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Robert B. Davis Institute for Learning. (1993). The infinite number line, Clip 4 of 4: Placing fractions and mixed numbers on the number line [video]. Retrieved from

B10B11B12-FRC-CMPRF-CLIP004

In the fourth clip researcher Carolyn Maher asked the students to each come up and take a turn placing some fractions on the number line. She called on Gregory to come up first. He placed a one half on the number line half way between zero and one. Laura then came up to the projector and placed one fourth midway between the zero and one half. Brian commented that she should put it on the one half if they were going from zero to two. Alan agreed. The researcher replied that Laura wanted right where she had placed it. The researcher then asked if one half and one fourth could go on the same spot. Some students said it depended on what the whole was supposed to be. The researcher repeated her question asking if a fraction could have more than one place on the number line. She asked if the point named a quarter could be in two places on the number line. A student replied that there could be. Alan said that there could be infinite places because between any two points a person could divide into fourths. The researcher followed by saying that she wasn’t asking about dividing the line. Erik stated that it was impossible; that you would have an improper fraction when a person had two wholes each comprised of fourths would be eight fourths. Brian interjected that two wholes could be renamed a whole. Erik countered that they were two separate wholes. The researcher asked the class to establish a few things to make sure they were all in agreement. She asked the class if the number line ended or if it was continuous. The students replied that the number line went on and on. The researcher asked the students the same question in regard to rulers and Cuisenaire rods. The students replied that a ruler as well as the rods ended. The researcher stated that the ruler and Cuisenaire rods were segments. Jessica replied that it could go on and on if a person wanted them to go on and on. The researcher stated that right now these models are models whereas the number line goes on and on without stopping forever. The researcher asked if they could build such a model that went on and on forever. She stated that that was the idea. The researcher said that maybe it would help them to think of pieces of the number line and not to get the piece confused with the ruler or the rods. She continued and said that once a number name was given to a particular point then that point will always remain with the same name. She added that the question is where to fit the other fractions and how to give them names. Erik revisited his earlier argument. The researcher asked Erik what name he would give the midpoint between one and two. Erik stated that it would be one and one half. Erik countered that it would not be fourths. David said he placed one and one half midway between one and two on his paper. The researcher then asked the class where they place one and three fourths. Michael replied that a person would probably put it a little to the right of one and one half. The researcher asked by how much to the right. Jessica replied that it would go in the middle of one and one half and two. Meredith commented about two fourths. The researcher commented that she did not understand so Meredith came up to the overhead. Meredith said that if you had (one and) two fourths that it would be equal to one and one half. She then said if you had two more fourths it would be equal to two. The researcher asked Meredith if she had another fourth what she would have. Meredith said one and three fourths. Michael agreed. Later, he said that it would be a fourth because a half is two fourths and what they were looking at was a half of a half. The researcher told the students not confuse the lengths with the new number names and then asked the students to think about where two and one half would be placed. Alan stated that it would be behind the two. Kelly said that it would be a little bit past the two. The researcher asked how much past the two. Kelly replied half. The researcher asked if Kelly meant half-way. Kelly replied that she did mean half way. The researcher asked half of what. Kelly replied half of that ruler. Amy replied two and three. The researcher asked again where to put two and one half. Another student replied half of the ruler. David said six inches. At the end of the session the researcher asked the students to find as many fractions as possible between zero and two. She said she would see them at the end of next week. Erik walked up to the researcher at the end of the session as other were packing up to leave. He said that if a person had two wholes then there would be an improper fraction which would be eight fourths. The researcher followed with saying that he was giving her another name for two which would be eight fourths.

http://hdl.rutgers.edu/1782.1/rucore00000001201.Video.000067474

doi:10.7282/T3HD7T7M

The video is protected by copyright. It is available for reviewing and use within the Video Mosaic Collaborative (VMC) portal. Please contact the Robert B. Davis Institute for Learning (RBDIL) for further information about the use of this video.