### Standard 5: Algebra

In grades 9-12, the mathematics curriculum should include the continued study of algebraic concepts and methods so that students can:

• Represent situations that involve variable quantities with expressions, equations, inequalities, and matrices
• Benchmarks 11C (Common Themes: Constancy and Change)
Grades 6-8, page 274
Symbolic equations can be used to summarize how the quantity of something changes over time or in response to other changes.

Benchmarks 4F (The Physical Setting: Motion)
Grades 9-12, page 91
The change in motion of an object is proportional to the applied force and inversely proportional to the mass.

Benchmarks 4G (The Physical Setting: Forces of Nature)
Grades 9-12, page 96
Gravitational force is an attraction between masses. The strength of the force is proportional to the masses and weakens rapidly with increasing distance between them.

• Use tables and graphs as tools to interpret expressions, equations, and inequalities
• Benchmarks 12B (Habits of Mind: Computation and Estimation)
Grades 9-12, page 291
Use ratios and proportions, including constant rates, in appropriate problems.

Benchmarks 11C (Common Themes: Constancy and Change)
Grades 9-12, page 275
Graphs and equations are useful (and often equivalent) ways for depicting and analyzing patterns of change.

Benchmarks 9B (The Mathematical World: Symbolic Relationships)
Grades 9-12, page 220
Any mathematical model, graphic or algebraic, is limited in how well it can represent how the world works. The usefulness of a mathematical model for predicting may be limited by uncertainties in measurements, by neglect of some important influences , or by requiring too much computation.

• Operate on expressions and matrices, and solve equations and inequalities
• 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.

Benchmarks 9B (The Mathematical World: Symbolic Relationships)
Grades 9-12, page 221
When a relationship is represented in symbols, numbers can be substituted for all but one of the symbols and the possible value of the remaining symbol computed. Sometimes the relationship may be satisfied by one value, sometimes more than one, and sometimes maybe not at all.

Benchmarks 12B (The Mathematical World: Symbolic Relationships)
Grades 9-12, page 291
Find answers to problems by substituting numerical values in simple algebraic formulas and judge whether the answer is reasonable by reviewing the process and checking against typical values.

• Appreciate the power of mathematical abstraction and symbolism
• Benchmarks 2C (The Nature of Mathematics: Mathematical Inquiry)
Grades 9-12, page 38
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.

and so that, in addition, college-intending students can:
• Use matrices to solve linear systems
• Demonstrate technical facility with algebraic transformations, including techniques based on the theory of equations

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