High School 9-12

The fundamental purpose of this course is to formalize and extend the mathematics that students learned in the middle grades. Because it is built on the middle grades standards, this is a more ambitious version of Algebra I than has generally been offered. The critical areas deepen and extend understanding of linear and exponential relationships by contrasting them with each other and by applying linear models to data that exhibit a linear trend, and students engage in methods for analyzing, solving, and using quadratic functions. The Mathematical Practice Standards apply throughout the course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations.

Critical Area 1:

By the end of eighth grade, students have learned to solve linear equations in one variable and have applied graphical and algebraic methods to analyze and solve systems of linear equations in two variables. Now, students analyze and explain the process of solving an equation. Students develop fluency writing, interpreting, and translating between various forms of linear equations and inequalities, and using them to solve problems. They master the solution of linear equations and apply related solution techniques and the laws of exponents to the creation and solution of simple exponential equations.

Critical Area 2:

In earlier grades, students define, evaluate, and compare functions, and use them to model relationships between quantities. In this unit, students will learn function notation and develop the concepts of domain and range. They explore many examples of functions, including sequences; they interpret functions given graphically, numerically, symbolically, and verbally, translate between representations, and understand the limitations of various representations. Students build on and informally extend their understanding of integer exponents to consider exponential functions. They compare and contrast linear and exponential functions, distinguishing between additive and multiplicative change. Students explore systems of equations and inequalities, and they find and interpret their solutions. They interpret arithmetic sequences as linear functions and geometric sequences as exponential functions.

Critical Area 3:

This critical area builds upon students’ prior experiences with data, providing students with more formal means of assessing how a model fits data. Students use regression techniques to describe approximately linear relationships between quantities. They use graphical representations and knowledge of the context to make judgments about the appropriateness of linear models. With linear models, they look at residuals to analyze the goodness of fit.

Critical Area 4:

Students build on their knowledge from Critical Area 2, where they extended the laws of exponents to rational exponents. Students apply this new understanding of number and strengthen their ability to see structure in and create quadratic and exponential expressions. They create and solve equations, inequalities, and systems of equations involving quadratic expressions.

Critical Area 5:

Students consider quadratic functions, comparing the key characteristics of quadratic functions to those of linear and exponential functions. They select from among these functions to model phenomena. Students learn to anticipate the graph of a quadratic function by interpreting various forms of quadratic expressions. In particular, they identify the real solutions of a quadratic equation as the zeros of a related quadratic function. Students expand their experience with functions to include more specialized functions—absolute value, step, and those that are piecewise-defined.

Common Core State Standards for Mathematics – Appendix A, pg. 15

NUMBER AND QUANTITY

• The Real Number System:

1. Extend the properties of exponents to rational exponents.

2. Use properties of rational and irrational numbers.

• Quantities:

1. Reason quantitatively and use units to solve problems.
2. Use properties of rational and irrational numbers.

ALGEBRA

• Seeing Structure in Expressions:

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

• Arithmetic with Polynomials and Rational Expressions:

1. Perform arithmetic operations on polynomials.
2. Rewrite rational expressions.

• Creating Expressions:

1. Create equations that describe numbers or relationships.

• Reasoning with Equations and Inequalities:

1. Understand solving equations as a process of reasoning and explain the reasoning.
2. Solve equations and inequalities in one variable.
3. Solve systems of equations.
4. Represent and solve equations and inequalities graphically.

FUNCTIONS

• Interpreting Functions:

1. Understand the concept of a function and use function notation.
2. Interpret functions that arise in applications in terms of the context.
3. Analyze functions using different representations.

• Building Functions:

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

• Linear, (Quadratic), and Exponential Models:

1. Construct and compare linear, quadratic, and exponential models and solve problems.
2. Interpret expressions for functions in terms of the situation they model.

STATISTICS AND PROBABILITY

• Interpreting Categorical and Quantitative Data:

1. Summarize, represent, and interpret data on a single count or measurement variable.
2. Summarize, represent, and interpret data on two categorical and quantitative variables.
3. Interpret linear models.

Building on their work with linear, quadratic, and exponential functions, students extend their repertoire of functions to include polynomial, rational, and radical functions. Students work closely with the expressions that define the functions, and continue to expand and hone their abilities to model situations and to solve equations, including solving quadratic equations over the set of complex numbers and solving exponential equations using the properties of logarithms. The Mathematical Practice Standards apply throughout each course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations.

The critical areas for this course are as follows:

Critical Area 1:

The first critical area develops the structural similarities between the system of polynomials and the system of integers. Students draw on analogies between polynomial arithmetic and base-ten computation, focusing on properties of operations, particularly the distributive property. Students connect 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 unit culminates with the fundamental theorem of algebra. A central theme of this unit is that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers.

Critical Area 2:

Building on their previous work with functions, and on their work with trigonometric ratios and circles in Geometry, students now use the coordinate plane to extend trigonometry to model periodic phenomena.

Critical Area 3:

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 unit. 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.

Critical Area 4:

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.

