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 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 and 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 arithmetic and geometric sequences. They'll also explore the properties of the real number system and use their knowledge of quadratics as a stepping stone to investigate the complex numbers.

Students deepen their knowledge of geometry, including proof, circles, and parabolas in the coordinate plane. They will further develop their understanding of trigonometry, including 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 and describing their roots and end-behaviors.
• 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 these functions with 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 sequences to model situations in context.

Geometry

• Proving the isosceles triangle theorem, its converse, and other results related to isosceles triangles.
• Prove the centroid and centroid ratio theorems and derive the formula for the coordinates of a triangle's centroid.
• Proving properties of parallelograms and theorems relating parallelograms and rectangles.
• Finding equations of circles and parabolas in a coordinate plane.
• Determining properties of circles and parabolas by completing the square.
• Finding equations of tangent lines to circles.
• Using the properties of the perpendicular bisectors of diameters and chords to solve problems in the coordinate plane.
• Applying Thales' theorem in the coordinate plane.
• Understanding the equation of a parabola, 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 trigonometric 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, and 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.