Algebra II is an advanced mathematics course that deepens students’ understanding of algebraic concepts introduced in Algebra I and prepares them for higher-level coursework such as Precalculus, Statistics, and college-level mathematics. Algebra II broadens the mathematical tools students have at their disposal, allowing them to model and solve more complex real-world problems across a wide range of disciplines.
The course emphasizes both theoretical understanding and practical problem solving. Students will explore complex numbers, polynomials, rational expressions, radical functions, quadratic functions, exponential and logarithmic functions, sequences and series, and probability and statistics. Through this course, students will learn to:
- Extend their ability to manipulate and solve a wide variety of equations and inequalities
- Analyze the behavior of functions and interpret graphs in greater depth
- Model real-world phenomena using mathematical functions and structures
- Apply knowledge of algebraic techniques to understand and solve systems of equations and inequalities
- Develop fluency with transformations and inverses of functions
Students build on their logical reasoning and abstract thinking skills as they learn to generalize patterns, make connections between representations, and approach increasingly complex problems with confidence. Algebra II is vital preparation for fields that require strong mathematical reasoning such as computer science, engineering, economics, and the life sciences.
At 2Sigma School, Algebra II incorporates project-based learning that connects advanced algebraic concepts to real-world applications. Students may work on projects such as modeling population growth with exponential functions, analyzing data sets with statistical tools, or exploring finance through exponential and logarithmic relationships. These projects help students apply their learning in meaningful, tangible ways.
Students should have successfully completed both Algebra I and Geometry before enrolling in Algebra II. The highest performing students enjoy working with abstract concepts, solving multi-step problems, and seeking out patterns and relationships. Algebra II pushes students beyond solving simple problems to building mathematical models that describe, predict, and explain complex systems.
Course Outline
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Sequences and Functions
This unit revisits functions using sequences as a specific example, defining them as a type of function with inputs from a subset of integers. Students identify arithmetic and geometric sequences, relating them to linear and exponential functions learned previously. They learn to represent sequences through writing terms, interpreting and creating tables and graphs, and using recursive function notation. Students develop explicit expressions for the nth term of a sequence by recognizing patterns, and they model real-world situations with sequences, focusing on appropriate domains and precision. The unit concludes by introducing scenarios involving the sum of a finite sequence, which will be explored further in a later unit.
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Polynomials and Rational Functions
This unit expands upon students' knowledge of linear and quadratic functions by introducing polynomials of higher degree and their unique graphical features, such as end behavior and the impact of multiplicity on zeros. Students will learn to connect the factored form of a polynomial to its graph, identifying intercepts and sketching curves. The unit also covers polynomial division and the Remainder Theorem, providing tools to rewrite polynomial equations in factored form. Finally, students will explore rational functions, their asymptotic behavior, and the process of solving rational equations, concluding with an examination of polynomial identities and their application to geometric sequences.
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Polynomial Functions
This unit extends students' knowledge of exponents and radicals to include rational exponents and introduces complex numbers to solve equations, including quadratics, that have no real solutions. Students review exponent rules and learn to apply them to expressions with rational exponents, connecting these concepts to radicals. The unit explores square and cube roots as solutions to equations, emphasizing the crucial distinction between positive and negative square roots and the implications of this when solving equations. Students are then introduced to imaginary and complex numbers, learning to perform operations with them and use them to find complex solutions to quadratic equations through methods like completing the square and the quadratic formula.
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Exponential Functions and Equations
Building on prior knowledge of exponential functions with integer domains, this unit examines exponential functions with real number domains. Students will learn how to write, interpret, and evaluate these functions. The second part of the unit introduces base-2 and base-10 logarithms to express exponents in exponential equations. Students will then utilize logarithms to solve these equations and analyze exponential functions. The mathematical constant e is presented as a tool for modeling continuous growth situations, leading to the study of natural logarithms and an introduction to logarithmic functions.
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Transformations of Functions
This unit examines how functions can be modified to suit specific situations, a key aspect of mathematical modeling (MP4). Students will connect different representations as they translate, reflect, and scale various types of functions. Throughout the unit, students will refine their language to accurately describe these transformations. The unit concludes with students applying transformations to different functions in order to model a real-world data set.
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Trigonometric Functions
In this unit, students are introduced to trigonometric functions. While students have explored various function types with different key features previously, they now encounter periodic functions for the first time; these are functions whose output values repeat at regular intervals. Students begin by studying circular motion and apply right triangle trigonometry to determine the coordinates of points on a circle. The unit circle is introduced, and students analyze the symmetry of its coordinates, developing an understanding of radian angles in relation to the fact that a full circle has an angle of 2π. From the unit circle, the domain of cosine, sine, and tangent are expanded, and students begin to conceptualize them as functions. Students graph these functions, using their knowledge of the unit circle, and expand the domain of the functions a second time to include angles beyond 2π and less than 0. The second half of this unit builds directly on the work of the previous unit by having students apply their knowledge of transformations to trigonometric functions and use these functions to model periodic situations.
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Statistical Inferences
Students build on their previous understanding of collecting samples and using them to estimate population characteristics. This unit expands on those concepts by exploring the normal distribution and its application in providing estimates with a margin of error. The unit also covers experimental, observational, and survey studies. For experimental studies, data analysis methods using randomization distributions and normal distribution modeling are examined. The importance of random selection in gathering survey and observational study samples, and the importance of random assignment in experimental studies is emphasized. The unit concludes by exploring methods to analyze results from these various study types. Data from surveys and observational studies that employ random samples are used to estimate population means and proportions, incorporating a margin of error.
To take any of our courses, students must be familiar with opening a browser, navigating to a website, and joining a Zoom meeting.
Students must have a quiet place to study and participate in the class for the duration of the class. Some students may prefer a headset to isolate any background noise and help them focus in class.
Most course lectures and content may be viewed on mobile devices but programming assignments and certain quizzes require a desktop or laptop computer.
Students are required to have their camera on at all times during the class, unless they have an explicit exception approved by their parent or legal guardian.
Our technology requirements are similar to that of most Online classes.
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A desktop or laptop computer running Windows (PC), Mac OS (Mac), or Chrome OS (Chromebook).
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Students must be able to run a Zoom Client. |
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A working microphone, speaker, webcam, and an external mouse. |
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A high-speed internet connection with at least 15mbps download speed (check your Internet speed). |
This course includes several timed tests where you will be asked to complete a given number of questions within a 1-3 hour time limit. These tests are designed to keep you competitively prepared but you can take them as often as you like. We do not proctor
these exams, neither do we require that you install special lockdown browser.
In today's environment, when students have access to multiple devices, most attempts to avoid cheating in online exams are symbolic. Our exams are meant to encourage you to learn and push yourself using an honor system.
We do assign a grade at the end of the year based on a number of criteria which includes class participation, completion of assignments, and performance in the tests. We do not reveal the exact formula to minimize students' incentive to optimize for a
higher grade.
We believe that your grade in the course should reflect how well you have learnt the skills, and a couple of timed-tests, while traditional, aren't the best way to evaluate your learning.