Standard 14: Mathematical Structure

In grades 9-12, the mathematics curriculum should include the study of mathematical structure so that all students can:
• Compare and contrast the real number system and its various subsystems with regard to their structural characteristics
• Benchmarks 9A (The Mathematical World: Numbers)
Grades 9-12, page 214
Comparison of numbers of very different size can be made approximately by expressing them as nearest powers of 10.

Benchmarks 9A (The Mathematical World: Numbers)
Grades 9-12, page 214
When calculations are made with measurements, a small error in the measurements may lead to a large error in the results.

Benchmarks 9A (The Mathematical World: Numbers)
Grades 9-12, page 214
The effects of uncertainties in measurements on a computed result can be estimated.

Benchmarks 12B (Habits of Mind: Computation and Estimation)
Grades 9-12, page 291
Express and compare very small and very large numbers using powers-of-ten notation.

Benchmarks 12B (Habits of Mind: Computation and Estimation)
Grades 9-12, page 291
Trace the source of any large disparity between an estimate and the calculated answer.

Benchmarks 12B (Habits of Mind: Computation and Estimation)
Grades 9-12, page 291
Recall immediately the relations among 10, 100, 1000, 1 million, and 1 billion (knowing, for example that 1 million is a thousand thousands).

Benchmarks 12B (Habits of Mind: Computation and Estimation)
Grades 9-12, page 291
Consider the possible effects of measurement errors on calculations.

• Understand the logic of algebraic procedures
• Benchmarks 9B (The Mathematical World: Symbolic Relationships)
Grades 9-12, page 220
Symbolic statements can be manipulated by rules of mathematical logic to produce other statements of the same relationship, which may show some interesting aspect more clearly. Symbolic statements can be combined to look for values of variables that will satisfy all of them at the same time.

• Appreciate that seemingly different mathematical systems may be essentially the same
• Benchmarks 9A (The Mathematical World: Numbers)
Grades 9-12, page 214
Numbers can be written with bases different from ten (which people probably use because of their 10 fingers). The simplest base, 2, uses just two symbols (1 and 0, or on and off)

and so that, in addition, college-intending students can:
• Develop the complex number system and demonstrate facility with its operations
• Prove elementary theorems within various mathematical structures, such as groups and fields
• Develop an understanding of the nature and purpose of axiomatic systems