Visualizing halves - The clinical interview

The student is given a sheet of diagrams and asked to identify which diagrams have exactly half shaded in.

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Transcript

Wheatley: I have a page here and would you tell me which of these show a half? Let’s just talk about…yes…tell me which ones show half.
Student: That’s a half, that’s a half, that’s a half….
Wheatley: You skipped over this one.
Student: Yes.
Wheatley: It doesn’t show a half?
Student: No
Wheatley: Go ahead.
Student: Half, half, half
Wheatley: OK now. You skipped over this one. You said this one is not t half.
Student: Right
Wheatley: OK. Why not?
Student: Because those are equal and those are equal, but the area of that is not going to be the area of that.
Wheatley: I understand. OK, how about his one and why is it not half?
Student: It would have been a half if that one was on a slant with it. But, it was straight so…
Wheatley: So, this side has more?
Student: Yes.
Wheatley: OK, let’s look at this one now. I am real interested in why you’d want to…So that does not have a half?
Student: Yeah it does. When I glanced at it I didn’t…
Wheatley: OK, so tell me how did you decide?
Student: If I were to slide the…if I were to translate the blank side over it would make a triangle the same size as that.
Wheatley: Oh, so moving that over, it sort of inverted the triangle this way. What about this one?
Student: That one….it almost looks like it could be.
Wheatley: Any way you could….
Student: Cause that’s the square and a square. It is hard to tell that is really…
Wheatley: OK
Student: It doesn’t look like it if it were just to be that. It looks like that the blank has more area.
Wheatley: More area than this piece?
Student: Yeah, more area than just his part in there.
Wheatley: OK. Now over here you just took the square.
Student: Yeah
Wheatley: So
Student: Well, yeah it does. If there was a line right there. So, that is too.
Wheatley: So that is a half also. How about this one?
Student: No
Wheatley: Why not?
Student: Because this side has more area. Because if it were on a slant like that one…or like that , you know.
Wheatley: Um hmm. Very helpful. Yes, very nice.

Analysis

The student has a fairly good concept of one-half. However, she does not consider h to show one-half. This parallelogram is partitioned with a vertical line that is not parallel to the sides of the figure. This causes her problems. She is unable to see that the left side could be rotated and placed directly on the unshaded part. Perhaps her concept of rotation is not well developed.

Her response to m is interesting. At first, she rejects the figure as having one-half shaded. However, when I call her attention to it, she provides an insightful explanation involving a translation of the unshaded triangle on the right over to the left to form an equilateral triangle congruent to the shaded region.