 6 is the same as 6 groups of 8 and the class was split on their response, so they set off to their journals.

The yes’s went with multiplication expressions representing the same product and commutative property:

I loved this no because the picture changes:

I am not sure about this argument but I would love to talk to the student a bit more about the bottom part!

After that talk, I am excited to see what these guys do when they actually start their multiplication unit!

# Which One Doesn’t Belong? Place Value

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

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

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

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

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

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

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

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

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

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

The third grade team is planning for a dot image number talk that focuses on this standard:

“Apply properties of operations as strategies to multiply and divide.2Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)”

Before this talk the students have been doing work with equal groups and are moving into array work with the arranging chairs activity in Investigations. They have also been doing dot images with smaller groups and have noticed the commutative property as arranging the same dots into different-sized groups.

These are the three images we are playing around with and anticipating which would would draw out the most interesting strategies based on the properties. We are thinking of having a journal entry afterwards to see if students make any connections between the strategies.

So if you feel like playing around with some dot images and doing some math, I would love anyone’s thoughts on which image you would choose and why!

The start of my planning….

My new thoughts on these images and responses…

After chatting with a few friends yesterday and thinking about which image would elicit the most expressions that could allow students to see some connections between the properties of operations, I am thinking about some changes to the images (in orange).

In image 1, I am wondering if we should split each group of 8 into fours but leave a bigger space between the top four groups and bottom four groups. It may allow students to better see the 4’s and then group them as 8’s and at the same time thinking about “doubling” the top group to get the total because of symmetry. They could then explore ideas like (4 x 4) + (4 x 4) = 4 x 4 x 2  or (4 x 2) x 4 = (4 x 4) x 2 [associative property] or 8 x 4 = (4×4) + (4 x 4) [distributive property] or any fun mix of them. If we leave it as it is, I think it may be hard to move them past 4 x 8, skip counting by 8’s or using 2’s.

In the second image, I love the structure of it but am wondering how students could use that 4 in the middle aside from just adding it on each time? Will we just end up with a lot of expressions with “+4” at the end? I am wondering what would happen if we adding an extra group of four next to it? Would students see the structure of a 5 and double it in some way? (5×4)x2 = 10 x 4 or 5x(2×4)=(5×4)x2 [associative] or 5 x 8 = 10 x 4 [doubling/halving] or 2×4 + 2×4 + 2×4 + 2×4 + 2×4 = 10 x 4

Then what question to pose at the end? Do we ask them to freely choose two expressions and explain how they are equal? or Do we choose the two we want them to compare? Do we have the dot image printed at the top of the page for them to use in their entry?

~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

# 3rd Grade Dot Image Number Talk

Since the 3rd graders are entering their multiplication unit, I find it the perfect time for some dot images!! I used the image below as a quick image in which I ask them to think about how many dots they saw and how they saw them. Quick images are so great for pushing students to visualize the dots and move beyond counting by ones and twos. I flashed the image for about 3 seconds, gave students time to think, and then gave them one more quick look at the image to check and/or revise their thinking.

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

# Dots, Dots, and More Dots…the Planning Stage

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…

1. Could they come up with an estimate after one flash of the image or two?
2. 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?
3. What did they look for on the second flash? Were they looking for the missing pieces first or second?
4. 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….