Category: Grade 4

Categories
Fractions Grade 4 Numbers & Operations -- Fractions

Fraction bar as an operation

Misconceptions surrounding fractions are so well-studied that I feel a bit ridiculous sharing anything about them. Anyway…

I was chatting with this kid who was having a bunch of trouble with written fraction notation. She had been correctly solving problems that involved language such as “shade in four out of seven pieces” or “divide this shape into eighths,” but got stuck when she reached a problem that asked her to “shade in 4/6 of the shape.”

Alice: Oh, so that’s 5.

Me: Can you explain why?

Alice: Because it’s not six sixths.

Me: So, not quite.

Alice: Oh, it’s 2. Because that’s 6-4.

Me: …

Alice: Or it’s 10?

Me: See…

Alice: I’m really confused here. What’s the answer?

There’s no puzzles or misunderstandings here. Alice thought that the fraction symbol was an operation between the numbers 4 and 6. And of course she did. Every other time that she’s seen two numbers and a symbol before she’s been asked to produce a third number. This is new ground for her.

I’ve been taking the advice of Brilliant Commenters Fawn, Jenny and Avery and using the language of “out of” to bridge the gap for this kid.

 

Categories
Decimals Fractions Numbers & Operations -- Fractions

A 4th Grader Who Prefers Decimals to Fractions

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This student — let’s call her Alice — is in 4th Grade. She did some work with fractions in 3rd Grade, but clearly isn’t comfortable with them.

I went over to Alice and noticed that she wrote “0.5” for point A. I asked her to read that number, and she said “a half.” Then I drew a half-filled circle and I asked Alice to tell me what fraction of the circle was filled in. She said “a half.”

Me: Can you write “a half” as a fraction?

Alice: Why do you have to? This way is so much easier.

[I show her how I write a half.]

Alice: Oh, a one and a two.

[I draw two more circles, one with a quarter filled in, the other with three quarters filled in.]

Me: What part of the circle is filled in in these two circles?

Alice: A quarter. Three quarters.

Me: How would you write those numbers down.

Alice: Umm…so this would be one-four?

Me: Yes, though I’d read this as one-fourth.

Alice: And this would be one-three.

This is interesting in all sorts of ways. First, because you can really see in Alice’s work the difference between written and spoken language. Alice can tell you what a half is. She can even tell you how much is shaded in on the other circles, but she can’t write it. Attention needs to be given to both verbal and written language, and we teachers tend to focus on our students written work.

Also, “one-four” and “one-three”? That’s so interesting. Alice sees “three” as the most important part of “three quarters,” and tentatively thinks that fractions are just always “one-something.” That’s a pretty strong tell.

The other remarkable thing is how strongly Alice prefers decimal representations to fractions. Alice showed this preference consistently in her problem solving.

IMG_3252

 

The kindly Professor Danielson argues that, in a curriculum, fractions ought to precede decimals. But it’s also true that decimals are addictive. In my high school classes, kids use their calculators to transform fractions to decimals as a defensive measure. You know the easiest way to help (most) kids solve equations with fractions? Point out that they can convert those fractions to decimals.

Decimals are absolutely enticing to people, even to this kid who is just getting started in this whole mess.

Categories
Geometry Grade 4

Area is like Perimeter

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I’m pretty sure that I know what’s going on in this Geometry pre-assessment. (Thanks for the submission, RG!)

Notice that line there? The kid is counting squares. The dimensions of this rectangle are 10 by 6, and the kid is adding up 10, 10, 6 and 6 to get 32.

To me this is directly related to another perimeter/area mistake that I documented a few months ago. (Also related might be this.)

Categories
Geometry Geometry

Triangles and 3-gons

Triangles and 3sided

 

Two 4th Grade girls were playing “Guess My Rule.” All the green shapes fit Molly’s rule, all the red ones were excluded by her rule. Christi was trying to figure out the rule.

Christi: Oh, I’ve got it! They’re all triangles.

Molly: Nope!

Christi:  But, look, it’s true! They’ve all got three sides.

Molly:  They all have three sides, but they’re not all triangles.

Christi:  WTF?

Molly:  Yeah, L has three sides, but it’s not a triangle.

L

Why not? What makes this seem untriangley to Molly?

