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