Combinatorics and Reasoning: Representing, Justifying and by Carolyn A. Maher (auth.), Dr. Carolyn A. Maher, Dr. Arthur

By Carolyn A. Maher (auth.), Dr. Carolyn A. Maher, Dr. Arthur B. Powell, Dr. Elizabeth B. Uptegrove (eds.)

Combinatorics and Reasoning: Representing, Justifying and development Isomorphisms relies at the accomplishments of a cohort crew of newcomers from first grade via highschool and past, targeting their paintings on a collection of combinatorics projects. by way of learning those scholars, the Editors achieve perception into the principles of facts construction, the instruments and environments essential to make connections, actions to increase and generalize combinatoric studying, or even discover implications of this studying at the undergraduate point. This quantity underscores the facility of getting to easy principles in construction arguments; it exhibits the significance of delivering possibilities for the co-construction of data via teams of inexperienced persons; and it demonstrates the price of cautious building of applicable projects. furthermore, it files how reasoning that takes the shape of facts evolves with young ones and discusses the stipulations for aiding scholar reasoning.

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Example text

One blue, and two blues, but Milin just said you don’t have all two blues, and you said that – why is that? All right, so show me another two blues. With them stuck together, because that’s what I am doing. In that case, no. Okay, so now what are you doing, Stephanie? What if you just had two blues and they weren’t stuck together, you could – But that’s what I’m doing. I’m doing the blues stuck together. Okay. Then we have three blues, which you can only make one of. Then you want two blues stuck apart – not stuck apart; took apart.

The shirts and jeans task (above) introduces the fundamental counting principle, a key idea in combinatorics. 3 Representations as Tools for Building Arguments 19 Fig. 1 A diagram and an organized list for displaying the shirts and jeans solution In solving this problem, students may abstract the mathematics underlying the real-world situation; they may come to realize that the number of combinations of shirts and jeans is equivalent to the product of the number of shirts and the number of jeans.

Yeah, yeah. 4 Stephanie’s Sharing Milin’s Family Tree Finally, during a whole class discussion, Stephanie confidently explained the reasoning behind her doubling pattern to her classmates as shown in Fig. 15. STEPHANIE: I have one red, okay? And I have a yellow and from each of these you can make two because all you have to do is you add on . . you can add on a red to a red and a yellow to a red . . and for the yellow you can add on a red to the yellow and a yellow to the yellow, okay? Fig.

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