Tag Archives: Dot Images

3rd Grade Dot Image

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!

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The start of my planning….

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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?

So much to think about..


1st Grade Dot Addition

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.

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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

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Image 2 – See if students decompose to form equivalent expressions

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

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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:

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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!!


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:

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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

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….

Connecting the Dots in 1st Grade Math Centers

As many elementary teachers know all too well, effective Math Centers take A LOT of planning and preparation. Are all of the activity manipulatives available to students? Are the directions clear for students? Are the game boards laminated? Are the ipods/ipads charged? and on and on and on….

Last week, I realized that sometimes simpler is better. A handful of my 1st graders, who have a very strong place value sense and can mentally add and subtract 2-digit numbers, have been asking (hounding) me to teach them multiplication. I struggled with this for a few days because I didn’t want to just tell them that multiplication was “groups of” or take out the tiles for array building quite yet. It wanted it to develop from something more natural, something they were used to seeing but just in a different light.

This group of students is familiar with dot images since we do number talks with them often, focusing on addition equations and properties of operations. I put the following dot image on the board:

ImageThumbs went up (our signal for having an answer) and they all agreed on the answer of 36. Then I asked them write down all of the equations they could for finding the answers. Not the main point of this post, however when a student says I knew that if it was four 10s, it would be 40 so I took away one from each group to get 36, I can’t help but get goose bumps:)

I recorded their answers on the board and then chose to focus on 9 + 9 + 9 + 9 = 36. I asked them to explain that equation to me. One student said there was 9 in each bunch (close enough to “group” so i jumped on it). I explained that this is an example of when we can write this same problem as multiplication. “This is four groups of nine, so we can write that as 4 x 9.” Their reaction “That’s it? That’s Easy” Priceless. We did a few more together before the class ended.

The next time we met, I wanted to give them a chance to do some work in partners so I could walk around and listen to each of the conversations. I tried to plan an activity that would allow me to see their thoughts on multiplication and if any of it really “stuck” with them. I racked my brain, and the internet, for something that would be engaging and fun for them, until I just decided to give them a dot image and see what happened!

Here are some of the results:
Dot Image:


Student Work:

ImageDot Image: ImageStudent Work: (I was bummed, his second equation is wrong bc he forgot the middle two 6’s, but the rest is amazing!)Image

Least prep ever for a math center with the most amazing results! Demonstrates the relationship between addition and multiplication and has the properties of operations all over it! I am almost convinced you could teach K-1 math class with dot images, ten frames and number lines!

Mathematically Yours,


Isn’t Math Really Just How You Look At It?

Over the past year and a half, I have attended numerous CCSS trainings, read the standards and examined the CCSS learning trajectories. It is evident there is an emphasis placed on understanding of the properties of operations in the elementary grades. I don’t know about anyone else, but I remember it being taught to me as a lesson: Commutative Property is a+b=b+a… and such. No meaning behind it, simply some symbols, that if you could memorize and recite each, you were considered successful (as far as grades were concerned) in math class.

Fast forward to my second year as a K-5 math specialist. Having taught nothing below 5th grade in my previous 15 years in education, I am slowly wrapping my head around the depth of conceptual knowledge in grades K-1.  I always knew K-1 was very “hands-on” but I have to admit, I really did not understand the complexity and beauty in the way kindergarteners “see” math until this year.

The other day I did a number talk with a class of kindergarten students. I displayed various dot images with anywhere from  5-10 dots arranged in different patterns. My goal was to have students subitizing the dot patterns and writing addition equations to match the groupings.

I flashed the first dot image on the smartboard for @ 2 seconds and the students wrote the number of dots they saw on their dry erase board. Students shared their answer with a partner and showed me their boards. I put the image back up and asked how they saw (visualized) the dots.  We talked about different groupings, circled the dots for each, and practiced writing a couple equations together.

Feeling confident about the goals i had set for the number talk, i began to rethink them a bit after the following image:


Students quickly shared the answer of seven and then I asked, “How did you see the dots?”

The first student said,”I saw 2, 1 ,1,1, 2.” I had him circle the dots the way he saw them on the SMARTBoard and asked the students to write an equation for that grouping. Many successfully wrote a version (with some backwards 2s) of 2 + 1 +1+ 2+1=7. As I was looking around, I noticed one little girl had written all of the possible ways to arrange the 2s and 1s in the equation on her dry erase board. I realized at that moment, THIS is the commutative property in action! We shared all of the equations and I wrote them on the Smartboard.  I posed the wondering to the class: How can these equations look different but still have the same answer? They talked to their neighbor and the common response was because no dots left the picture…not exactly what I was looking for, but good answer.  I thought maybe it was too many numbers in the equation to see the commutative property or i just asked the question wrong, so i continued.

I asked for another way they saw it. Tons of thumbs went up (this is our sign for having a strategy) and the next student came to the board and circled 5 and 2. She knew it was a five, she explained because of a dice and she just knew two (there was the subitizing i wanted, but at this point we were going deeper). I asked students to write an equation for that grouping. They shared with their partner and we recorded 2+5=7 and 5+2=7. I was excited because two students had already written both equations on their boards before the share out. Now I posed the same type of question, worded differently, “What do you notice about the two equations we just wrote?”

I got responses like:
“The have the same numbers”
“Seven is at the end”
“Seven is the answer”
“He took my eraser” (all a part of the kindergarten learning curve)
“5,2,7 are there, mixed up”
I went with that  comment and pressed further… “So how can the 5 and 2 be mixed up and still have the same answer?”

After a  minute or two, one little girl said, “It’s just how you look at it. From that way (she pointed left) it is 2 then 5. If you look that way (she pointed right) it is 5 then 2.”

So there you have it teachers…the commutative property is “just the way you look at it.” Simple and beautiful.