Applying Reasoning and Proof
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Think about the Fraction Tracks game and consider the following questions. Select "Show Answer" to reveal our commentary.

 Question: How might the fraction pieces help students clarify their thinking and develop some conjectures about equivalent fractions? Show Answer
 Our Answer: For students who do not see some of the possible combinations, the fraction pieces allow them to find other combinations they can use in the game. The pieces also help students clarify their understanding and give them a visual model of the concept.
 Question: How does having students share strategies support the development of reasoning and proof? Show Answer
 Our Answer: Students bring the activity to understanding by using their own language, they expand on ideas by using their own examples, and they question and challenge one another in a different way to bring meaning and understanding to the mathematics of the activity.
 Question: How might this activity be extended for students who show an understanding of equivalent fractions? Show Answer
 Our Answer: There are many ways to extend this activity. One might be to extend the board to 2. Also, tenths could be removed or twelfths could be added to the board.
 Question: In what ways does this task encourage reasoning and proof? Show Answer
 Our Answer: This task expects students to apply and extend their previous experience with equivalent fractions to a new task. The expectation is that students will share their reasoning and will need to prove that their reasoning is accurate. The task also expects students to refine their thinking to help them develop understanding of the concept of equivalent fractions.
 Question: How does Ms. Paster encourage students to share their reasoning and thinking with one another? Show Answer
 Our Answer: Her questions lead students to include an explanation of their reasoning so that their partners also understand what they are thinking.
 Question: What characteristics of a classroom that encourages reasoning and proof are present in this lesson? Show Answer
 Our Answer: Students work in groups to play the game, talk about their strategies, explain their reasoning to one another, and then write down their ideas. The expectation is for students to be able to explain and justify their reasoning.
 Question: How does this activity show how each of the process standards can be integrated into a single activity? Show Answer
 Our Answer: Problem solving is enveloped in the task. Communication, in the form of the students writing and talking about their reasoning and justifying their thinking, is apparent in all parts of the class. Students connect their previous experience with equivalent fractions to a new situation. Students represent mathematical ideas through the use of models (fraction pieces and game boards).

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