Complex Numbers

This math unit begins with a deep dive into the fundamentals and properties of complex numbers, starting with rewriting roots involving negative radicands into complex number expressions and simplifying such forms. As students progress, they engage in operations of complex numbers including subtraction, division (focusing on imaginary parts), and the evaluation of powers of the imaginary unit \(i\). A gradual transition into geometric representations occurs as students practice calculating the absolute value of complex numbers on the complex plane. Further advancement in the unit includes converting complex numbers between different forms; initially from polar (in radians) to rectangular form, and from rectangular back to polar form in degrees. The unit also involves converting complex numbers from exponential form expressed in radians to rectangular form, utilizing Euler's formula. Towards the unit's end, a shift to pure geometry is observed where students apply the Pythagorean Theorem to calculate the length of a hypotenuse in decimal form, thereby bridging complex number theory with practical geometry applications. This progression allows students to not only understand but also manipulate complex numbers in various mathematical contexts and representations.Skills you will learn include:

• Meaning of i
• Complex plane
• Polar coordinates

This math unit progresses through a focused exploration of sinusoidal functions, starting by understanding how individual parameters like amplitude, period, and phase shift influence the function's graph. It begins with basic exercises that require students to match sinusoidal functions and their parameters to corresponding graphs. As the unit progresses, the complexity increases as students deal with multiple parameters simultaneously, analyzing how these combine to transform functions graphically and algebraically. Students then move to reverse tasks, where they identify sinusoidal functions from graphs and deduce parameters from functions. Ultimately, the unit covers up to four parameters, enhancing the learners' ability to handle more complex transformations and deepening their understanding of how variations in amplitude, period, phase shift, and vertical shift affect the mathematical expressions and visual representations of sinusoidal functions. The unit is grounded in developing analytical skills necessary for function transformations, composition, and inversion, which is crucial for higher mathematics and applied sciences.Skills you will learn include:

• Sinusoidal functions
• Phase shift
• Period
• Amplitude

This math unit develops students' skills in trigonometry, specifically focusing on solving triangles and calculating their areas using Heron's Formula. Initially, students learn to correctly set up the formula in various situations, enhancing their comprehension of selecting and manipulating the formula based on given triangle parameters. As the unit progresses, it emphasizes the full application of Heron's Formula, where students practice calculating the exact area of triangles. The unit gradually builds from understanding how to choose and set up the correct trigonometric approach to applying it meticulously to find precise solutions in geometric problem-solving scenarios. Through this approach, students enhance their proficiency in both foundational trigonometry skills and practical geometry applications.Skills you will learn include:

• Heron's formula
• Sin, cos, tan
• Solving triangles with trignometry

This math unit begins with foundational trigonometry, focusing initially on calculating trigonometric ratios from given values, which aids in understanding basic relationships in right triangles. The unit progresses, enabling learners to solve for unknown angles and side lengths using these ratios. From there, the unit advances to applying trigonometric principles to solving for both angles and sides directly from diagrams, increasing the complexity and application in real-world scenarios. As learners become proficient with foundational concepts, the unit introduces the Rule of Sines, progressing from setup to full calculations, allowing students to solve problems involving non-right triangles with higher complexity. Following this, the Rule of Cosines is taught, similarly progressing from selecting the correct formula to performing full calculations for side lengths, addressing problems involving angles not covered by the Rule of Sines. The unit culminates in combining trigonometric knowledge to calculate the area of a triangle using the sine function, emphasizing practical uses of trigonometry in determining areas, an essential skill in geometric problem-solving.Skills you will learn include:

• Law of Sines
• Law of Cosines
• Solving triangles and triangle area with trignometry

This math unit progresses through various trigonometry skills starting with foundational knowledge and advancing to more complex applications. Initially, students learn to calculate trigonometric ratios using given values of sides in right triangles, laying the groundwork for understanding sine, cosine, and tangent relationships. They then apply these concepts to solve for unknown angles and side lengths, using side measurements and angles. As the unit progresses, students tackle more challenging problems including solving for angles and side lengths from diagrams, enhancing their capability to interpret visual information and apply trigonometric principles practically. Later, the focus shifts to calculating angles and determining trigonometric ratios from specific angles provided in degrees, deepening their comprehension of trigonometric identities. Eventually, students learn to approximate these ratios and calculate sides in decimals, practicing how to work with more precise values and solve trigonometric relationships in various forms. This comprehensive approach builds a solid foundation in trigonometry, equipping students with essential skills for advanced mathematical studies.Skills you will learn include:

• Sin, cos, tan
• Solving triangles with trignometry
• Solving angles
• Solving side lengths

