This "10th Grade Math: Number Systems" module is a foundational module for understanding how to work with all the number systems encountered in algebraic and geometric computation at the high school level. It reinforces the theory of computation with fractions and extends that to variable expressions in rational form. The module deals extensively with conceptual understanding of square roots and how to simplify them and manipulate expressions containing square roots without relying on technology. The module culminates with an explanation of complex numbers in various representations and emphasizes the utility of complex numbers.
The focus of this course is on conceptual understanding and reasoning, not memorization or routine calculation. Problems will be embedded in real contexts. The exercise sets are designed to become progressively more challenging so that completion of the problems will generate new learning and not just check for understanding.
This module includes
• 8 LESSONS IN 10 VIDEOS
• 8 PROBLEM SETS WITH COUNTERPART ANSWER SHEETS
• OVER 135 PRACTICE EXERCISES/PROBLEMS
• 12 EXERCISE REVIEW VIDEOS (with step-by-step explanation how to reach an answer and suggested alternate approaches)
• A CUMULATIVE UNIT EXAM (40+ QUESTIONS) IN 2 PARTS WITH COUNTERPART ANSWER KEY AND SOLUTIONS
• JUST UNDER 7 HOURS OF VIDEO-LESSONS
Upon completion of this course, students will demonstrate facility with systems of numbers beyond simple integer and decimal values. Students will be able to distinguish and classify integers, rational numbers, irrational numbers, real numbers, imaginary numbers, and complex numbers. Students will master the properties and operations of fractions as rational numbers and be able to extend these properties to rational expressions including variable and polynomial expressions.
Students will be able to identify the properties of irrational numbers and formalize the rules for working with square roots of positive real numbers. Students will be able to multiply roots and provide solutions to equations in simplified radical form. Students will apply the use of square roots in common geometric contexts involving rectangles and right triangles, especially the isosceles right triangle, the 30-60-90 right triangle, and Pythagorean triples. Students will also be able to apply the rules of multiplying roots to rationalize the denominator of rational expressions.
Students will develop the concept of the imaginary number i and the set of imaginary and complex numbers. Students will be able to add and subtract complex numbers as algebraic expressions. Students will also multiply complex numbers, use complex conjugates to rationalize expressions with complex denominators, and divide by complex numbers. Students will be able to represent complex numbers as points in the complex plane with a real axis and imaginary axis, in both rectangular and polar forms. Students will be use vectors to represent complex numbers in the plane in order to add, subtract, and multiply complex numbers. Students will be able to compute the magnitude (absolute value or length) of a number in the complex plane and the argument (directional angle of rotation) of a complex number. These explorations will include a brief application of the tangent of an angle, an introduction to the Fundamental Theorem of Algebra and the existence of complex roots, and connections to real-world applications in alternating current circuit design.
This video-course is primarily intended as a foundational module for a full course in 10th grade mathematics in an integrated algebraic and geometric approach. However, it is appropriate for students in 8th-11th grade in any algebra-intensive course.
Students taking this module will need to have completed some previous algebra. This module assumes facility using the Pythagorean Theorem, the ability to multiply binomial expressions, and previous experience factoring quadratics. Experience with special right triangles will be helpful to the student. Students need to be able to locate points in the coordinate plane.
Lesson 1: Rational and Irrational Numbers
Understand that rational numbers can be expressed in the form p/q, where p and q are members of Z, the integers.
Distinguish rational and irrational numbers
Express numbers in the form p/q, if rational.
Use set notation to demonstrate the hierarchy of integers (Z), rationals (Q), and irrationals (Q') within the real numbers (R).
Lesson 2: Add and Subtract Rational Expressions
Add and subtract simple common fractions
Adding rational expressions with a common denominator
Add and subtract rational expressions with denominators that are relatively prime
Add and subtract rationa expressions with denominators that share a common factor
Lesson 3: Multiply and Divide Rational Expressions
Division and multiplication as related operations
Multiplying rational expressions
Dividing rational expressions by inverting the divisor
Undefined expressions and division by zero
Lesson 4: Rationalizing the Denominator
Definition of the square root, properties of square roots defined
Multiplying radical expressions
Simplifying radical expressions by factoring
Rationalizing the denominator by multiplying a simple root
Applications to geometry and special right triangles
Rationalizing the denominator by multiplying by the conjugate
Lesson 5: Introduction to Complex Numbers and Adding and Subtracting Complex Numbers
Defining the imaginary number i
Constructing complex numbers in the form a + bi
The hierarchy of sets of numbers including real, imaginary, and complex numbers
Review: Identifying numbers by set: integer (Z), rational (Q), irrational (Q'), real (R), imaginary (I), or complex (C)
Adding and subtracting complex numbers by combining like terms (horizontally or vertically)
Applications (RLC series circuits)
Lesson 6: Multiplying and Dividing Complex Numbers
Repeated multiplication by i
Simplifying radicals containing a negative radicand
Squaring terms with i
Multiplying a complex number by a scalar
Multiplying a complex number by a complex number with various models
Multiplying complex conjugates
Using the complex conjugate to divide complex numbers
Solving equations with radicals
Applilcations (RLC parallel circuits)
Lesson 7: Complex Numbers Represented as Points in the Complex Plane
The real and imaginary axes
Points in the rectangular coordinate system as complex numbers
Complex numbers as vectors in the plane
The magnitude (absolute value) of a complex number, |z|, as the length of the vector
The argument (direction) of a complex number as the angular rotation of the vector
Points in the polar coordinate system as complex numbers
Lesson 8: Operations with Complex Numbers in the Complex Plane
Adding complex numbers by adding vectors
Subtracting complex numbers by adding inverse vectors
Multiplication by i as a 90° rotation about the origin
Scalar multiplication as a dilation of the vector
Multiplying two complex vectors: the effects on both magnitude and direction
Converting from polar form to rectangular form
Solving roots of unity, plotting the solutions, and the Fundamental Theorem of Algebra
Common Core State Standards Alignment
This module directly, but not exhaustively, addresses many of the standards in the CCSS Math High School: Number and Quantity category.
According to the CCSS, much of the content on rational and irrational numbers should have been mastered in the middle grades. However, many high school students need further exposure to and practice with rational and irrational numbers. Conceptual understanding of fractions (rational numbers) is often lacking, and computation with irrational numbers (especially square roots) is often strictly calculator-based. This module revisits those standards with more emphasis on understanding than use of rote algorithms or the calculator as the source of answers.
These standards relating to complex numbers receive significant emphasis:
CCSS-Math-HS Numbers and Quantity: The Complex Number System:
A1: Know there is a complex number i such that i2 = -1, and every complex number has the form a + bi with a and b real.
A2: Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
A3: Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
B4: Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
B5: Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.
C9: Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
These standards relating to vectors are introduced in relation to complex numbers expressed as vectors in the complex plane:
CCSS-Math-HS Numbers and Quantity: Vector and Matrix Quantities:
A1: Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).
B4: Add and subtract vectors.
B5: Multiply a vector by a scalar.