(Relevant: Young Children’s Ideas About Shapes)

Categories
Geometry Geometry Measurement & Data

Weird Conversation About Perimeter

I gave the 4th Graders meter sticks today, and (of course) they did all sorts of weirdo things with them. Drumming, whacking things, marching while cradling the meter stick like a rifle.

They were supposed to be measuring the perimeter of the classroom.

One kiddo seemed to be trying to poke the ceiling, but he seemed to be doing it with enough care that I thought he might be measuring something.

—

Me: Wha?

Him: I’m measuring that jut in the ceiling.

Me: Why?

Him: Because it’s in the way, you’d have to follow it if you were walking on the ceiling.

(Note: I’d previously described perimeter in terms of path. The perimeter is the path you take around some region.)

Me: But you wouldn’t have to go around that thing if you were walking around the room this way, while standing on the floor.

Him: But you would if you were on the ceiling.

Me: But we’re not measuring the path on the ceiling, we’re going on the floor.

Him: Oh, I thought that we were measuring the perimeter of the whole room.

—

Now, maybe he was just being a punk because he wanted an excuse to poke the ceiling with a long stick. Maybe, though, he had a really interesting interpretation of perimeter, as all the paths that you take around a room. After all, there’s some ambiguity in the way I talked about the perimeter of the room, since the room is a 3D object, and perimeter is usually applied to objects in the plane.

That ambiguity, though, is a feature, not a flaw of the task assigned. Too many perimeter problems that I see young kids do only take place around rectangles or other polygonal shapes. This conversation with the kid was a really interesting one because it pushed on the messy process of finding 2D ways of seeing our 3D world.

Categories
Geometry Geometry Geometry

“Perimeter is the space outside of a shape.”

I was hanging out with some 3rd Graders today. Their task was to find the area and perimeter of some shapes. This girl was working with a shape like this one, except not quite this one:

array1

In particular, the shape she was working with had a perimeter of 22, and an area of 21. She, however, had counted an area of 21 for both. She had already called me over once to ask whether you count each of the sides of a corner square in the perimeter. My Spider Sense went flaring, but I wasn’t sure how to help, so I told her that each side did count for the perimeter. She seemed suspicious, but went with it. Then she called me over again.

—

Girl: Wait…how is it possible for the perimeter and the area to be the same?

Me: That’s a really interesting question. I’m curious: what’s perimeter, anyway?

Girl: Well… Perimeter…It’s a thing, but it’s outside of something.

Me: Hehe. That’s not a bad start…

Girl: OK, so perimeter is the space outside of a shape. Area is the space inside a shape.

Me: Cool, that’s very interesting!

 

At this point I’ve sort of constructed a theory about her initial question. Maybe she’s thinking of perimeter as 3D space, instead of lined space. That could explain her confusion about the space around the shape being equal to the space in the shape. Maybe that’s also the source of her doubt about double counting the sides of a square in the perimeter. (Though that doesn’t fit in super-well.)

I decided to push on her definition.

Me: So, perimeter is the space outside of a shape. So is this all the perimeter? [I drew a shape and shaded in the area around it.]

array 3

 

Girl: No! No, it couldn’t be, because then that would go on for all of this space. It’s more like this:

array2

Me: Interesting! I have a question about your picture. Is it important that the lines stick out of the shape? Could you have drawn it where the lines don’t extend out of it?

Girl: No, it’s important that they stick out.

—

At this point I didn’t know exactly what to do, so I just tried to explain that perimeter and area measure different things. I gave concrete examples of perimeter (“It’s like a fence”), but I didn’t really feel like the explanations stuck with her.

I’m curious to hear all of your thoughts on this interaction. My takeaway is a curricular one. Area and perimeter are different concepts, and they don’t necessarily benefit from being presented together and in contrast with each other. Maybe it’s better to introduce each individually, and only play them off each other after students have a solid notion of each concept’s meaning.

Categories
Multiplication Numbers & Operations in Base 10

6 x __ = 240

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You can’t say that the kid is incapable of understanding what the box means here. Still, in the space of one line, it slipped through her fingers.

Is this connected to the way kids inconsistently treat exponents? I’m struggling to articulate a general principle, but it goes something like “Operations defined in terms of others are strongly associated with their parent operation, to the point that students often perform the parent in place of the derivative operation. As a result, students should always be introduced to a new operation in its own context, not in terms of other operations, whenever possible.”

Thoughts?

Categories
Division Numbers & Operations in Base 10

91 mushrooms, 7 people

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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.