Applying Reasoning and Proof
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Reflect on each of the following questions about Ms. Olivas's class, and then select "Show Answer" to reveal our response.

 Question: What conjectures do the students make? How do they test their conjectures? Show Answer
 Sample Answer: In the beginning, when considering four students, they briefly test the conjecture that you multiply the number of students by 2, which turns out not to be true. Later, one group made two rows of 24 cubes and concluded that there would be 48 exchanges; however, after the teacher asked a question, they reconsidered and decided that this was not a conjecture they wanted to justify.
 Question: Some of the students became confused when trying to represent the situation with blocks, whereas Josefina, who made a diagram, was successful. How did different representations affect students' reasoning about the problem? Show Answer
 Sample Answer: The data tables and stick figures seemed helpful, at least when thinking about a small number of students. For larger numbers, Josefina made rectangles with names on them that clearly stood for every member of the class. This representation illustrated that each student should expect to receive 23 cards, which made the justification for her solution method (adding 23 on the calculator 24 times) more convincing and easier for other students to follow. Finally, using cubes or pattern blocks to show 23 times 24 was cumbersome and may have even hindered the solution in some cases. Choosing appropriate units is important.
 Question: What characteristics make this task appropriate for promoting reasoning and justification in the classroom? Show Answer
 Sample Answer: The problem is engaging and has a variety of solution methods. The open nature of the teacher's instructions made it more probable that a variety of conjectures would be made and that a variety of ideas would be debated and justified.
 Question: Did the students give convincing arguments, or make a start toward convincing arguments? In what way did the stick figure drawing on the board help the students make a convincing argument? Show Answer
 Sample Answer: The stick figures of four children made it possible for the students to label the figures with letters and to record the number of cards given and received for each figure. The figures also helped make it clear that all possibilities had been found and that each child gave three cards. Josefina extended this idea with a more abstract drawing that helped her justify her solution by writing every child's name on a square, with "23" recorded on the square. In general, the students took small steps toward making convincing arguments but were unable to solve the complex problem independent of teacher support.
 Question: How much of the reasoning was done independently by the students, and how much was supported by the teacher? Show Answer
 Sample Answer: We only viewed a short segment of the lesson, but the students seemed to be doing most of the reasoning and were inventing their own problem-solving methods. The teacher asked occasional questions to prompt students to explain their thinking but did not appear to suggest methods or correct the students. You may want to review the video at the point where Ms. Olivas asks, "If you started making exchanges, would there be 24 exchanges?" to see how that question prompted several of the students to reconsider their thinking and to share their reasoning with one another.
 Question: How did the teacher give evidence of the use of inductive reasoning? Show Answer
 Sample Answer: She talked about the relationship in general terms and based it on the multiplicative relationship, rather than on a few examples. This type of general rule is usually found through inductive reasoning, by observing relationships across columns in a table. In inductive reasoning, a conjecture is made about the relationship or rule that relates one value to the other. The conjecture is tested further, and a reason is given as to why it should always hold. This type of reasoning usually requires strong number sense and the ability to organize one's work in order to consider and compare several cases. Notice that the suggested rule would be an easy one to apply if 1,000 students were exchanging cards, and also fairly easy to reconstruct if it is forgotten, once it is connected to the idea of multiplying the number of card-givers by the number of cards that each gives.

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