Integrated Math III

As part of this comprehensive course, students generalize their knowledge of quadratics to higher-order polynomials. This includes factoring polynomials using various methods, solving polynomial equations, and dividing polynomials. Students will also grasp fundamental polynomial theorems such as the factor and remainder theorems, learn how to plot polynomials and describe their properties.

Students will learn to solve both exponential and logarithmic equations and further develop their aptitude for modeling scenarios with exponential functions, such as calculating compound interest. Additionally, they will develop fluency in radical, rational expressions and equations, including manipulating and simplifying expressions, solving equations, and plotting graphs.

Building upon the foundations laid in Integrated Math II, students will strengthen their knowledge of sequences to include arithmetic and geometric sequences. They'll also explore properties of the real number system, and use their knowledge of quadratics as a stepping stone to investigate the fascinating world of complex numbers.

Students deepen their knowledge of geometry to include circles and parabolas in the coordinate plane. They will further develop their understanding of trigonometry to include the law of sines and cosines, graphs and properties of trigonometric functions.

Polynomials

• Factoring differences of squares, differences of cubes, biquadratic expressions, and using greatest common factors.
• Solving polynomial equations by factoring.
• Dividing polynomials using algebraic manipulation, synthetic division and long division.
• The factor, remainder, and rational roots theorems.
• Understanding and applying the concept of root multiplicity.
• Graphing polynomials.
• Finding the least common multiple of two polynomials
• Describing numerical relationships (e.g., generating Pythagorean triples) using polynomial identities.

Radicals & Rationals

• Simplifying rational and radical expressions using polynomial factorization.
• Adding, subtracting, multiplying, and dividing rational expressions.
• Graphs of reciprocal and radical functions, including performing graph transformations, describing their properties, computing inverses, and calculating intersections of reciprocal functions and lines.
• Rationalizing denominators in algebraic expressions.
• Solving advanced radical equations.

Exponentials & Logarithms

• Solving exponential and logarithmic equations, including cases with different bases, using the laws of logarithms, and the zero-product property.
• Graphing logarithmic functions.
• Computing inverses of exponential and logarithmic functions.
• Using exponential functions to model real-world situations, such as calculating compound interest and cases with continuous compounding.

Number Systems

• Describing the hierarchical structure of the real number system and classifying numbers accordingly.
• Expressing repeating decimals as fractions.
• Understanding the properties of sums and products of rational and irrational numbers.
• Addition, subtraction, and multiplication of complex numbers.
• Understanding the cyclic property of the imaginary unit and solving related problems.
• Solving quadratic equations with complex roots.

Sequences

• Constructing an explicit formula for the nth term of an arithmetic or geometric sequence.
• Converting between explicit and recursive formulas for arithmetic and geometric sequences.
• Solving mathematical problems related to arithmetic and geometric sequences (e.g., finding the common difference of an arithmetic sequence given two non-consecutive terms).
• Using arithmetic sequences to model situations in context.

Conic Sections

• Finding equations of circles and parabolas in a coordinate plane.
• Determining properties of circles and parabolas by completing the square.
• Understanding the focus-directrix property of a parabola and using this to construct a parabola's equation.
• Calculating points of intersection of circles and parabolas with lines.

Trigonometry

• Understanding the law of sines and cosines and using them to solve real-world problems.
• Calculating the area of an arbitrary triangle or quadrilateral using trigonometry.
• Graphing transformed trigonometric and reciprocal functions and describing properties such as amplitude, periodicity, and evenness/oddness.
• Finding expressions for zeros and extrema of transformed trigonometric functions.
• Using trigonometric functions to model real-world situations.

Probability & Statistics

• Conditional probability, the law of total probability, tree diagrams, the addition law, independent and mutually exclusive events.
• The rules of sum and product, ordering objects, permutations, combinations, and calculating probabilities using combinatorics.
• Analyzing data sets by sampling, computing their mean, variance, and standard deviation.