Description: Understanding and performing operations with complex numbers, including addition, subtraction, multiplication, and division.

Key Concepts: Representing complex numbers in standard form, using the imaginary unit \(i\), and solving equations involving complex numbers.

Examples:

• Simplify the expression \( (2 + 3i) + (4 - 5i) \).
• Multiply and simplify \( (1 + 2i)(3 - 4i) \).
• Solve quadratic equations that have complex solutions.

Description: Understanding and applying principles of geometry, including angles, shapes, area, and volume.

Key Concepts: Calculating the area and perimeter of various shapes, using the Pythagorean theorem, and solving problems involving angles and parallel lines.

Examples:

• Calculate the area of a triangle given its base and height.
• Use the Pythagorean theorem to find the length of a side in a right triangle.
• Determine the angles formed by parallel lines and a transversal.

Description: Understanding and applying trigonometric concepts, including sine, cosine, tangent, and their applications to right triangles.

Key Concepts: Calculating the sine, cosine, and tangent of an angle, solving right triangles using trigonometric ratios, and applying the unit circle.

Examples:

• Find the sine, cosine, and tangent of a given angle in a right triangle.
• Solve for an unknown side or angle in a right triangle using trigonometric ratios.
• Use the unit circle to determine trigonometric values for standard angles.

Description: Understanding and analyzing exponential functions, including growth and decay.

Key Concepts: Identifying key features such as asymptotes, intercepts, and interpreting real-world applications.

Examples:

• Graph \(y = 2^x\) and describe its behavior.
• Interpret the meaning of the parameters in the exponential function \(y = 3(1.05)^x\).

Description: Understanding and manipulating polynomial expressions, including addition, subtraction, multiplication, and division.

Key Concepts: Polynomial identities, synthetic division, and factoring.

Examples:

• Simplify \((x^2 + 3x + 2) - (x^2 - 2x - 4)\).
• Factor \(x^3 - 27\) using polynomial identities.

Description: Understanding and solving quadratic equations algebraically and graphically.

Key Concepts: Factoring, quadratic formula, completing the square, and properties of parabolas.

Examples:

• Solve \(x^2 - 5x + 6 = 0\) by factoring.
• Use the quadratic formula to solve \(x^2 + 3x - 4 = 0\).

Description: Understanding and solving problems involving rational expressions, including simplification, addition, subtraction, multiplication, and division.

Key Concepts: Finding common denominators, simplifying complex fractions, and solving rational equations.

Examples:

• Simplify \(rac{1}{x} + rac{1}{x+2}\).
• Solve \(rac{2x}{x^2 - 4} = rac{3}{x-2}\).

Description: Understanding and using function notation, interpreting the meaning of function values in real-world contexts.

Key Concepts: Evaluating functions, interpreting graphs, and analyzing the behavior of functions.

Examples:

• Evaluate a function \(f(x)\) for a specific value of \(x\).
• Interpret the meaning of a point on the graph of a function in a real-world context.
• Analyze the behavior of a function as \(x\) approaches a certain value.

Description: Creating, interpreting, and comparing linear, quadratic, and exponential models to represent real-world situations.

Key Concepts: Identifying the appropriate model for a given situation, interpreting the parameters of the model, and predicting future values.

Examples:

• Write and interpret the equation of a line that best fits a set of data points.
• Compare the growth rates of exponential and linear models.
• Predict the future value of an investment using an exponential model.

Description: Applying geometric concepts to model real-world situations, including the use of area, volume, and surface area formulas.

Key Concepts: Using geometric shapes to model physical objects, calculating areas, volumes, and other measurements based on real-world contexts.

Examples:

• Calculate the volume of a cylinder to determine the capacity of a container.
• Use the surface area formula to estimate the amount of paint needed for a project.
• Model the trajectory of an object using quadratic functions.

Description: Interpreting and analyzing data presented in tables, graphs, and charts to solve problems.

Key Concepts: Understanding trends, making predictions, and drawing conclusions based on data.

Examples:

• Analyze a graph to determine the relationship between two variables.
• Interpret a table to calculate percentages or averages.

Description: Understanding and applying basic concepts of probability and statistics, including measures of central tendency and dispersion.

Key Concepts: Calculating mean, median, mode, standard deviation, and interpreting probability in different contexts.

Examples:

• Calculate the mean and standard deviation of a data set.
• Determine the probability of a specific outcome in a given scenario.

Description: Solving problems involving ratios, proportions, and percentages.

Key Concepts: Setting up and solving proportions, calculating percentages, and applying proportional reasoning in various contexts.

Examples:

• Solve for \(x\) in the proportion \(rac{a}{b} = rac{x}{d}\).
• Calculate the percentage increase from one value to another.

Description: Understanding and performing operations with complex numbers.

