- Subchapter A
- Patterns and Relationships
- For Grades: K-2
- Learning Goal 1
- Circles, squares, triangles, and other shapes can be found in nature and in things that people build.

- Learning Goal 2
- Patterns can be made by putting different shapes together or taking them apart. Patterns may show up in nature and in the things people make.

- Learning Goal 3
- Things move, or can be made to move, along straight, curved, circular, back-and-forth, and jagged paths.

- Learning Goal 1
- For Grades: 3-5
- Learning Goal 1
- Mathematics is the study of quantity and shape and is useful for describing events and solving practical problems.

- Learning Goal 2
- Mathematical ideas can be represented concretely, graphically, or symbolically.

- Learning Goal 1
- For Grades: 6-8
- Learning Goal 1
- Usually there is no one right way to solve a mathematical problem; different methods have different advantages and disadvantages.

- Learning Goal 2
- Logical connections can be found between different parts of mathematics.

- Learning Goal 1
- For Grades: 9-12
- Learning Goal 1
- Mathematics is the study of quantities and shapes, the patterns and relationships between quantities or shapes, and operations on either quantities or shapes. Some of these relationships involve natural phenomena, while others deal with abstractions not tied to the physical world.

- Learning Goal 2
- As in other sciences, simplicity is one of the highest values in mathematics. Some mathematicians try to identify the smallest set of rules from which many other propositions can be logically derived.

- Learning Goal 3
- Theories and applications in mathematical work influence each other. Sometimes a practical problem leads to the development of new mathematical theories; often mathematics developed for its own sake turns out to have practical applications.

- Learning Goal 1

- For Grades: K-2

- Patterns and Relationships
- Subchapter B
- Mathematics, Science, and Technology
- For Grades: 6-8
- Learning Goal 1
- Mathematics is helpful in almost every kind of human endeavorâ€”from laying bricks to prescribing medicine or drawing a face.

- Learning Goal 1
- For Grades: 9-12
- Learning Goal 1
- Mathematical modeling aids in technological design by simulating how a proposed system might behave.

- Learning Goal 2
- 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.

- Learning Goal 3
- Mathematics provides a precise language to describe objects and events and the relationships among them. In addition, mathematics provides tools for solving problems, analyzing data, and making logical arguments.

- Learning Goal 4a
- Developments in science or technology often stimulate innovations in mathematics by presenting new kinds of problems to be solved.

- Learning Goal 4b
- The development of computer technology (which itself relies on mathematics) has generated new kinds of problems and methods of work in mathematics.

- Learning Goal 5
- Developments in mathematics often stimulate innovations in science and technology.

- Learning Goal 6
- Mathematics is useful in business, industry, music, historical scholarship, politics, sports, medicine, agriculture, engineering, and the social and natural sciences.

- Learning Goal 1

- For Grades: 6-8

- Mathematics, Science, and Technology
- Subchapter C
- Mathematical Inquiry
- For Grades: K-2
- Learning Goal 1
- Numbers and shapes can be used to tell about things.

- Learning Goal 1
- For Grades: 3-5
- Learning Goal 1
- Quantities and shapes can be used to describe objects and events in the world around us.

- Learning Goal 2
- In using mathematics, choices have to be made about what operations will give the best results. Results should always be judged by whether they make sense and are useful.

- Learning Goal 1
- For Grades: 6-8
- Learning Goal 1
- Mathematicians often represent things with abstract ideas, such as numbers or perfectly straight lines, and then work with those ideas alone. The "things" from which they abstract can be ideas themselves (for example, a proposition about "all equal-sided triangles" or "all odd numbers").

- Learning Goal 2a
- When mathematicians use logical rules to work with representations of things, the results may not be entirely valid for the things themselves.

- Learning Goal 2b
- Using mathematics to solve a problem requires choosing what mathematics to use; probably making some simplifying assumptions, estimates, or approximations; doing computations; and then checking to see whether the answer makes sense.

- Learning Goal 1
- For Grades: 9-12
- Learning Goal 1
- 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. The more interesting the results, the better. The only limit on the set of rules is that they should not contradict one another.

- Learning Goal 2
- Much of the work of mathematicians involves a modeling cycle, consisting 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. The actual thinking need not follow this order.

- Learning Goal 3
- To be able to use and interpret mathematics well, it is necessary to be concerned with more than the mathematical validity of abstract operations and to take into account how well they correspond to the properties of the things represented.

- Learning Goal 1

- For Grades: K-2

- Mathematical Inquiry