Algebra II

Course Description:

 Algebra II is a course that uses problem situations, physical models, and appropriate technology to extend algebraic thinking and engage student reasoning. Problem-solving situations will provide all students an environment that promotes communication and fosters connections within mathematics, to other disciplines, and to the real world. Students will use physical models to represent, explore, and develop abstract concepts. The use of appropriate technology will help students apply mathematics in an increasingly technological world. The concepts emphasized in the course include analysis of “family of functions,” solving systems of equations, graphing, data analysis, and logarithmic and exponential functions.

Standard Number 1.0: Number and Operations

 Students will recognize, represent, model, and apply real numbers and operations and will demonstrate an understanding of properties and operations of the complex number system.

Learning Expectations:

 The student will:

1.1     demonstrate an understanding of the subsets, elements, properties, and operations of the complex number system;

1.2     connect physical, graphical, verbal, and symbolic representations of real numbers;

1.3     articulate, model, and apply the concept of inverse (e.g. opposites, reciprocals, and powers and roots);

1.4     describe, model, and apply inverse operations;

1.5     connect physical, graphical, verbal, and symbolic representations of absolute value;

1.6     use a variety of notations appropriately (e.g. logarithmic, factorial, sigma, delta, radical);

1.7     perform operations on algebraic expressions and informally justify the procedures chosen.

PerformanceIndicatorsState:

As documented through state assessment –

At Level 1, the student is able to

·         order a given set of real numbers;

·         identify the reciprocal of a real number;

·         multiply two polynomials with each factor having no more than two terms.

At Level 2, the student is able to

·         perform basic operations using complex numbers (i.e., addition, subtraction, and multiplication);

·         select a graph that represents an absolute value equation on a coordinate plane;

·         identify the exponential form of a logarithmic expression and vice versa;

·         simplify expressions with rational and negative exponents;

·         add, subtract, and multiply algebraic expressions.

At Level 3, the student is able to

·         determine the conjugate of a complex number.

Performance Indicators Teacher:

As documented through teacher assessment –

At Level 1, the student is able to

·         probe the relationships among various subsets of the real-number system;

·         explore various representations of absolute value on a number line;

·         use ratios and proportions to represent real-world problems;

·         use estimation to determine a reasonable solution for a tedious arithmetic computation of a real-world situation that may involve unit conversions;

·         investigate product and factoring patterns of polynomials;

·         compare and contrast the GCF and the LCM of a set of algebraic expressions;

·         add, subtract, and perform scalar multiplication on matrices using appropriate technology.

At Level 2, the student is able to

·         analyze the relationships among sets of numbers using a Venn diagram of the complex number system;

·         use delta notation to represent the rate of change in a real-world situation;

·         use the inverse notation of powers and roots;

·         perform basic operations on rational algebraic expressions.

At Level 3, the student is able to

·         justify the procedures chosen when performing operations on algebraic expressions and equations;

·         use factorial notation for coefficients in a binomial expansion;

·         determine the multiplicative inverse of a complex number;

·         formulate the representation of a series using sigma notation.

Sample Task: Students design and build a simple fractal from available materials.

Linkages: Mathematics – Estimation, Measurement, and Computation. Make connections to concept mapping in literature, language arts, and social studies. Connect estimation and computation strategies to business and finance.

Standard 2.0: Algebra

Students will describe, extend, analyze, and create a wide variety of patterns and functions using appropriate materials and representations in real-world problem solving, and will demonstrate an understanding of the behavior of a variety of functions and their graphs.

Learning Expectations:

The student will:

2.1     analyze mathematical patterns related to algebra and geometry in real-world problem solving;

2.2     use algebraic thinking to generalize a pattern by expressing the pattern in functional notation;

2.3     solve linear systems using a variety of techniques, including matrices;

2.4     communicate the meaning of variables in algebraic expressions, equations, and inequalities;

2.5     manipulate the algebraic functions with constants and analyze graphs to describe the behavior of functions;

2.6     apply the concept of rate of change;

2.7     identify and represent a variety of functions (e.g. linear, quadratic, cubic);

2.8     identify, describe, and articulate the characteristics and the parameters of a parent function;

2.9     interpret results of algebraic procedures;

2.10  apply the concept of variable in simplifying algebraic expressions, solving equations, and solving inequalities;

2.11  interpret graphs that depict real-world phenomena;

2.12  model real-world phenomena using functions and graphs;

2.13  describe the domain and range of functions and articulate restrictions imposed either by the operations or by the real-life situations which the functions represent;

2.14  use linear programming to solve real-world problems.

