Month: October 2013

Categories
Uncategorized

Collaborative Math Mistake

Jason Ermer has a really cool thing going with Collaborative Mathematics. He wrote to me with an interesting student response that he got, and asked if I’d be interested in sharing it. Good news: I am!

First, here’s the problem:

Then we’ve got the student response.

OK, bad news, I can’t embed this video. Since this is the internet, this may be the point where you’re all like “Oh yeah like I’m going to click on a thing to get to some other interesting thing. Hell, I don’t even know if this other thing is interesting.” Well I’m vouching for this thing. It is interesting. Clicky clicky.

Did you click? Science tells us that some of you didn’t. If you didn’t, then here’s my summary of what the kid said:

“8 is 1000 in binary, so you’ll land on the same finger that you’ll land on when you’re at your eighth finger.”

(See? I told you. I bet now you’re regretting not clicking on the thing.)

Jason is going to write a few posts thinking about this student’s response, but let’s give him a head start on that discussion here.

What strikes you about this kid’s response? Please share any questions that you’ve got about the response. I’m sure we can rope Jason into lurking in the comments.

Categories
Grade 3 Numbers & Operations in Base 10 Place Value

103 > 130 and Other Mistakes

Open thread. Talk about what you find interesting here, or anything. I’ll kick things off in the comments.

Categories
Measurement & Data Rates

Miles per hour vs. Miles in one hour

Here’s a short mistake that I came across today that I found interesting.

I was chatting with a 5th Grader. The question was, “What do you think is your top speed?”

Her: I don’t know how fast I run.

Me: Well, you know here is how fast I walk. [Walks.] I think that’s about 3 miles per hour.

Her: OK, well maybe I can run 6 miles an hour.

Other Kid: You can run way faster than that. You can run 15 miles an hour.

Her: Well, yeah, for a little bit. But I couldn’t run 15 miles in one hour. I’d get tired.

—

I don’t give enough thought to miles per hour. It’s really an abstraction of realistic rates, rates that you could actually use. Like, if it takes me 3.9 seconds, on average, to add a paperclip to a chain, then I can use that to realistically figure out how many paperclips I could chain together in 5 minutes. But miles per hour — at least in the context of running — isn’t realistic in that way. It’s a concept that imagines a world that pays attention to my current speed but strips away all the reality of exhaustion and physical limits.

In the future I’m going to try to be more sensitive and explicit about this when talking about miles per hour with little kids.

Thoughts about rates and the units we use would be very, very welcome. Share interesting anecdotes in the comments, please.

Categories
Algebra in Geometry Expressing Geometric Properties with Equations Vertical Angles

Verticle Angles

Nora

 

[Re: the title, I know I’m not the only one who has seen “vertical” misspelled in every possible way.]

It’s easy to say that this is sloppiness on the part of the student. And maybe it is. But it’s the sort of sloppy mistake that I would rarely make, and that beginners often make, which leads me to think that there’s something else going on here as well.

We’ve talked about reading on this site before, and it’s something that I don’t know a ton about.  But it seems to me that part of “looking for and making use of structure” is something like what I’m trying to get at. If you’re really experienced at math, then you start seeing a problem like this as rigidly structured into two separate and equal expressions. I’d bet that for a student that doesn’t have a lot of experience with these sorts of problems that sort of structure is less apparent, and this sort of mistake is less apparent.

Did that make any sense?

[Thanks to Nora for the submission!]

Categories
Division Numbers & Operations in Base 10

91 mushrooms, 7 people

IMG_3025

 

Me: What’s 91 divided by 7?

Her: [Draws hands on board.]

Me: What are these for?

Her: For counting.

My move was to nail the question down on a context and ask her the question again.

Me: Hold on. Let’s make up a division story for this question. Let’s say that 7 people are equally sharing 91 crackers.

Her: Can we change it to mushrooms?

Me: Sure.

