In our math PDC group we decided to use string of math equations with our students across the district. The goal was to investigate what students know about the properties of operations and how they use them across the grade levels. The number stings were inspired by @jamestanton (I think…) and then modified to be appropriate for different grade levels.
I accidentally gave the original number string to my fourthies and it was very difficult (obviously, Ms. Miner!). But we talked about it for so long! And the kids were excited and asking questions and taking out manipulative to assist in their explanations and thinking. I couldn’t stop there. I decided to pull a small group of kids and present part of the “correct” number string to them.
The equation we worked on was (7×8) + (8×3). This is the record of their thinking, it’s hard to see what they did first, but I will try to talk about that below.
This is the audio of our math talk, it’s long and I didn’t trim it so there are a few interruptions. I have pulled out some of the things that stood out to me and wrote about them below.
Students immediately wanted to solve the (7×8). They were unsure of this math fact, so they broke it into (7×4) + (7×4) and got 56. They knew 8×3 = 24. They added 56+24 to get 80. Awesome! I could tell right away that they are comfortable with decomposing into smaller, more friendly math facts. They are still thinking additively. They were very sure of this answer, and they arrived at it with very little scaffolding from me.
** It was cool that they essentially did what I was asking them to do in reverse. They took 7×8 and broke it down into (7×4) + (7×4). I was asking them to take (7×8) + (8×3) and put it back together (10×8). They didn’t see this connection, but I think we can get there!
I decided to push them. I asked them to think about another way to solve (5:00 in clip). One student (Gavin) suggested switching the factors and multiplying 7×3 and 8×8 and then adding. They were surprised to get an answer that was close, but they knew it wasn’t right. And you can hear one little girl (Emerson) at the start tell him right away that he can’t do that because of the parentheses.
So I asked for another way. I prompted them to think about it in terms of “groups of.” I asked if they could see “7 groups of 3” (from Gavin’s thinking above) in the original problem (7×8). Now here they go on a little tangent about seeing groups within groups, which is very cool and I want to explore later. I think they are at the beginning stages of understanding the associative property but are still thinking additively and so it gets a little jumbled.
Gavin notices the 10, but he still wants to do 8×8 and then subtract 10. At this point I started to erase the parentheses and that’s when Emerson chimes in again. But I got stuck, and quickly put them back on! I think this problem does NEED parentheses. You can come up with multiple answers if they aren’t there, right?
At around 8:15 you can hear me start to lead them to the idea of 10×8. I may not have continued on this path, but I wanted to explore the extent of their knowledge of the properties and so I wondered if they would even be able to see the 10×8 if I presented it to them. At 8:48 you can hear the students start to attempt to explain why it works. Emerson thinks I left out one of the 8s, and Molly and Gavin try really hard to explain where that 8 is. I was impressed that Molly was trying to explain that even though there are two 8s written in the original problem, there are more than two 8s… there are seven 8s and then three more 8s. I should have cued into that more than I did at the time. We then got out some cubes to see if that would help with the explanation. They built and aligned them to look like two separate arrays; 7×8 and 3×8.
Eventually at the end of the video you can hear Emerson ask about the parentheses again. This is where I realized how hung up on the parentheses she was. This is when I started to wonder a few things:
First, it was a content question. What is the reason we can add 7 and 3 before doing the multiplication? And second, how do I get this student to think flexibly about the parentheses that she is so certain about using? Without overgeneralizing that they can ALWAYS be used flexibly?
I checked in with the math coach (Sarah Caban) and she tweeted out for help! Thanks #MTBOS for all of the responses. Check out this collaborative googledoc started by @davidwees! https://docs.google.com/document/d/1N9WvU9ypZCfVM3pJXDapHv0Pssw95hgVcRElMmZF8NE/edit?usp=sharing
I am super excited to continue this exploration with my kiddos! I have so many ideas about where to head next, I wish I could jump into our multiplication/division unit now!