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A. Patterns and Relationships
By the end of 12th grade, students should know that:
- Mathematics is the study of any patterns or relationships, whereas natural science is concerned only with those patterns that are relevant to the observable world.
- As in other sciences, simplicity is one of the highest values in mathematics.
- Theories and applications in mathematical work influence each other.
- New mathematics continues to be invented, and connections between different parts of mathematics continue to be found.
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B. Mathematics, Science, and Technology
By the end of 12th grade, students should know that:
- Mathematical modeling aids in technological design by simulating how a proposed system would theoretically behave.
- Mathematics and science as enterprises share many values and features: belief in order, ideals of honesty and openness, the importance of criticism by colleagues, and the essential role played by imagination.
- Mathematics provides a precise language for science and technology—to describe objects and events, to characterize relationships between variables, and to argue logically.
- Developments in science or technology often stimulate innovations in mathematics by presenting new kinds of problems to be solved.
- Developments in mathematics often stimulate innovations in science and technology.
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C. Mathematical Inquiry
By the end of 12th grade, students should know that:
- Some work in mathematics is much like a game mathematicians choose an interesting set of rules and then play according to those rules to see what can happen.
- Much of the work of mathematicians involves a modeling cycle, which consists of three steps: (1) using abstractions to represent things or ideas, (2) manipulating the abstractions according to some logical rules, and (3) checking how well the results match the original things or ideas.
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