Common Core State Standards for Mathematics – Appendix A, pg. 36

NUMBER AND QUANTITY

• The Real Number System:

1. Extend the properties of exponents to rational exponents.

2. Use properties of rational and irrational numbers.

• The Complex Number System:

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

Vector and Matrix Quantities:

3. Perform operations on matrices and use matrices in applications.

ALGEBRA

• Seeing Structure in Expressions:

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

• Arithmetic with Polynomials and Rational Expressions:

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

• Creating Expressions:

1. Create equations that describe numbers or relationships.

• Reasoning with Equations and Inequalities:

1. Understand solving equations as a process of reasoning and explain the reasoning.
2. Solve equations and inequalities in one variable.
3. Solve systems of equations.
4. Represent and solve equations and inequalities graphically.

FUNCTIONS

• Interpreting Functions:

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

• Building Functions:

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

• Linear, Quadratic, and Exponential Models:

1. Construct and compare linear, quadratic, and exponential models and solve problems.
2. Interpret expressions for functions in terms of the situation they model.

• Trigonometric Functions:

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

GEOMETRY

• Similarity, Right Triangles, and Trigonometry:

3. Define trigonometric ratios and solve problems involving right triangles.

STATISTICS AND PROBABILITY

• Interpreting Categorical and Quantitative Data:

1. Summarize, represent, and interpret data on a single count or measurement variable.
2. Summarize, represent, and interpret data on two categorical and quantitative variables.
3. Interpret linear models.

• Making Inferences and Justifying Conclusions:

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

• Conditional Probability and the Rules of Probability:

2. Use the rules of probability to compute probabilities of compound events in a uniform probability model.

Common Core State Standards for Mathematics – Appendix A, pgs. 82-9

The fundamental purpose of the course in Geometry is to formalize and extend students’ geometric experiences from the middle grades. Students explore more complex geometric situations and deepen their explanations of geometric relationships, moving towards formal mathematical arguments. Important differences exist between this Geometry course and the historical approach taken in Geometry classes. For example, transformations are emphasized early in this course. Close attention should be paid to the introductory content for the Geometry conceptual category found in the high school CCSS. The Mathematical Practice Standards apply throughout each course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations. The critical areas, organized into six units, are as follows:

Critical Area 1:

In previous grades, students were asked to draw triangles based on given measurements. They also have prior experience with rigid motions: translations, reflections, and rotations, and have used these to develop notions about what it means for two objects to be congruent. In this unit, students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. They use triangle congruence as a familiar foundation for the development of formal proof. Students prove theorems—using a variety of formats—and solve problems about triangles, quadrilaterals, and other polygons. They apply reasoning to complete geometric constructions and explain why they work.

Critical Area 2:

Students apply their earlier experience with dilations and proportional reasoning to build a formal understanding of similarity. They identify criteria for similarity of triangles, use similarity to solve problems, and apply similarity in right triangles to understand right triangle trigonometry, with particular attention to special right triangles and the Pythagorean theorem. Students develop the Laws of Sines and Cosines in order to find missing measures of general (not necessarily right) triangles, building on students’ work with quadratic equations done in the first course. They are able to distinguish whether three given measures (angles or sides) define 0, 1, 2, or infinitely many triangles.

Critical Area 3:

Students’ experience with two-dimensional and three-dimensional objects is extended to include informal explanations of circumference, area, and volume formulas. Additionally, students apply their knowledge of two-dimensional shapes to consider the shapes of cross-sections and the result of rotating a two-dimensional object about a line.

Critical Area 4:

Building on their work with the Pythagorean theorem in 8th grade to find distances, students use a rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines, which relates back to work done in the first course. Students continue their study of quadratics by connecting the geometric and algebraic definitions of the parabola.

Critical Area 5:

In this unit, students prove basic theorems about circles, such as a tangent line is perpendicular to a radius, inscribed angle theorem, and theorems about chords, secants, and tangents dealing with segment lengths and angle measures. They study relationships among segments on chords, secants, and tangents as an application of similarity. In the Cartesian coordinate system, students use the distance formula to write the equation of a circle when given the radius and the coordinates of its center. Given an equation of a circle, they draw the graph in the coordinate plane, and apply techniques for solving quadratic equations, which relates back to work done in the first course, to determine intersections between lines and circles or parabolas and between two circles.

Critical Area 6:

Building on probability concepts that began in the middle grades, students use the language of set theory to expand their ability to compute and interpret theoretical and experimental probabilities for compound events, attending to mutually exclusive events, independent events, and conditional probability. Students should make use of geometric probability models wherever possible. They use probability to make informed decisions.

Common Core State Standards for Mathematics – Appendix A, pg. 27

GEOMETRY

• Congruence:

1. Experiment with transformations in the plane
2. Understand congruence in terms of rigid motions
3. Prove geometric theorems
4. Make geometric constructions

• Similarity, Right Triangles, and Trigonometry:

1. Understand similarity in terms of similarity transformations
2. Prove theorems involving similarity

• Circles:

1. Understand and apply theorems about circles
2. Find arc lengths and areas of sectors of circles

• Expressing Geometric Properties with Equations:

1. Translate between geometric description and the equation for a conic section
2. Use coordinates to prove simple geometric theorems algebraically

• Geometric Measurement and Dimension:

1. Explain volume formulas and use them to solve problems
2. Visualize relationships between two-dimensional and three-dimensional objects

• Modeling with Geometry:

1. Apply geometry concepts in modeling situations

STATISTICS AND PROBABILITY

• Conditional Probability and the Rules of Probability:

1. Understand independence and conditional probability and use them to interpret data
2. Use the rules of probability to compute probabilities of compound events in a uniform probability model

Common Core State Standards for Mathematics, pg. 75