Making Math Real: Algebra II
18 half-day DL Course | 9:00 am to 1:00 pm daily [Pacific Time]
Registration Fee: $2249 for tuition & materials binder (shipped)
MANDATORY PREREQUISITES: All course participants must have successfully completed the Overview, 9 Lines Intensive, Fractions, Decimals and Advanced Place Value, Pre-Algebra, and Algebra I.
STRONGLY RECOMMENDED: The 4 Operations & the 400 Math Facts, Geometry Part I, and Geometry Part II.
IMPORTANT PREREQUISITE NOTE: The 9 Lines Multiplication Facts Acquisition and Application Strategy is an integral part of all the major MMR seminars, The 4 Operations & the 400 Math Facts; Fractions, Decimals, and Advanced Place Value; Pre-Algebra; and Algebra I. Therefore, it is a now a prerequisite for all participants to have completed the 9 Lines Intensive course prior to taking the other major MMR seminars.
COURSE DESCRIPTION: In the developmentally appropriate scope and sequence starting with pre-algebra, algebra II follows algebra I and high school geometry. The developmental focus of pre-algebra and algebra I is analysis and the developmental focus of geometry is synthesis, thereby making algebra II the first significant mathematical experience of applying analysis-synthesis to activate and sustain working memory for math.
The development of analysis-synthesis has been activated across the grades starting in kindergarten and occurs any time students are connecting the symbolic details (numerical or otherwise) to the big picture of comprehension; and from the big picture of comprehension back to the component symbols that comprise an expression or equation. For example in the elementary grades, analysis-synthesis occurs every time we prompt students to explain how they see the concrete model and the symbols are telling the same story. In addition, analysis-synthesis is the principal developmental foundation for solving word problems. Although analysis-synthesis has been developing along the way, algebra II is the first time analysis-synthesis is the main developmental emphasis.
The development of analysis from the successful completion of pre-algebra and algebra I refers to the activations and applications of detail analysis. Detail analysis is an executive function-based sensory-cognitive development that directly supports the ability to activate and sustain working memory from the interrelationship of the symbolic details, which is the basis for the successful processing of algebraic expressions and equations. For example, detail analysis makes it possible to perceive which of the 7 factoring styles is indicated from either product form or factored form, if the slope of the line is positive, negative, 0, or undefined from slope-intercept form, to identify which form of the linear equation is given: standard form, slope-intercept form, or point-slope form, etc.
The development of synthesis from the successful completion of geometry refers to the activations and applications of synthesizing the big picture from partially given information. In supporting the development of deductive reasoning, specifically for solving deductive proofs, the application practice of geometry provides partial information about certain relationships, within or in conjunction with, lines, angles, polygons and/or circles, from which the student must systematically deduce all pertinent information about those relationships to reach specific conclusions (incontrovertible proof) about those relationships. For example, synthesis makes it possible for students to deduce two triangles are congruent based on the partially provided information that two corresponding sides and a corresponding included angle of the two triangles are congruent, therefore guaranteeing that all six corresponding relationships of sides and angles are congruent.
The ability to synthesize partially given information and expand it to comprehensive proof means that all component details comprising the big picture have been accounted for, and before writing or expressing any solution or conclusion, it must be a fact, not an assumption. The ability to be certain about a solution prior to expressing it defines the successful experience of math: “I know that I know”.
Analysis-synthesis is necessary for successfully activating and sustaining working memory for algebra II content. The principal content focus of algebra II is functions, their graphs, and their applications, and the overall function of algebra II is to prepare students for pre-calculus. This presents a significant extension of both algebra I and geometry content, and analysis-synthesis is activated to support the perceptual connections necessary to decode and encode the equations and their graphs, the graphs and their equations, and especially for all the applications of word problems. Within the first three units, algebra II content will require full activation of all the algebra I content: every component detail of simplifying expressions, solving equations in one variable through level XIV, solving linear equations graphically and algebraically for single equations and two equations using systems of equations as well as systems of inequalities, and more than 12 categories of algebra I word problems.
All of the algebra II content will be structured through functions, their graphs, transformations of the parent functions, and their applications. Topics include comprehensive units on:
- Functions and relations, linear and absolute value functions including linear programming and solving systems for three equations and three variables
- Quadratic functions and parabolas including the factoring of prime products: completing the square and the quadratic formula, imaginary numbers, complex numbers and their graphs, and real and imaginary roots
- Polynomial functions of higher degree, the shape of their graphs and end behavior, factoring polynomials of higher degree including real and complex solutions, polynomial long division and synthetic division
- Rational functions including their transformations to polynomial rational form; graph behavior: asymptotes and holes, and direct and inverse variation
- Applications of functions including piecewise functions, the four operations with functions, composition of functions, inverse functions, and even and odd functions
- Square root functions, radical expressions and equations, exponential expressions and equations, rational exponents, and transformations between radical and exponential forms
- Exponential and logarithmic functions, expressions, and equations including the natural exponent and the natural logarithm: problem solving with continuous growth and decay, simple and compound interest, half-life and doubling
- Arithmetic and geometric sequences and series, sigma notation, and infinite sums
- Synthesis applications of all the algebra II functions
And if time permits:
- Probability, using the counting principle for permutations and combinations.
SPECIAL NOTE: Since this course will use all of the algebra I content immediately, you are strongly encouraged to have fully integrated all of the math content in the Making Math Real: Algebra I course and have it fully refreshed and activated to receive full developmental benefit from the Making Math Real: Algebra II.
This course is designed for educational therapists, special educators, and elementary, secondary, and community college teachers. These techniques are designed to reach the full diversity of learning styles. Extensive color-coding is a critical element of the structure. Please bring 4 colored ball point pens or pencils in blue, green, red and black.