# The Brick Problem: Three Cases

### What is the weight of the whole brick?

Data Case Margo

Initial Calculations: The Larger Visual #### The Video #### Transcribed Speech from the Video

The problem is asking scales are balanced with a whole brick on one side and an exact half of exactly the same brick, plus a 3-pound weight on the other.

What is the weight of the whole brick?

Step one says interpret the word problem visually

This is my scale that I’ve drawn. On the left side I’ve drawn a whole brick. And, uh, on the other side of the scale, which is supposed to be balanced. I’ve drawn—I brought to scale on half of the brick on the left side plus the three-pound weight on the right side, and this illustration is supposed to show that balance on both sides so we have a picture. Step two is to substitute the values for each brick on the left side and the right side of the scale so it will be balanced. And I chose simple numbers, one For my whole brick on my left side, but I made a mistake because half of one is point five plus three is three point five, And one does not equal three-point five, But that’s the idea and once you figure out what number is on the left equal half of the number on the left plus three Then that is the value–Then you’re solving um The weight of the whole brick

### Data Case Sandra

#### Initial Calculations

Case Sandra: The Larger Visual The Video ### Transcribed Speech from the Video:

Researcher comments are in parenthesis and are in italics; other words and/or phrases in parenthesis were difficult to decipher from the recording and may not be accurate. A— “dash” indicates an abrupt change of thought or realization.

The weight of a brick

Scales are balanced with a whole brick on one side and an exact half of exactly the same brick, plus a 3-pound weight on the other.

What is the weight of the whole brick?

So first I did step one I drew a picture to look like a brick; then I read the word problem and matched the numbers on the side of the brick. Well with then the brick what I do actually, I put one up here, and one down right here cause this I want to be two whole sides So I put the one there cause this side is by one half and when I read the words on this side, it said add three  But I times three—no, uh, but it said two. Then I multiplied one half times three, and it gave me three-sixths so I divided. So here you go, one half times three Equals three-sixths, so I did three sixths divided by one half which shows right—(stick)  (there’s a word here I can’t catch when she slaps the poster here. Is the poster-paper coming unglued?) three sixths divided by one half, and that gave me point twenty-five. So I took three-sixths from what I got earlier, which is one half—no I got point two five, that’s it (I mean), so (I did) three-sixths times point two five, it gave me one twenty-five. So, um, basically, I did the weight of the whole brick was point one twenty five and that’s what I wrote the weight of the brick.

Interviewer: Do you see anything now that might be weak point? What do you think is a weak point? Anything?

Sandra: yeah I feel like right there when I multiplied three, when it said add three In the word problem I multiplied

Interviewer: It should have been an addition there?

Sandra: Yeah it should have been

Interviewer: I wonder what caused the confusion

Do you think it’s the shape of that…?

Sandra: No I feel like cause I’m not used to multiplying again so I added; So this one gonna be worse cause I’m used to the word problem when you multiply, and they say add

So um—

Interviewer: that’s where you think the error might be

Shandra: yeah

Interviewer: Ok so I’m working on this with a math professor too, and we’re breaking down some of that stuff. Good; Excellent; Super; that’s it

### Data Case Isabella

#### Initial Calculations: Case Isabella: The Larger Visual Case Isabella: The Video #### Transcribed Speech from the Video

Researcher yells action

My problem is called Equal Bricks The question said scales are balanced with a whole brick on one side and an exact half of exactly the same brick, plus a 3-pound weight on the other. What is the weight of the whole brick? So the first thing I did was draw a picture of a scale that showed each side We knew that there was one brick on one side, and the other had half the same—pound of brick on the other side plus three. I put B because B equals a brick in pounds. So the first step was to draw a picture and identify variables or variable. Step two was to make an equation,so I put B equals half B plus three. So then I have to solve for B. Remember what you do to one side you have to do to the other side, and to bring all like or the same variables to the same and solve it. So what I did was subtract three on both sides so I got B minus three equals one-half B; multiply that by two Uh—on both sides to get B by itself over here. So I have two B minus six equals B. Then I subtracted B on both sides: ended up with zero. So you can solve with B. So you have two minus B minus six equals zero, which simplifies out to B minus six equals zero. So then I solve for B add six to both sides and B equals six. So the Brick equals six pounds So then I plug the answer back into the initial equation; this one for verification. So right here I have B equals one-half B plus three; six equals six over two plus three. six equals six Ta Da.

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## A Model of a Shared Attentional Frame

The model below of a Shared Attentional Frame is an adaptation of Tomasello’s (2003) rendering of the “Structure of a linguistic symbol” (p. 29) and a “Joint attentional frame” (p. 26).

This model has evolved from the ideas of Tomasello (2003), McCafferty (2002), and many others.

McCafferty, S. (2002). Gesture and creating zones of proximal development for second language learning. The Modern Language Journal, 86, 192-202.

Tomasello, M. (2003). Constructing a language: A usage-based theory of language acquisition. Cambridge, MA: Harbard University Press. 