- Circles, squares, triangles, and other shapes can be found in things in nature...
- Patterns can be made by putting different shapes together or...
- Things move, or can be made to move, along straight, curved, circular...
- Mathematics is the study of many kinds of patterns, including numbers...
- Mathematical ideas can be represented concretely, graphically...
- Usually there is no one right way to solve a mathematical problem...
- Logical connections can be found between different parts of mathematics.
- Mathematics is the study of any patterns or relationships, whereas natural...
- 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...

**3-5**

**6-8**

**9-12**

- Mathematics is helpful in almost every kind of human endeavor...
- Mathematical modeling aids in technological design by simulating...
- Mathematics and science as enterprises share many values and features...
- Mathematics provides a precise language for science and technology...
- Developments in science or technology often stimulate innovations...
- Developments in mathematics often stimulate innovations in science...

**3-5**

**6-8**

**9-12**

- Numbers and shapes can be used to tell about things.
- Numbers and shapes--and operations on them--help to describe...
- In using mathematics, choices have to be made about what operations...
- Mathematicians often represent things with abstract ideas, such as numbers...
- When mathematicians use logical rules to work with representations...
- Some work in mathematics is much like a game--mathematicians choose...
- Much of the work of mathematicians involves a modeling cycle...

**3-5**

**6-8**

**9-12**