# MATHEMATICS

The mathematics program is designed to fulfil these goals:

To develop in students an appreciation for the essential character of mathematics as an intellectual discipline

To foster development of logical reasoning skills, both inductive and deductive, and the ability to use these skills in general problem solving tasks beyond the mathematics classroom.

To give students a mathematical foundation that is sufficient for all levels of further study

To help students master technical skills that will be needed in their future studies

## MATHEMATICS

FCA offers up to three levels of full year courses. All students are placed in a level based on the recommendation of their current math teacher.

FCA math courses, except Geometry (all levels) and Algebra I require a TI-83 or TI-84 series graphing calculator. FCA does not provide calculators to students.

### MATHEMATICS I

The fundamental purpose of the Mathematics I course is to formalize and extend students’ understanding of linear functions and their applications. The critical

topics of study deepen and extend understanding of linear relationships—in part, by contrasting them with exponential phenomena and, in part, by applying linear models to data that exhibit a linear trend. Mathematics I uses proper- ties and theorems involving congruent figures to deepen and extend geometric knowledge gained in prior grade levels. The courses in the Integrated Pathway follow the structure introduced in the K–8 grade levels of the California Common Core State Standards for Mathematics (CA CCSSM); they present mathematics as a coherent subject and blend standards from different conceptual categories.

The standards in the integrated Mathematics I course come from the following conceptual categories: Modeling, Functions, Number and Quantity, Algebra, Geometry, and Statistics and Probability. The content of the course is explained below according to these conceptual categories, but teachers and administrators alike should note that the standards are not listed here in the order in which they should be taught. Moreover, the standards are not topics to be checked off after being covered in isolated units of instruction; rather, they provide content to be developed throughout the school year through rich instructional experiences.

### MATHEMATICS II

The Mathematics II course focuses on quadratic expressions, equations, and functions and on comparing the characteristics and behavior of these

expressions, equations, and functions to those of linear and exponential relationships from Mathematics I. The need for extending the set of rational numbers arises, and students are introduced to real and complex numbers. Links between probability and data are explored through conditional probability and counting methods and involve the use of probability and data in making and evaluating decisions. The study of similarity leads to an understanding of right-triangle trigonometry and connects to quadratics through Pythagorean relationships. Circles, with their quadratic algebraic representations, finish out the course. The courses in the Integrated Pathway follow the structure introduced in the K–8 grade levels of the California Com- mon Core State Standards for Mathematics (CA CCSSM); they present mathematics as a coherent subject and blend standards from different conceptual categories.

The standards in the integrated Mathematics II course come from the following conceptual categories: Modeling, Functions, Number and Quantity, Algebra, Geometry, and Statistics and Probability. The course content is explained below according to these conceptual categories, but teachers and administrators alike should note that the standards are not listed here in the order in which they should be taught. Moreover, the standards are not topics to be checked off after being covered in isolated units of instruction; rather, they provide content to be developed throughout the school year through rich instructional experiences.

### MATHEMATICS III

In the Mathematics III course, students expand their repertoire of functions to include polynomial, rational, and radical functions. They also expand their study

of right-triangle trigonometry to include general triangles. And, finally, students bring together all of their experience with functions and geometry to create models and solve contextual problems. The courses in the Integrated Pathway follow the structure introduced in the K–8 grade levels of the California Common Core State Standards for Mathematics (CA CCSSM); they present mathematics as a coherent subject and blend standards from different conceptual categories.

The standards in the integrated Mathematics III course come from the following conceptual categories: Modeling, Functions, Number and Quantity, Algebra, Geometry, and Statistics and Probability. The course content is explained below according to these conceptual categories, but teachers and administrators alike should note that the standards are not listed here in the order in which they should be taught. Moreover, the standards are not topics to be checked off after being covered in isolated units of instruction; rather, they provide content to be developed throughout the school year through rich instructional experiences.

### PRE-CALCULUS

Precalculus combines concepts of trigonometry, geome- try, and algebra that are needed to prepare students for the study of calculus. The course strengthens

students’ conceptual understanding of problems and math- ematical reasoning in solving problems. Facility with these topics is especially important for students who intend to study calculus, physics, other sciences, and engineering in college. The main topics in the Precalculus course are com- plex numbers, rational functions, trigonometric functions and their inverses, inverse functions, vectors and matrices, and parametric and polar curves. Because the standards that comprise this course are mostly (+) standards, students who enroll in Precalculus should have met the college- and career-ready standards of the previous courses in the Inte- grated Pathway or Traditional Pathway. It is recommended that students complete Precalculus before taking an Ad- vanced Placement calculus course.

### CALCULUS

When taught in high school, calculus should be presented with the same level of depth and

rigor as are entry-level college and university calculus courses. The content standards presented in

this chapter outline a complete college curriculum in one- variable calculus. Many high school programs may have insufficient time to cover all of the following content in a typical academic year. For example, some districts may treat differential equations lightly and spend substantial time on infinite sequences and series; other districts may do the opposite. Consideration of the College Board syllabi for the Calculus AB and Calculus BC sections of the Advanced Placement Examination in Mathematics may be helpful in making curricular decisions.† Calculus is a widely applied area of mathematics and involves a beautiful intrinsic theory. Students who master this content will be exposed to both aspects of the subject.

The sample problems included in this chapter are meant to illustrate and clarify the content standards. Some problems are written in a form that can be used directly with students; others will need to be modified before they are used with students.

### STATISTICS

Statistics and Probability offers students an alternative to Precalculus as a fourth high school mathematics course. In the Statistics and Probability

course, students continue to develop a more formal and precise understanding of statistical inference, which requires a deeper understanding of probability. Students learn that formal inference procedures are designed for studies in which the sampling or assignment of treatments was random, and these procedures may be less applicable

to non-randomized observational studies. Probability is still viewed as long-run relative frequency, but the emphasis now shifts to conditional probability and independence, and basic rules for calculating probabilities of compound events. In the plus (+) standards are the Multiplication Rule, probability distributions, and their expected values. Probability is presented as an essential tool for decision making in a world of uncertainty.

The course may be taught as either a one-semester (half- year) course or a full-year course. Supplementing a one- semester course with additional modeling experiences can extend it to a full-year course.