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1、ATTEND TO PRECISION: FRACTIONS AND GEOBOARDS CONSTANCE RICHARDSON: What does half mean to you? What would you tell me? STUDENT: Two equal parts. RICHARDSON: Two equal parts, all right. Everyone understand what she means by equal? All right, then give me what you mean by equal. When two things are th
2、e same size. Ah, theres another word. So equal and same size. Today we want you to think about halves. And were going to use the geoboards to show you youre going to show me, actually how many ways can we make half. LIFE LeGEROS: The standard for mathematical practice “attending to precision“ is les
3、s about getting the right answer than about clearly communicating thinking. Here we see a classroom thinking about geometry, which is a fantastic subject for providing opportunities for clearly communicating thinking and using related skills such as developing definitions and using mathematical lang
4、uage. The only way to get it in half was to cut it in half or put a rubber band around it. Okay, all right. If I put it here, holding the rubber band down, Im in half. STUDENTS: No. RICHARDSON: Why not? He said put a rubber band around it and put it in half. STUDENT: Put it more to the side a little
5、 bit. RICHARDSON: Want it more to the side this way a little bit? Okay, now were at half. STUDENTS: No. Do you see what I mean? Youve got. when I say a half, I need to clearly understand what I mean by a half. And Tom. am I doing what Tom asked me to do? All right, are we at a half yet? STUDENTS: No
6、. RICHARDSON: How do you know that? PJ? STUDENT: Because theres five rows of pegs up there and if you put it in the third, theres going to be two on each side. RICHARDSON: Oh, so were thinking a little bit. You counted there are five rows of pegs here and you said if I put it on the third one. STUDE
7、NT: The third one going down. RICHARDSON: Third one going down such as this. Thats half and then you put another one around the third. RICHARDSON: And put another one around the other third. And are we at half? STUDENTS: Yes. RICHARDSON: How do we know that, besides PJ counting over and saying, “Wai
8、t a minute, is this half?“ or did he use. what term did he use for this one? Middle. RICHARDSON: Its in the middle. If we take five and go in the middle, we now are halfway there. So how could we prove that without just counting in the middle? Theyre both the same size or you could use a ruler. RICH
9、ARDSON: All right, Corey says that theyre both the same size. Do they look the same size to you? Is there proof that theyre the same size? STUDENTS: No. RICHARDSON: Corey gave us a way we could prove that by doing what? We could actually measure it, but we really dont need to do that. Can anyone see
10、 a shape in here that we could use to actually measure? A square. RICHARDSON: A square. Is that a square? Could we use that shape to tell us if this is equal? STUDENTS: Yes. RICHARDSON: All right. How many squares would fit in each of them and see if one has more than the other. I could see how many
11、 squares would fit in each of them and see if one has more than the other. If this is truly a half, how many squares should appear in the other half? STUDENTS: Eight. RICHARDSON: Eight. LeGEROS: In this lesson so far, weve seen the teacher asking the class to attend to precision in two important way
12、s. First of all, shes asking the students to be very careful in explaining their thinking clearly. This has mostly been done around the mechanics about how to actually move the rubber band to represent one half. Second of all, shes asking the students to attend to precision in terms of the definitio
13、n of one half. So theyve actually reached, at this point, a consensus around what one half of a geoboard is. By using this concrete understanding of one half in the context of a geoboard of eight units, the students will now be free to explore ideas around area. RICHARDSON: Now, if you guys can prov
14、e to us that this is actually equal to eight squares. this one, you say, is only going to make one. STUDENT: Thats not going to make a square. Its going to make a triangle going like that. STUDENT: No, if you flip these two around, it would be a square but it would be bigger. Thered be a link. two o
15、f these, you can flip these and put these together. RICHARDSON: Marvin, do me a favor. Use a rubber band and make this shape here. STUDENT: See, thats what I was trying to say. It would be bigger than these squares right here. RICHARDSON: See that shape? Can you try and make it for me? Is this shape
16、 the same as that one? STUDENT: Yeah. RICHARDSON: Are these two shapes the same? So. now Marvin, if you made the same shape on that side. Uh-oh, Billys shaking his head now. His theorys working for him. Okay, and then what? STUDENT: Make it come down. RICHARDSON: Ah, now, have we agreed that these two together make this? STUDENTS: Yeah. RICHARDSON: Now what does that prove? STUDENT: That theres tw
