Observing Student Reasoning and Proof
 Introduction | Sums of Numbers | Problem Reflection #1 | Products of Numbers | Problem Reflection #2 | Classroom Practice | Observe a Classroom | Your Journal

Think about the student work you just observed, and reflect on the following questions. After you've formulated your own answers, select "Show Answer" to see our response.

 Question: How does this problem provide an opportunity for students to learn about square numbers through reasoning? Show Answer
 Sample Answer: While Kaylee's conjecture may seem obvious to adults and older students, it helps the students make connections between a "square number," such as 5 times 5 equals 25, and a geometric model of a square with 5 rows of 5 square units per row. It also helps the students be more precise and careful as they make observations. You may notice that this is setting the stage for a far less obvious conjecture that is encountered in middle school, namely, that the difference between two consecutive square numbers, n2 and (n + 1)2 is 2n + 1. For example, the difference between 192 and 202 is 2 x 19 + 1. By definition, 2n + 1 is an odd number. As you look at the sequence of consecutive squares and compute the differences between each consecutive pair, you always get odd numbers: 1, 3, 5, 7, . . . 39, 41, etc.
 Question: How does Kaylee justify her conjecture? What is the limitation of this conjecture? Show Answer
 Sample Answer: Kaylee uses visual representation to explain and justify her reasoning. However, she does not mention the one extra square unit that has to be added in addition to one row and one column.
 Question: How does this problem encourage students to make thoughtful efforts to communicate their reasoning to their classmates? Show Answer
 Sample Answer: Because geometric representation of squares was used, the students are able to talk about the number of square units in one row to represent one factor, and the number of rows of square units for the other factor. They were able to continue their discussion by talking about using different colors of square units to help illustrate the difference between a 3-by-3 square and a 4-by-4 square.

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