This math unit begins with teaching students to recognize and select basic trigonometric identities based on variable side ratios of right triangles, such as sine, cosine, and tangent. Progressing further, the unit emphasizes the calculation of angles using inverse trigonometric functions (both in inverse notation and arc notation) from given ratios. As students become proficient in determining angles from ratios, they move on to learn how to calculate specific trigonometric ratios from given angles, which deepens their understanding of the relationship between angles and side lengths in triangles. Moving towards more advanced applications, the unit covers the reciprocal process: calculating angles given specific trigonometric ratios in various notational formats. It also includes practice solving for angles using custom ratio fractions alongside trigonometric identities. Ultimately, the later sections of the unit require students to apply their knowledge to solve for unknown sides and trigonometric ratios using side lengths provided in diagrams or worded contexts. The unit concludes by refining skills in interpreting side lengths directly from diagrams and enhancing proficiency in trigonometric applications framed within practical and increasingly complex geometric scenarios.Skills you will learn include:

• Working with ratios
• Sin, cos, tan
• Solving triangles with trignometry

This math unit begins by developing a foundational understanding of the coordinate plane and quadrant identification using degrees and radians, which is crucial for studying trigonometry. Initially, students learn to determine which quadrant specific angles in degrees and radians belong to. This progresses into deeper exploration of trigonometric ratios like sine, cosine, and tangent, focusing on their signs based on quadrant location. The unit then enhances this knowledge by linking visual representations of quadrants with specific trigonometric ratios and their signs, both denoted in radians and degrees. As the unit advances, students apply their learning to solve more complex problems involving multiple trigonometric identities and their interrelationships within the unit circle. This progression from basic quadrant identification to a nuanced understanding of trigonometric identities and application in various quadrants supports the development of the skills needed to handle advanced trigonometry problems effectively.Skills you will learn include:

• Trig identities
• Quadrants
• Signs of trig identies by quadrant

In this math unit, students engage in a comprehensive exploration of trigonometric identities, focusing on the relationships and transformations between various trigonometric functions. Initially, the unit delves into using Pythagorean identities to convert between cosine and sine values, enhancing understanding of how these functions interrelate across different quadrants, whether represented as ratios, radians, or degrees. As the unit progresses, it expands to include solving for secant from tangent values and vice versa, requiring students to apply their knowledge to more complex scenarios and different quadrant settings. Further advancement in the unit explores conversions between cosecant and cotangent functions, challenging students to manipulate and apply trigonometric identities without direct formulas, across various quadrant representations. The unit culminates in exercises that focus on Pythagorean ratio cancellation, encouraging students to simplify complex trigonometric expressions. This progression finely hones their problem-solving skills and deepens their understanding of trigonometric relationships and identities in advanced trigonometry.Skills you will learn include:

• Trig identities
• Pythagorean identities
• Trig identity problem manipulation
• Cos and Sin squared cancellation problems

This math unit centers around the mastery of trigonometric identities using the Pythagorean theorem, progressively teaching students to manipulate and solve for various trigonometric function values based on given conditions. Initially, students work with basic identities to convert between sine and cosine values, incorporating quadrant implications to determine signs. The unit advances to more complex transformations between secant, cosecant, and tangent functions, initially using identities explicitly, then moving to contexts where identities are applied indirectly. As students progress, they are challenged to handle trigonometric functions without direct use of identities, promoting a deeper understanding of underlying relationships. Further complexity is added by involving different function conversions in varying quadrants, affecting the trigonometric results' signs. The final part of the unit involves simplifying complex trigonometric expressions using Pythagorean Ratio Cancellation techniques, both with and without direct identity applications. This progression sharpens problem-solving skills and fluency in navigating advanced trigonometric concepts, preparing students for higher-level mathematics.Skills you will learn include:

• Trig identities
• Pythagorean identities
• Trig identity problem manipulation
• Cos and Sin squared cancellation problems

This math unit advances through various aspects of trigonometry, beginning with foundational skills in identifying the signs of trigonometric ratios in different quadrants, then applying these skills to solve problems involving specific angle measures. As students progress, they delve into the unit circle and apply the Pythagorean identity to convert dimensional measurements into sine and cosine values using radians. The unit further develops with detailed exploration and verification of trigonometric identities involving the Pythagorean theorem, such as confirming the relationships between squared functions of sine, cosine, tangent, secant, cosecant, and cotangent. Toward the end of the unit, students apply these identities in more complex scenarios, solving for one trigonometric function using another while considering the quadrant's influence on the sign of the result. The culmination of the unit challenges students with problems that require advanced manipulation and simplification of trigonometric expressions, essential for higher mathematics and calculus.Skills you will learn include:

• Trig identities
• Pythagorean identities
• Trig identity problem manipulation

This math unit initiates learners into the application and manipulation of trigonometric identities starting with foundational concepts and advancing into sophisticated manipulations. Students first engage with trigonometric ratios and their signs across the four quadrants, enhancing their understanding of how these signs affect the computation of trig values in various quadrant scenarios. Progression in this unit takes learners into more detailed applications, involving challenging tasks of verifying the correctness of angle computations and trigonometric values at specific quadrant locations. The unit then elevates to exploring trigonometric identities, starting with simpler formulae and extending into more complex identities such as double angle, half-angle, and power-reducing identities. Each subsequent topic delves deeper into verifying the truth of intricate trigonometric identities including sum/difference identities and transformations involving sums to products and products to sums using variables expressed in Greek letters. By the culmination of this unit, students develop a nuanced understanding and ability to manipulate trigonometric identities essential for higher-level mathematics.Skills you will learn include:

