So, my blog site is more than a little dusty. My kids began playing sports this year. I had no idea how my family's lives would change. I never would have dreamt we would have sports 6 days a week!
I have had so much fun with my first graders over the past few weeks! Our mathematical goal over the past few weeks was to come to an understanding about the relationship between addition and subtraction. I remember years ago just giving my kids the publisher's curriculum worksheets that the blank equations where kids just had to plug and chug and telling them what to do and why it made sense. My poor little kiddos of course copied what I did. Some were able to do other problems alone "successfully" and others, not so much...
The last few years, I have worked to become much better. I created tasks for students to work within contexts, to
notice that there's something going on with the numbers in the equations, and that it keeps happening over and over again. We eventually came to a discussion about whether or not we think it would work every time, to use the same three numbers in all the equations.
This year, I began the same way. Working within a context, allowing students to
notice and
wonder. I recorded equations from student representations on chart paper and left them up in the room so each day we can refer back to them. Leaving the work up, gives me the opportunity to say, "Hey wait, that's weird, we just noticed that yesterday... and the day before, and the day before, and last week..."
Last week we were at a point where we had collected a lot of evidence and I think to do that again would be pointless.
We began our lesson today by focusing on all the posters we had made about the equations we noticed. I asked students to again, talk about what we noticed. I told them, that I thought it was time to move from noticing and wondering to proving. I gave students a few problems within contexts. (Put together/take apart change unknown worked quite well) Students focused on coming up with as many equations as they could. (Weeks have gone by where students created their own representations and created equations. Now we are ready for the grade level math.) They partner talked with neighbors to see if they were missing any. Then, we came to consensus over all of the equations. We did this about three or four times. Then, we tried it with naked numbers (no context). Again, our results were as predicted.

My recordings of what the students came up with. 
One student finally said, "Okay, we get it..." Me:"What do you get?" Student: "This is always going to work. The numbers are connected." Me: "Find out what your partner thinks about what he said." A few students shared that some of the numbers are parts and some are whole. Me: "So what you're saying is, there is a relationship between these numbers?" Class: "Yes!" Me: "Let's assume you're right... How does understanding that relationship help you? Find out what your partner thinks." Students pretty much said that it makes it easier. Some students weren't sure. So I wrote an equation that is above their grade level standard and out of their comfort zone purposefully.
232  17 = 215 (Kids looked relieved after I wrote the answer.)
Then I wrote
232  215 = _____ The kids yelled, "17!" in just a couple seconds.
Me: "How were you able to figure that out so quickly?" The kids described that the first one helped them.
Then I wrote
17 + ____ = 232 Again, students responded 215 with ease.
Then I wrote
215 + ____ = 232 Even more students responded 17 right away.
Me: "Let me ask again a little differently... How can you use what you know about this (pointing to the first equation), as a strategy to help you solve these? Find out what your parter thinks."
Students talked with their partners. I had them switch and they talked to a different partner. Sadly, we were already late to recess at this point so I had to let them go. I would have loved to collect a reflective math journal right at the end of the lesson. I think that is where we should pick up tomorrow.
What I like about what I did with my kids this year, is that we really sat in the pocket of proving our conjecture before just saying that we accept that this will always work. We worked collaboratively to prove that this is true, and that we discussed and worked through how we can use this knowledge as a strategy to help us. I don't think my kids last year really got the "So what?" part of the understanding.
As always, I want to keep working on my questioning, so if you have any tips, please share!
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