PerformanceIndicatorsState:

As documented through state assessment –

At Level 1, the student is able to

·         translate a verbal sentence into an algebraic equation and vice versa;

·         select the algebraic equation that generalizes the pattern represented by data in a given table;

·         solve multi-step (more than two steps) linear equations (one set of parentheses on each side of the equations and/or variables on both sides);

·         select the graph that represents a given linear function expressed in slope-intercept form;

·         select the graph that models a given real-world situation (i.e., linear and non-linear);

·         identify the graphical representation of the solution to a one-variable inequality on a number line.

Level 2, the student is able to

·         select functional notation to generalize a given numeric pattern;

·         solve one-variable linear equations with rational expressions;

·         select the graph of a two-variable inequality;

·         determine the domain of polynomial, rational, square root, exponential and logarithmic functions;

·         determine the range of a wide variety of functions given a graph;

·         solve a system of linear equations with 2 variables (e.g. substitution, elimination, Cramer’s Rule, and graphing);

·         apply properties of logarithms to simplify a logarithmic expression;

·        identify matrices that model given real-world situations.

At Level 3, the student is able to

·         determine the inverse of a logarithmic function given its graph.

Performance Indicators Teacher:

As documented through teacher observation –

At Level 1, the student is able to

·         explain what the changes in slope of a non-linear graph representing a real-world situation;

·         analyze mathematical patterns related to algebra and geometry in real-world problem solving.

At Level 2, the student is able to

·         collect real-world data to make generalizations;

·         apply results of algebraic procedures to real-world situations;

·         use a variety of methods to solve linear systems in two and three variables (e.g., elimination, substitution, Cramer’s Rule, matrices, and graphing);

·         explain the restrictions on the variable in a radical equation;

·         choose an appropriate method to find the roots of a quadratic equation (e.g. completing the square, quadratic formula, factoring, or graphing calculator);

·         solve quadratic inequalities;

·         construct matrices given real world situation.

At Level 3, the student is able to

·         evaluate the graph of a function to determine if it is periodic;

·         sketch a system of linear inequalities and determine the maximum or minimum value of the related function;

·         justify the procedures chosen when performing operations on algebraic expressions and equations;

·         find the maximum or minimum value given the graph of the feasible region of the real world linear programming application;

·         determine all the roots of a higher order polynomial (i.e., Descartes’ Rule of Signs, Rational Root Theorem, and Synthetic Division).

Sample Task: Examine patterns found in Pascal’s Triangle.

Linkages: Mathematics: Statistics and Probability. Data analysis and pattern recognition in science.

Standard 3.0: Geometry

Students will investigate, model, and apply geometric properties and relationships.

 Learning Expectations:

The student will:

3.1     apply geometric properties, formulas, and relationships to solve real-world problems;

3.2     justify conclusions using deductive reasoning;

3.3     use inductive reasoning to make conjectures;

3.4     communicate position using spatial sense with two- and three-dimensional coordinate systems;

3.5     perform a given transformation and predict the results of the transformation.

PerformanceIndicatorsState:

As documented through state assessment –

At Level 1, the student is able to

·         apply the given Pythagorean Theorem to real-world problems.

At Level 2, the student is able to

·         predict the graphical transformation that occurs when coefficients and/or constants of given function are changed (no trigonometric or logarithmic functions);

·         apply proportion and the concepts of similar triangles to solve real world problems.

.

At Level 3, the student is able to

·         describe the transformation that has changed a “parent function” to the given related function (e.g., right shift of 3 units, reflection in the x-axis).

Performance Indicators Teacher:

As documented through teacher observation –

At Level 1, the student is able to

·         estimate the irrational solution of a real-world problem using the Pythagorean Theorem.

At Level 2, the student is able to

·         apply the distance formula to obtain the equation of a circle in order to solve real-world problems;

·         use deductive reasoning to draw conclusions.

At Level 3, the student is able to

·         use matrices to find the area of a triangle on a coordinate plane;

·         investigate and explore the conics section.

Sample Task:

Students use properties of similar triangles to determine the height of objects that are difficult to measure.

