- Model situations using oral, written, concrete, pictorial, graphical, and algebraic methods

- Reflect on and clarify their own thinking about mathematical ideas and situations
- Develop common understandings of mathematical ideas, including the role of definitions
- Use the skills of reading, listening, and viewing to interpret and evaluate mathematical ideas
- Discuss mathematical ideas and make conjectures and convincing arguments
- Appreciate the value of mathematical notation and its role in the development of mathematical ideas

Grades 3-5, page 268

Geometric figures, number sequences, graphs, diagrams, sketches, number lines, maps, and stories can be used to represent objects, events, and processes in the real world, although such representations can never be exact in every detail.

*Benchmarks* 11B (Common Themes: Models)

Grades 3-5, page 268

Seeing how a model works after changes are made to it may suggest how
the real thing would work if the same thing were done to it.

*Benchmarks* 11B (Common Themes: Models)

Grades 6-8, page 269

Models are often used to think about processes that happen too slowly,
too quickly, or on too small a scale to observe directly, or that are too
vast to be changed deliberately, or that are potentially dangerous.

*Benchmarks* 11B (Common Themes: Models)

Grades 6-8, page 269

Mathematical models can be displayed on a computer then modified to
see what happens.

Grades 6-8, page 269

Different models can be used to represent the same thing. What kind of a model to use and how complex it should be depends on its purpose. The usefulness of a model may be limited if it is too simple or if it is needlessly complicated. Choosing a useful model is one of the instances in which intuition and creativity come into play in science, mathematics, and engineering.

*Benchmarks* 9E (The Mathematical World: Reasoning)

Grades 6-8, page 233

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.

*Benchmarks* 12D (Habits of Mind: Communication Skills)

Grades 6-8, page 297

Locate information in reference books, back issues of newspapers and
magazines, compact disks, and computer databases.

*Benchmarks* 12D (Habits of Mind: Communication Skills)

Grades 6-8, page 297

Understand writing that incorporates circle charts, bar and line graphs,
two-way data tables, diagrams, and symbols.

*Benchmarks* 12D (Habits of Mind: Communication Skills)

Grades 6-8, page 297

Find and describe locations on maps with rectangular and polar coordinates

*Benchmarks* 1C (The Nature of Science: The Science Enterprise)

Grades 3-5, page 16

Clear communication is an essential part of doing science. It enables
scientists to inform others about their work, expose their ideas to criticism
by other scientists, and stay informed about scientific discoveries around
the world.

*Benchmarks* 1B (The Nature of Science: Scientific Inquiry)

Grades 3-5, page 11

Scientists do not pay much attention to claims about how something
they know about works unless the claims are backed up with evidence that
can be confirmed and with a logical argument.

*Benchmarks* 2C (The Nature of Mathematics: Mathematical Inquiry)

Grades 6-8, page 37

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").