Key Concepts: Adding, subtracting, multiplying, and dividing complex numbers, representing complex numbers in the complex plane, solving quadratic equations with complex solutions.

Examples:

• Simplify \( (3 + 2i) + (4 - 5i) \).
• Multiply \( (2 + 3i)(1 - 4i) \).
• Solve \( x^2 + 4 = 0 \) for \( x \) in the complex number system.

Description: Understanding and analyzing various types of functions, including their properties and transformations.

Key Concepts: Interpreting function notation, analyzing function graphs, performing transformations such as translations, reflections, and dilations, understanding inverse functions.

Examples:

• Evaluate \( f(x) = 2x^2 - 3x + 5 \) at \( x = 3 \).
• Identify the domain and range of the function \( g(x) = \sqrt{x - 1} \).
• Determine the inverse of the function \( h(x) = rac{2x + 3}{x - 4} \).

Description: Understanding and solving problems involving geometric shapes, properties, and theorems.

Key Concepts: Calculating areas, perimeters, and volumes of various shapes, understanding the properties of triangles, circles, and polygons, applying the Pythagorean theorem.

Examples:

• Calculate the area of a triangle with base 5 and height 3.
• Determine the volume of a cylinder with radius 4 and height 10.
• Apply the Pythagorean theorem to find the length of the hypotenuse of a right triangle with legs of length 6 and 8.

Description: Understanding the relationships between the sides and angles of triangles, and solving problems involving trigonometric functions.

Key Concepts: Using sine, cosine, and tangent ratios, solving right triangles, applying the unit circle, understanding trigonometric identities.

Examples:

• Calculate the sine, cosine, and tangent of a given angle in a right triangle.
• Solve for the missing side of a right triangle using trigonometric ratios.
• Use the Pythagorean identity \( \sin^2( heta) + \cos^2( heta) = 1 \) to find trigonometric values.

Description: Understanding and solving problems involving exponential growth and decay.

Key Concepts: Modeling real-world scenarios with exponential functions, solving exponential equations, understanding the properties of exponential growth and decay.

Examples:

• Solve \(2^x = 8\).
• Model population growth using the function \(P(t) = P_0 e^{rt}\).

Description: Solving and interpreting polynomial equations of degree greater than two.

Key Concepts: Factoring higher-order polynomials, understanding the Fundamental Theorem of Algebra, solving cubic and quartic equations.

Examples:

• Factor and solve \(x^3 - 3x^2 - 4x + 12 = 0\).
• Graph polynomial functions and analyze their behavior.

Description: Performing operations on polynomials, including addition, subtraction, multiplication, and division.

Key Concepts: Simplifying polynomial expressions, performing polynomial long division and synthetic division, understanding polynomial identities.

Examples:

• Simplify \((x^2 + 2x)(x - 3)\).
• Divide \(x^3 + 2x^2 - 5x - 6\) by \(x - 2\).

Description: Solving quadratic equations by various methods (factoring, completing the square, quadratic formula).

Key Concepts: Understanding the properties of quadratic functions, finding roots/zeros, analyzing parabolas.

Examples:

• Solve \(x^2 - 4x + 4 = 0\) by factoring.
• Use the quadratic formula to find the roots of \(x^2 - 2x - 3 = 0\).

Description: Understanding and solving linear equations and inequalities in one or more variables.

Key Concepts: Solving for variables, interpreting solutions, graphing linear equations, understanding slope and intercepts.

Examples:

• Solve \(2x + 3 = 7\)
• Graph \(y = 2x - 4\)
• Solve and graph \(3x - 5 > 1\)

Description: Interpreting and building linear functions from different representations (e.g., tables, graphs, equations).

Key Concepts: Identifying slope and y-intercept, constructing functions from data, analyzing function behavior.

Examples:

• Write the equation of a line given a point and the slope.
• Interpret the slope and y-intercept from a real-world context.

Description: Solving systems of linear equations and inequalities, both algebraically and graphically.

Key Concepts: Solving by substitution and elimination, interpreting solutions graphically, analyzing consistent, inconsistent, and dependent systems.

Examples:

• Solve the system \(egin{cases} x + y = 10 \ 2x - y = 3 \end{cases}\)
• Graph the system and find the solution region: \(y > 2x - 3\) and \(y \leq -x + 5\)

Description: Analyzing and interpreting data from tables, graphs, and charts.

Key Concepts: Understanding different types of data displays, interpreting trends and patterns, calculating central tendency measures (mean, median, mode).

Examples:

• Read and interpret data from a bar graph.
• Calculate the mean of a given data set.

Description: Working with percentages in various contexts, including percentage increase/decrease and finding the whole from a percentage.

Key Concepts: Calculating percentage change, understanding percentage applications in financial contexts, converting between percentages, fractions, and decimals.