Linkages:

Research and discuss geometric applications such as art.

Standard 4.0: Measurement

The student will understand and be able to apply the units, systems and processes of measurement.

Learning Expectations:

The student will:

4.1apply measurement concepts and relationships in algebraic and geometric problem-solving situations;

4.2apply appropriate techniques, tools, and formulas to determine measurements.

PerformanceIndicatorsState:

As documented through state assessment –

At Level 1, the student is able to

·         select the area representation for a given product of two binomials.

At Level 2, the student is able to

·         apply the given formula to find area and circumference of circles, area and perimeter of polygons, and volume of regular solids;

At Level 3, the student is able to

·         solve real world problems given logarithmic and exponential formulas (e.g. Ph scale, Richter scale.).

Performance Indicators Teacher:

As documented through teacher observation –

At Level 1, the student is able to

·         select the appropriate unit of measure given the real world situation.

At Level 2, the student is able to

·         use appropriate measurements in collecting data for a real world situation.

There are no teacher performance indicators at Level 3.

Sample Task:

Students construct designs using basic geometric constructions. Then they transfer the design to a piece of 8" X 11" pane of plexiglass and paint the pane to create a “stained glass.” Students construct one of the regular 3-dimensional solid and compute the volume and surface area.

Linkages:

Mathematics – Geometry and Number & Operations. Surveying, construction, and architecture. .Mosaic Tiling.

Standard 5.0:  Data Analysis and Probability

The student will collect, organize, represent, and interpret data; make and evaluate inferences and predictions; present and evaluate arguments based on data analysis; and model situations to determine theoretical and experimental probabilities. 

Learning Expectations:

The student will:

5.1    understand concept of randomness in sampling;

5.2    apply appropriate technology in data collection and analysis;

5.3    apply counting principles of permutations and combinations using appropriate technology;

5.4     apply theoretical and experimental probability to analyze the likelihood of an event;

5.5    collect, represent, and describe linear and nonlinear data sets developed from real world;

5.6    make predictions from a data set using curve fitting with appropriate technology;

5.7    make inferences about a data set using appropriate measures of  central tendency and dispersion;

5.8    describe and apply the normal distribution and its properties;

5.9    identify mutually exclusive and non-mutually exclusive events;

5.10    analyze the probability of dependent events and of independent events;

5.11    use simulations to estimate probability;

5.12    choose, construct, and analyze appropriate graphical representations for a data set;

5.13    analyze the validity of statistical conclusions and the use, misuse, and abuse of data.

PerformanceIndicatorsState:

As documented through state assessment –

At Level 1, the student is able to

·         make a prediction from the graph of a real-world data set;

·         determine the measures of central tendency for a given set of real-world data;

·         choose the matching linear graph when given a set of ordered pairs representing real-world data.

At Level 2, the student is able to

·         categorize the correlation of a scatterplot using real-world data (i.e., positive, negative, strong, or weak);

·         determine the number of possible outcomes for a given experiment (i.e. the multiplication counting principle, permutations, or combinations);

·         determine the theoretical probability of a simple event for a given situation;

·         determine the theoretical probability of a compound event (i.e., dependent or independent, union and intersection).

At Level 3, the student is able to

·        find the equation for the line of best fit given a scatterplot depicting real-world data.

Performance Indicators Teacher:

As documented through teacher observation –

At Level 1, the student is able to

·         analyze student-collected data to make predications or generalizations.

At Level 2, the student is able to

·         use simulations to help predict the probability of a given situations;

·         determine the theoretical probability of mutually exclusive events for a given situation;

·         analyze theoretical or experimental probability to determine the likelihood of an event;

·         analyze data using linear and quadratic functions using the appropriate technology;

·         analyze the validity of statistical conclusions and the use, misuse, and abuse of data;

·         identify the mean and the standard deviation given the graph of a normal distribution.

At Level 3, the student is able to

·         use the measure of central tendency which best represents the given real-world data set given a distribution curve.

Sample Task:

Students analyze real-world data collected from the newspaper and explore and report the uses, misuses, and abuses of reported statistical data. Students search the internet to collect age and market value of a selected vehicle over a specific period of time. They use a graphing calculator to create a scatterplot and construct a line of best fit to predict the depreciation of the vehicle.

Linkages:

Sports; social studies; economics.