**Benchmark
9E: The Mathematical World : Reasoning**
**(grades 6-8, page 233)**
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. (1 of 6)
Standard 3-1, page 81, Grades 5-8

Recognize and apply deductive and inductive reasoning
Standard 3-3 page 143, Grades 9-12

Follow logical arguments

Standard 3-4 page 143, Grades 9-12

Judge the validity of arguments

Practical reasoning, such as diagnosing or troubleshooting almost anything,
may require many-step, branching logic. Because computers can keep track
of complicated logic, as well as a lot of information, they are useful
in a lot of problem-solving situations. (2 of 6)
Standard 3-5, page 81, Grades 5-8

Appreciate the pervasive use and power of reasoning as a part of mathematics
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. (3 of 6)
Standard 7-3 page 41, Grades K-4

Recognize that a wide variety of problem structures can be represented
by a single operation
Standard 2-3, page 78, Grades 5-8

Develop common understandings of mathematical ideas, including the
role of definitions

Standard 3-3, page 81, Grades 5-8

Make and evaluate mathematical conjectures and arguments

Standard 2-2 page 140, Grades 9-12

Formulate mathematical definitions and express generalizations discovered
through investigations

Standard 3-1 page 143, Grades 9-12

Make and test conjectures

People are using incorrect logic when they make a statement such as "If
A is true, then B is true; but A isn't true, therefore B isn't true either."
(4 of 6)
Standard 3-3 page 81, Grades 5-8

Make and evaluate mathematical conjectures and arguments
A single example can never prove that something is true, but sometimes
a single example can prove that something is not true. (5 of 6)
Standard 3-2, page 81, Grades 5-8

Understand and apply reasoning processes, with special attention to
spatial reasoning and reasoning with proportions and graphs
Standard 3-2 page 143, Grades 9-12

Formulate counterexamples

An analogy has some likenesses to but also some differences from the real
thing. (6 of 6)
Standard 3-2, page 81, Grades 5-8

Understand and apply reasoning processes, with special attention to
spatial reasoning and reasoning with proportions and graphs