And she starts counting on the hands. She hadn’t done this for smaller numbers, like 30 divided by 3. There she articulated that 30 divided by 3 is 10, because 3 times 10 is 30. That doesn’t seem to be on her mind right now, so I try to ask a suggestive question.

Me: [Draws 7 stick figures.] Here are the 7 people. They don’t have any arms though.

Her: Can you make one super tall and one super short?

Me: Not this time. They’re all the same armless height. Anyway, how many mushrooms can we definitely give to each person?

Her: 10.

Me: Cool, and that would take care of a bunch of the mushrooms. That would take care of 70 of the mushrooms. And how many left would there be for us to take care of?

Her: 21.

Me: Nice. So, how many more mushrooms can we give to each person?

And then she goes back to her hands and does a bunch of counting. I interrupt her and ask her whether we could give them each 4. She says no, after some thought. She says that it would have to be more than 2. It takes a little bit of thinking before she tries and confirms that 3 works.

—

I think that this picture, and this dialogue, captures an important step in learning multiplication and division, and how awkward it all is.

I’m very new to all of this, so I’d appreciate some comments. As is our custom on this site, here are a few prompts:

  • Umm…how did that dialogue go? What worked? What could’ve gone better, in your view?
  • I feel like there’s some wisdom here about how people learn division and multiplication that I’m not able to articulate particularly well. Maybe you can?
  • How do you ween kids off of relatively slow and sloppy methods like counting?

Looking forward to your thoughts.

 

Categories
Decimals Division Fractions Numbers & Operations -- Fractions Numbers & Operations in Base 10

44 crackers shared equally with 8 people…

Remainder5

 

Lots of good stuff going on here. But I don’t think I entirely understand where 1/8 came from, though I get how that gets turned into 5.8.

Remainder4

 

Remainder3

 

Remainder2

 

Remainder1

 

Remainder6

 

[I never know whether to include all the mistakes from a class set or just a few. I feel as if it’s helpful to include more mistakes, but sometimes overwhelming. My solution today is to post one especially cool mistake largely, and the others smallerly. Let me know whether that works.]

Categories
Division Numbers & Operations in Base 10

Lots of correct ways to do division

Div6

Div1 Div2 Div3 Div4 Div5

 

No mistakes here, but I’m sort of blown away by the variety of division techniques I saw in yesterday’s problem set. Thought that it was worth sharing.

Categories
Congruence Proofs

If ABCD is a parallelogram…Prove that angle A is congruent to angle E

IMG_3019 IMG_3020 IMG_3021 IMG_3022 IMG_3023 IMG_3024

These are from my classroom, and a little bit of context might be helpful:

  • This is from a very strong Honors class.
  • Proof, right now, means something less formal in our Geometry class than it might mean in yours’. Proof doesn’t mean “Statements/Reasons.” Proof means offering an explanation for why something is true.
  • We do this because it’s just as rigorous without crushing the souls of anyone in the classroom. Look at the array and variety of reasoning going on in these proofs. By keeping things less formal, we’ve got enough breathing room to actually do some Geometric thinking.
  • As the year has been going on, though, we’re getting more and more rigorous. These exercises help reveal some sloppiness in the kids reasoning. These proofs fail by their own standards of explanation. I’m thinking that I’ll be printing these out and handing them back for discussion. This is exactly what my English teachers friend does with essays.

Feel free to comment wildly here, either on my standards, some of my bulletted statements, or about any of the student work.

Categories
Multiplication Numbers & Operations in Base 10

9 times 13 is 121

IMG_3017

 

The kid’s handwriting is hard to read, so I want to point you towards 9 times 13, near the top of this page.

I think that this is a great example of a mistake that you can feel fairly good about. Your thoughts, on any of his work?

Categories
Equations of Parallel and Perpendicular Lines Linear, Quadratic, and Exponential Models* slope

Slope of two parallel lines

Tina says: “Two students have done this so far. Not a mistake, but still curious what these kids are thinking:”

2013-10-06 18.53.11 (1024x768)

She’s talking about the 4/6 thingy. Any ideas, people?