Examples:

• Calculate a 15% discount on a $50 item.
• Find the original price if an item is sold for $120 after a 20% increase.

Description: Understanding basic concepts of probability and statistical measures.

Key Concepts: Calculating probabilities, understanding random events, interpreting standard deviations and distributions.

Examples:

• Determine the probability of drawing a red card from a standard deck of cards.
• Interpret the significance of a standard deviation in a data set.

Description: Solving problems involving ratios, rates, and proportional relationships.

Key Concepts: Understanding unit rates, setting up and solving proportions, applying proportional reasoning in various contexts.

Examples:

• Calculate the unit rate in a given context.
• Solve for \(x\) in the proportion \(rac{3}{4} = rac{x}{8}\).

Description: Evaluating the effectiveness of arguments in a text.

Key Concepts: Identifying claims, evidence, and reasoning, analyzing the effectiveness of arguments, understanding rhetorical strategies.

Examples:

• Identify the main claim in an argumentative essay.
• Evaluate the strength of evidence provided to support a claim.
• Analyze the use of rhetorical strategies in a persuasive text.

Description: Understanding the author's perspective, purpose, and how they influence the text.

Key Concepts: Identifying the author's point of view, understanding the purpose of the text, analyzing how point of view and purpose affect content and style.

Examples:

• Identify the author's point of view in an opinion piece.
• Analyze the purpose of a scientific article.

Description: Understanding how the structure of a text contributes to its meaning and effectiveness.

Key Concepts: Analyzing the organization of paragraphs and sections, understanding the role of introductions and conclusions, recognizing structural elements like chronology and comparison.

Examples:

• Identify the organizational pattern of a passage (e.g., cause and effect, problem and solution).
• Analyze how the structure of a paragraph supports its main idea.

Description: Understanding the effect of specific words and phrases used by the author.

Key Concepts: Interpreting connotations, analyzing the impact of word choice on meaning and tone, understanding figurative language.

Examples:

• Determine the meaning of a word in context.
• Analyze how word choice affects the tone of a passage.
• Identify examples of figurative language and their effects.

Description: Improving the development of ideas in a text through appropriate detail, support, and focus.

Key Concepts: Adding relevant details, strengthening support, ensuring focus on the main idea.

Examples:

• Add relevant details to support a main idea.
• Eliminate unnecessary information that distracts from the main focus.

Description: Enhancing the clarity and impact of language through word choice, style, and tone.

Key Concepts: Choosing precise words, maintaining an appropriate tone, varying sentence structures.

Examples:

• Choose precise and appropriate words to convey a message.
• Maintain a consistent tone throughout a passage.
• Vary sentence structures to enhance readability.

Description: Improving the structure and coherence of a text.

Key Concepts: Organizing paragraphs logically, using transitions effectively, ensuring coherence and flow.

Examples:

• Reorganize paragraphs to improve logical flow.
• Use transitions to connect ideas effectively.

Description: Analyzing and understanding the central themes and ideas in a passage.

Key Concepts: Identifying themes, understanding their development, summarizing thematic elements.

Examples:

• Identify the theme of a historical document.
• Analyze how a theme develops over the course of a passage.
• Summarize the central ideas presented in a scientific article.

Description: Understanding and interpreting the explicit meaning of the text.

Key Concepts: Identifying main ideas, summarizing key details, recognizing explicit statements in the text.

Examples:

• Identify the main idea of a paragraph.
• Summarize the key points of an informational text.
• Locate explicit statements that support the main idea.

Description: Condensing information from texts to capture the essential points.

Key Concepts: Summarizing paragraphs, chapters, or entire texts, identifying key details that support the summary.

Examples:

• Summarize the main events of a narrative passage.
• Identify key details that support the main idea in an expository text.

Description: Recognizing and analyzing relationships between individuals, events, and ideas in texts.

Key Concepts: Understanding cause and effect, comparing and contrasting ideas, analyzing relationships and interactions.

Examples:

• Identify cause-and-effect relationships in a historical text.
• Compare and contrast different viewpoints presented in an argumentative essay.

Description: Applying correct punctuation rules to ensure clarity and readability.

Key Concepts: Using commas to separate items in a series, setting off nonrestrictive clauses, using semicolons to connect closely related independent clauses, applying correct end punctuation.

Examples:

• Use commas to separate items in a list.
• Correctly punctuate nonrestrictive and restrictive clauses.
• Use a semicolon to connect two independent clauses.

Description: Applying correct usage of standard English conventions.

Key Concepts: Correcting inappropriate verb tense, ensuring subject-verb agreement, using correct pronoun-antecedent agreement, avoiding double negatives.

Examples:

• Ensure correct verb tense in a sentence.
• Identify and correct errors in subject-verb agreement.
• Use the correct form of pronouns in sentences.