Today, I had one of those opening activities that was planned as a 10-minute math talk, turn into something much more than that. I am always excited when the students drive the conversation that way, however today, I felt at a bit of a loss for the “right” question in our discussion.
We started with an Estimation and Number Sense activity in Investigations in which I flip four digit cards that form a 2-digit by 2-digit multiplication problem. Students estimate a product, we discuss strategies and whether our actual answer will be more or less than our estimate and why. It always sounds so simple in the planning part.
I flipped cards for the problem 81 x 82. Thumbs went up right away (our symbol for having an answer and strategy in our Number Talks) and we got the initial estimate of 6,400 because 80 x 80 was 6,400. A few students explained and revoiced that our actual answer would be more than that because we rounded both numbers down. One student said she had the actual answer of 6,642 and proceeded to take us through the partial products she used. So far, so good. Then a student says he used the estimate to get an actual answer, but came up with 6,644. He said since he only used 80 x 80, he needed one more group of 82 and two more groups of 81, which gave him 82 + 162 = 244. He added that to the 6400 and arrived at 6644. This is our board:
Students are nodding their head in agreement but then start to wonder why it is not matching the answer when we found partial products. It is off by 2, so some students think it is a calculation mistake somewhere but soon realize it is all correct. I send them back to their journals because they want to know which is right and I am not telling them. We come back together in a couple minutes and everyone has 6,642, even the student who gave me the 6,644. So, I ask them, “Then why isn’t “Billy’s” strategy working?” because that is really the fun part :).
This is where I was at a loss for a good question. Everyone could prove to me why the answer they got was right, but I didn’t know what to ask them in that moment without completely giving them the solution path. It is SO hard to question students. I say it all of the time and today, no matter how prepared I was, I was at a loss for a good question. Don’t get me wrong we shared some great work…
I got a example of why it doesn’t work when you have another problem…
I got the student who gave me 6,644, trying to compare what he did to what he knows the answer should be…
But even after we went through our volume lesson and they left me for the day, I still am thinking about what I could have asked them to push their thinking. I was feeling, in that moment, that certain questions would have pushed quite a few students to disengage while at the same time, I was not wanting to let it go. They left not knowing why that was happening, which I am completely fine with, but I don’t want it to be a missed opportunity either. I am thinking that tomorrow I will try to connect it to an area model another student had used to solve it and see how that compares to the problems that “Billy” used to get 6644.
I am up for any and all suggestions as to how to pull this conversation together. What would you have asked in that moment? What would you follow it up with tomorrow?
What if you related it back to the partial products answer, especially if she used an area method?
Did she use (80 + 1) x (80 + 2) = 80*80 + 1*80 + 2*80 + 1*2?
If so, you could draw it out and have them color each of Billy’s terms a different color: the 80 x 80 could be green, the 2 x 81 could be red, and the 1 x 82 could be yellow.
The 1 x 2 would end up orange because Billy double-counted it. 🙂
Thank you Julie! The coloring is a great visual! Thinking of putting these unresolved problems in a bank for my math workshop days. This way, students can choose to revisit problems that may not engage the whole class.
I think your idea of posing it on a choice day in the context of making an area model is a GREAT idea. Proving it doesn’t work in all the different models of multiplying kids know would be another fun way to provide some choice as students explore. Why doesn’t it make sense when counting equal groups? Why doesn’t it make sense with area models? Can we prove it doesn’t make sense with a scaling model?
I also think your decision to stop and think of good questions before probing further, especially when you knew it wouldn’t hook all the kids, was so wise! How smart of you to have ways in your classroom for coming back to things, together or in a small group. It reminds me of what we talked about in our workshop: structuring lessons purposely for teacher reflection time midway through.
Thanks so much Max!! That makes me feel so much better when I was feeling like a dropped the ball today. I love the idea of having them explore when it doesn’t make sense! I am excited to start an “unresolved problems” center in my math workshop rotation bc I know a handful of students would love to revisit this!
Thanks Again, always appreciate your thoughts!
I’m with the team suggesting area models/arrays. Not only will they see why it doesn’t work, they’ll be able to SEE the 2. Such a great discussion!
I agree with the suggestions above for sure as a way to push them towards an understanding. However, I like letting them sit on it for a bit. Maybe not having the right question at that moment was a hidden blessing. Perhaps they will now live in this unknown – where is the two land – for a bit. And you can’t buy that kind of engagement!
Honestly – from my standpoint this is THE question Kristin, “Then why isn’t “Billy’s” strategy working?” Because of your skill in asking the right question at that moment, your kiddos are really “doing” math.
Love this post and thanks for sharing!! -Zak
I think that when students are operating under a misconception, then the question that will shine a light on that misconception and expose it is really, really hard to come by since their paths to try to answer the question will be framed within the misconception’s structure. I love the ideas about drawing it out… and … I would revisit this lots and lots of times — I might even give it a name… call it “distributing the extras” perhaps because it’s going to show up a few more times in their mathematical futures. Imagine the bliss of the college teacher who gets these students who know that (a+b)^2 isn’t a^2 + b^2 🙂
Thanks so much for your thoughts! I did revisit with the student today because he was still working on it. I asked him to look at his array for the correct answer and then highlight where his products showed up from his first answer: https://www.educreations.com/lesson/view/wrong-answer-problem/25889492/?ref=link
Check it out!
Dammit, thought I would be really clever and suggest drawing rectangles, and I see I’ve been beaten to it.
I’ve always felt that graphical illustrations are often underused in maths teaching, particularly in cases like this. Teachers tend not to default to drawing pictures because they are mathematicians, and drawing pictures is not a rigorous mathematical proof … but as a visual demonstration, to help kids grasp roughly *why* something is happening, it’s often just what the doctor ordered.
I agree Steve and sometimes creating that representation is more difficult because it is the why. I dont know if you checked out the link in the previous comment, but here is the follow up: https://www.educreations.com/lesson/view/wrong-answer-problem/25889492/?ref=link