• Trig identities
• Co-function identities
• Double angle identities
• Half angle identities
• Sum/difference identities
• Power reducing identities

This math unit begins by introducing students to the foundational concepts of trigonometry, focusing initially on understanding and using the basic trigonometric ratios—sine, cosine, and tangent—associated with right triangles. Early topics cover identifying trigonometric relationships and learning how to formulate correct trigonometric expressions based on given side lengths or angles. Progressively, the unit moves into more practical applications, teaching students to approximate these trigonometric ratios from diagrams and visual representations, further ingraining the fundamentals. The problems evolve to include calculating unknown angles from given side ratios, both in fraction and decimal forms, utilizing trig identities to facilitate these computations. Towards the latter part of the unit, the complexity increases as students apply their acquired skills to solve for unknown side lengths, angles from diagrams, and exact trigonometric values. Each successive topic builds on prior knowledge, culminating in more advanced practice that combines theoretical trigonometric principles with practical problem-solving skills, reinforcing understanding and application of trigonometry in geometric contexts.Skills you will learn include:

• Sin, cos, tan
• Solving triangles with trignometry

This math unit focuses on the classification and identification of various types of numbers, particularly real, imaginary, and complex numbers, advancing through a structured sequence of understanding and applying different concepts. Initially, the unit introduces learners to the classifications and definitions of numbers like real, imaginary, and complex. As students progress, they engage in activities that require them to distinguish these numbers based on their properties, using definitions and multiple-choice questions to solidify their understanding. The unit then moves into more complex exercises that involve the use of diagrams, true/false statements, and descriptions to further deepen the learners’ ability to classify and group numbers accurately. More advanced topics introduce set builder notation, requiring students to apply mathematical symbols and definitions to classify numbers into the correct categories. This progression from basic classification to detailed notation prepares students for a deeper understanding and use of number theory in complex mathematical contexts.Skills you will learn include:

• Classifying number types
• Complex, imaginary, real numbers
• Natural, whole, rational, irrational numbers
• Simplifying numbers

In this math unit, students begin by learning the fundamental skills of converting angles between degrees and radians using the unit circle. They start with basic conversions, practicing both from degrees to radians and vice versa. As they progress, students further explore these conversions with a wide range of specific angles, enhancing their fluency and comprehension of angular measurements in both units. Midway through the unit, the focus shifts towards applying this knowledge to calculate the trigonometric values sine and cosine for angles defined in both radians and degrees. Students practice identifying the coordinates of points on the unit circle based on these angles. This includes exercises that provide angles in radians and degrees, for which they have to determine the cosine and sine coordinates. Towards the unit's conclusion, learners refine their understanding of mapping these angles and trigonometric coordinates visually and numerically on the unit circle. This involves more complex problem sets where they are required to interpret trigonometric functions and their corresponding coordinates or visualize angles based on provided coordinates, solidifying their comprehension of trigonometry’s relationship with the unit circle.Skills you will learn include:

• Unit circle
• Radians and degrees
• Cos and Sin values

This math unit initially focuses on understanding and calculating ratios of line lengths in various geometric configurations, including parallel and right angle line displays. It introduces students to basic trigonometry through the practical application of calculating these ratios and progresses to more complex tasks. As the unit develops, students begin working with trigonometric identities and ratios such as sine, cosine, and tangent. They learn to solve for unknown angles based on given side lengths and to express relationships using trigonometric formulas. The later sections of the unit continue to emphasize interpreting and solving problems using trigonometric ratios and identities but introduce more complex applications, involving decimal representations and extracting trigonometric values from diagrams. The unit concludes with a comprehensive understanding of using trigonometric principles to solve geometric problems, setting a strong foundation in trigonometry by systematically building fluency from simple ratio calculations to complex trigonometric applications.Skills you will learn include:

• Sin, cos, tan
• Solving triangles with trignometry

This math unit begins with foundational trigonometry, starting with identifying different sides of a right triangle—opposite, adjacent, and hypotenuse. It progresses to understanding and labeling the side ratios in right triangles, using the SOHCAHTOA acronym to deepen the comprehension of trigonometric relationships such as sine, cosine, and tangent. As students become more comfortable with these definitions, they move on to applying and manipulating these ratios to solve for unknowns in trigonometric expressions and focusing on approximating trigonometric values from diagrams. The unit further advances into calculating and approximating side length ratios and explicit side lengths using trigonometric principles, scaling up to more complex applications where geometric figures and ratios are used extensively. Towards the end, the unit incorporates fact families within trigonometry to link geometric and algebraic representations, fostering a comprehensive grasp of the relationships between angles, sides, and trigonometric identities.Skills you will learn include:

• Ratios of line lengths
• Opposite, adjacent, hypotenuse
• Sin, cos, tan
• Solving triangles with trignometry