Making Math Real: Pre-Algebra

Making Math Real: Pre-Algebra

18 half-day DL Course  |  9:00am to 1:00pm daily*
*In response to COVID-19 and in support of shelter-in-place orders, the daily course schedule for distance learning (DL) classes will be half days from 9:00 AM – 1:00 PM PST, due to the extra cognitive demands of interacting with computer screens for extended periods of time, increasing the number of days from 10 “in-person” full-days, to 18 “online” half-days.
Registration Fee:  $1,999 for tuition & course reader (shipped)
5 optional academic units  
[see below]

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MANDATORY PREREQUISITES:  All course participants must have successfully completed the Overview, K-12 AND Fractions, Decimals, & Advanced Place Value prior to enrolling.

IMPORTANT NOTE:  The 9 Lines Multiplication Facts Acquisition and Application Strategy is an integral part of all the major MMR seminars, The 4 Operations and the 400 Math Facts; Fractions, Decimals, and Advanced Place Value; Pre-Algebra; and Algebra I.  Therefore, it is a significant benefit to all participants to have completed the 9 Lines Intensive course prior to taking the other major MMR seminars.

COURSE DESCRIPTION:  This 10-day course continues the sequence from the Overview, K-12, 4 Operations and the 400 Math Facts, and Fractions, Decimals and Advanced Place Value courses, and provides the essential development for transitioning students from elementary math to algebra including the full structure, increments, and scope and sequence for pre-algebra.

Within the K-12 scope and sequence, pre-algebra presents the beginning of the second major mathematical and developmental milestones spanning kindergarten through calculus. The first major development, typically within the domain of grades K- 5 math content, is the concept – procedure integration of the four operations through fractions and decimals and the 400 math facts. The developmental focus for elementary grades mathematics is symbol imaging, the sensory-cognitive development required for students to activate their comprehension-based, working memory perceptual picture of the math.  The development of symbol imaging is fostered through the simultaneous multisensory structure of guiding students systematically from the concrete to the abstract. The successful transfer of the concrete experiences of the math to their specific reconstruction in abstract symbolic form provides the developmental basis for students to activate their working memory picture from math symbols, which is a major function of symbol imaging.

Baseline readiness for pre-algebra is the integration of the four operations through fractions and decimals and the 400 math facts, because once beginning the algebra strand starting with pre-algebra, students will apply everything they have integrated with the four operations through fractions and decimals and the 400 math facts. All of pre-algebra and algebra 1 is combining and simplifying the four operations with integers and rational numbers in support of simplifying variable expressions using the order of operations and solving variable equations using the inverse order of operations.

As the second major developmental milestone within the K-12 scope and sequence, pre-algebra introduces a new basis for activating working memory: detail analysis. The activation of working memory using detail analysis is entirely distinct from the elementary grades use of symbol imaging. Symbol imaging activates the concrete experience of the math that is encoded in the math symbols, while detail analysis provides the new basis for activating working memory: the interrelationship of the symbolic details.  It is through the perception of the interrelationship of the symbolic details that we activate our comprehension of algebraic expressions and equations.  For example, this is how we know when an expression is in simplest form, the distinction between expressions and equations, when an equation is ready to solve, and when an equation is in solution form. It is critical to note that the successful development of detail analysis requires the successful baseline integration of symbol imaging from the elementary grades.

Pre-algebra is the necessary fundamental for algebra 1, and as such, provides the requisite preparation for all algebra 1 concept applications. Therefore, there are specific distinctions between pre-algebra and algebra 1, and it is strongly contraindicated to blend the two. The “pre-” in pre-algebra is about preparing students for most of what comprises algebraic problem solving: simplifying variable expressions, solving variable equations and graphing. Therefore, it is within the domain of pre-algebra that students learn the integer and rational numbers facts across the four operations, as these number facts will be applied throughout simplifying and solving for all algebra. The extent of simplifying expressions in pre-algebra is basic only, mostly to learn what constants, variables, expressions and equations are, the Order of Operations, GEMDAS, and the inverse order of operations, SADMEG, and not for combining like terms which is an early algebra 1 unit in which students learn what a polynomial is, the degree of the polynomial, and expressing polynomials in standard form.

For solving equations, pre-algebra covers only the first four levels of solving equations in one variable and the first four levels of solving inequalities in one variable. Levels five through fourteen of solving equations in one variable are all covered in algebra 1. The parameter that distinguishes the pre-algebra equation solving levels from the algebra 1 levels is that all pre-algebra level equations are ready to solve and all the algebra 1 levels require extra transformations and/or simplifying of expressions before the equations are ready to solve. The pre-algebra equation solving levels provide students with a solid foundation for all future equation solving: an equation is ready to solve when one variable expression in simplest form equals one constant expression in simplest form; and to be appropriate equations for pre-algebra, all variable expressions and all constant expressions are already in simplest form. Any variable or constant expressions requiring combining like terms, use of the distributive property, or having variable terms in both expressions are all exclusively part of the algebra 1 equation solving levels.

In the ratio, proportion, percent unit, pre-algebra covers the four levels of fraction, decimal, and percent equivalence, the introduction of ratios, rates and proportions, the three forms of percent problem solving, and solving proportions with a single variable and three constants. Solving percent change and proportions with binomial numerators and/or denominators is covered in algebra 1.

The pre-algebra probability unit covers all basic probability including theoretical and experimental probability, simple, multiple, and compound events, dependent and independent events, dependent and independent events with and without replacement, and the counting principle. Permutations (nPr) and combinations (nCr) are typically presented in second semester algebra 2.

The pre-algebra linear graphing unit starts with graphing on the x-axis and solving levels one through four of inequalities. Pre-algebra follows with graphing on the x-y Cartesian plane, by first introducing solving equations in two variables and connecting the infinite solutions generated by 2 variables in one equation to the graphical expression of the infinite solutions as infinite points comprising a line. From this experience students learn slope-intercept and standard forms of linear equations. The extent of the linear graphing unit for pre-algebra is students’ ability to graph any line with positive, negative, and special slopes from any given equation; and to generate the equation of any graphed line with positive, negative, and special slopes. Algebra 1 continues this development by teaching students to generate the equations of lines algebraically without using graphs.

TOPICS INCLUDE:

  • Four operations with integers
  • Four operations with rational numbers
  • Introducing constants and variables
  • Expressions across the four operations
  • The definition of an equation
  • Solving equations and inequalities through level IV
  • Number theory
  • Fraction-decimal-percent equivalence
  • Ratio, proportion, and percent problem solving
  • Linear graphing: introducing 2-variable linear equations
  • Introducing the x-y Cartesian plane, ordered pairs, plotting ordered pairs on the graph, and generating coordinates from the graph
  • Graphing lines for positive, negative, and special slopes
  • Generating equations for lines with positive, negative, and special slopes

This course is designed for educational therapists, special educators, elementary and secondary classroom teachers, and college professors. Parents and those who consider themselves non-math majors are especially encouraged to enroll.  Prior knowledge of algebra is not required.

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-color ball point pens or pencils in blue, green, red and black.

FROM THE DESK OF DAVID BERG, ET:
Published on: February 17th, 2018
Pre-Algebra: The Foundation of the Algebra Strand Through Calculus

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Optional Academic Credit Costs & Registration:  Optional units of credit are available for each course at the low cost of $134 per unit, paid directly to CSUEB Continuing Education. Credit registration forms, payment instructions, and the grading policy will be provided on the first day of class. All credit registration forms must be submitted with payment to the MMR instructor by the end of each course. No exceptions.

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