# Algebra II Critical Areas

CRITICAL AREA #1: Relate arithmetic of rational expressions to arithmetic of rational numbers
A central theme of this Model Algebra II course is that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers. Students explore the structural similarities between the system of polynomials and the system of integers. They draw on analogies between polynomial arithmetic and base-ten computation, focusing on properties of operations, particularly the distributive property. Connections are made between multiplication of polynomials with multiplication of multi-digit integers, and division of polynomials with long division of integers. Students identify zeros of polynomials, including complex zeros of quadratic polynomials, and make connections between zeros of polynomials and solutions of polynomial equations. The Fundamental Theorem of Algebra is examined.

The Domains and Clusters below relate to this Critical Area:

The Complex Number System

• Perform arithmetic operations with complex numbers.
• Use complex numbers in polynomial identities and equations.

Seeing Structure in Expressions

• Interpret the structure of expressions.
• Write expressions in equivalent forms to solve problems.

Arithmetic with Polynomials and Rational Expressions

• Perform arithmetic operations on polynomials.
• Understand the relationship between zeros and factors of polynomials.
• Use polynomial identities to solve problems.
• Rewrite rational expressions.

Creating Equations

• Create equations that describe numbers or relationships.

Reasoning with Equations and Inequalities

• Understand solving equations as a process of reasoning and explain the reasoning.
• Represent and solve equations and inequalities graphically.

CRITICAL AREA #2: Expand understandings of functions and graphing to include trigonometric functions
Building on their previous work with functions and on their work with trigonometric ratios and circles in the Model Geometry course, students now use the coordinate plane to extend trigonometry to model periodic phenomena.

The Domain and Clusters below relate to this Critical Area:

Trigonometric Functions

• Extend the domain of trigonometric functions using the unit circle.
• Model periodic phenomena with trigonometric functions.
• Prove and apply trigonometric identities.

CRITICAL AREA #3: Synthesize and generalize functions and extend understanding of exponential functions to logarithmic functions
Students synthesize and generalize what they have learned about a variety of function families. They extend their work with exponential functions to include solving exponential equations with logarithms. They explore the effects of transformations on graphs of diverse functions, including functions arising in an application, in order to abstract the general principle that transformations on a graph always have the same effect regardless of the type of the underlying function. They identify appropriate types of functions to model a situation, they adjust parameters to improve the model, and they compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit. The description of modeling as “the process of choosing and using mathematics and statistics to analyze empirical situations, to understand them better, and to make decisions” is at the heart of this Model Algebra II course. The narrative discussion and diagram of the modeling cycle should be considered when knowledge of functions, statistics, and geometry is applied in a modeling context.

The Domains and Clusters below relate to this Critical Area:

Interpreting Functions

• Interpret functions that arise in applications in terms of the context.
• Analyze functions using different representations.

Building Functions

• Build a function that models a relationship between two quantities.
• Build new functions from existing functions.

• Construct and compare linear, quadratic, and exponential models and solve problems.

CRITICAL AREA #4: Relate data display and summary statistics to probability and explore a variety of data collection methods
Students see how the visual displays and summary statistics they learned in earlier grades relate to different types of data and to probability distributions. They identify different ways of collecting data—including sample surveys, experiments, and simulations—and the role that randomness and careful design play in the conclusions that can be drawn.

The Domains and Clusters below relate to this Critical Area:

Interpreting Categorical and Quantitative Data

• Summarize, represent and interpret data on a single count or measurement variable.

Making Inferences and Justifying Conclusions

• Understand and evaluate random processes underlying statistical experiments.
• Make inferences and justify conclusions from sample surveys, experiments and observational studies.

DOMAINS AND CLUSTERS BEYOND THE CRITICAL AREAS OF FOCUS
All standards listed within the domains and clusters below are beyond the College and Career Readiness requirements of the 9-12 sequence.

Vector and Matrix Quantities

• Represent and model with vector quantities.
• Perform operations on matrices and use matrices in applications.

Using Probability to Make Decisions

• Use probability to evaluate outcomes of decisions.