### Standard 3: Mathematics as Reasoning

In grades 5-8, reasoning shall permeate the mathematics curriculum so that students can:
• Recognize and apply deductive and inductive reasoning

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Benchmarks 12E (Habits of Mind: Critical-Response Skills)
Recognize when comparisons might not be fair because some conditions are not kept the same.

Benchmarks 9E (The Mathematical World: Reasoning)
Some aspects of reasoning have fairly rigid rules for what makes sense; other aspects don’t. If people have rules that always hold, and good information about a particular situation, then logic can help them to figure out what is true about it. This kind of reasoning requires care in the use of key works such as if, and, not, or, all, and some. Reasoning by similarities can suggest ideas but can’t prove them one way or the other.

• Understand and apply reasoning processes, with special attention to spatial reasoning and reasoning with proportions and graphs
• Benchmarks 9E (The Mathematical World: Reasoning)
One way to make sense of something is to think how it is like something more familiar.

Benchmarks 9E (The Mathematical World: Reasoning)
An analogy has some likenesses to but also some differences from the real thing.

Benchmarks 9E (The Mathematical World: Reasoning)
A single example can never prove that something is true, but sometimes a single example can prove that something is not true.

• Make and evaluate mathematical conjectures and arguments
• Benchmarks 9E (The Mathematical World: Reasoning)
Reasoning can be distorted by strong feelings.

Benchmarks 12A
Offer reasons for their findings and consider reasons suggested by others.

Benchmarks 12E (Habits of Mind: Critical-Response Skills)
Question claims based on vague attributions (such as "Leading doctors say ...") or on statements made by celebrities or others outside the area of their particular expertise.

Benchmarks 9E (The Mathematical World: Reasoning)
Sometimes people invent a general rule to explain how something works by summarizing observations. But people tend to over generalize, imagining general rules on the basis of only a few observations.

• Validate their own thinking
• Benchmarks 2C (The Nature of Mathematics: Mathematical Inquiry)