**Making Math Real: Fractions, Decimals & Advanced Place Value**

**22 half-day Distance Learning Course | 9:00 am to 1:30 pm daily ** [Pacific Time]

** Registration Fee: $2,849 for tuition & materials binder (shipped)
6 optional academic units **[see below]

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**MANDATORY PREREQUISITE**: All students who enroll in this course must have completed the *Making Math Real: Overview K-12* and *9 Lines Intensive* courses prior to enrolling.

**IMPORTANT NOTE ABOUT THE 9 LINES: **It is now a mandatory prerequisite to have also completed *Making Math Real: 9 Lines Intensive *prior to enrolling in* Fractions. *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*, *Geometry Part 1 & Geometry Part 2**, **Algebra I *and *Algebra 2*. Therefore, it is a significant benefit to all participants to have completed the *9 Lines Intensive* course.

**PREREQUISITE FULFILLMENT NOTE:** Successful completion of the *Fractions* course fulfills the prerequisite for taking the *Pre-Algebra* course.

**COURSE DESCRIPTION:** Making Math Real is a simultaneous multisensory structured methodology wherein each current mathematical development directly builds the foundation for the next. Continuing the structure of concept-procedure integration and sensory-cognitive development, the 2 strands presented in the *Making Math Real: Overview*, this course provides the essential bridge linking the synthesis of the 4 operations and the 400 math facts to algebraic processing and development. Fractions and their related applications are required tools for successful mathematical processing through calculus.

Making Math Real: *Fractions, Decimals, and Advanced Place Value* presents one of the largest and most intensive volumes of content of all the Making Math Real series of courses. In addition, Making Math Real: *Fractions, Decimals, and Advanced Place Value* presents the mathematical content and development required for the successful transition from elementary grades mathematics to the algebra strand. For further reading on this subject, please see the following article (below): *Fractions, Decimals & Advanced Place Value – The Three Essential Elements of Algebra Readiness*

The baseline readiness from the elementary grades for pre-algebra is the four operations through fractions and decimals and the 400 math facts. Therefore, to be ready for algebra, students need all the content and development provided in Making Math Real: *The 4 Operations and the 400 Math Facts*, and to complete the preparation for algebra readiness, Making Math Real: *Fractions, Decimals, and Advanced Place Value*.

**FRACTIONS**

In the fractions section of Making Math Real: *Fractions, Decimals, and Advanced Place Value*, all conceptual applications covering K-5 (6) will be presented. Since starting my career in education almost 50 years ago, fractions concepts and applications have been *consistently* an extremely high failure rate for students in the US. From the research:

*In 1978, more than 20,000 U.S. eighth graders (13- and 14-year-olds) were asked for the problem 12/13 + 7/8 whether 1, 2, 19, or 21 was closest to the sum. Only 24% answered correctly (Carpenter, Corbitt, Kepner, Lindquist, & Reys, 1980). The difficulty was not limited to fractions; on a similar NAEP (National Assessment of Educational Progress) problem with decimals (“Is 3.04 x 5.3 closest to 1.6, 16, 160, or 1,600”), only 21% of eighth graders answered correctly (Carpenter, Lindquist, Matthews, & Silver, 1983).*

*These data were collected a long time ago. In the ensuing years, numerous researchers, government commissions, organizations of mathematics teachers, and textbook writers have recommended ways of improving learning of rational numbers …Performance was essentially unchanged; 27% correct in 2014, versus 24% in 1978, again closely approximating a chance level of accuracy. (*1)

**(1) **** Hard Lessons: Why Rational Number Arithmetic Is So Difficult for So Many People **Robert S. Siegler and Hugues Lortie-Forgues, Current Directions in Psychological Science, 2017, Vol. 26(4) 346–351

There is no educational justification for fractions to be “so difficult for so many people”. The solution to this problem is the simultaneous multisensory structured curriculum delivery that directly supports students’ abilities to activate and sustain their comprehension-based working memory picture of all the interconnected concepts and applications of fractions.

Course emphasis will be on all fraction concept-applications presented in their correct scope and sequence across the grades including:

- Developing fraction number sense
- The four concepts of fractions
- The definitions of “numerator” and “denominator”
- Comparing and ordering fractions mentally
- Perceiving fraction magnitudes
- Rounding to the nearest whole number and to the nearest half
- Lowest terms and equivalent fractions: generating equivalent fractions and simplifying equivalent fractions to lowest terms
- Renaming to equivalent fractions with least common denominators
- Addition and subtraction of proper fractions with like and unlike denominators
- Mixed and improper fractions
- Addition and subtraction of mixed fractions with like and unlike denominators with and without renaming
- Concepts of fraction multiplication: multiplying by less than 1
- Cross simplifying fractions for proper, improper, and mixed fractions
- Concept of fraction division: why we transform division of fractions to multiplication
- Division of fractions for proper, improper, and mixed fractions
- Comparing and ordering fractions mathematically
- Fractions on the number line

**DECIMALS**

Since decimals *are* fractions expressed in place value code (base 10 and powers of 10 codes), all decimal concept applications *must* follow the respective fractions concept applications within a developmentally and instructionally correct scope and sequence. The indicated time for introducing decimals and their concept applications is in 4th grade. Consequently, students should have previously integrated the fractions concept applications from K through mid-4th grades as the foundation for now expressing and applying decimals in their code format directly connected to the respective applications in fractions format.

Course emphasis will be on all decimal concept-applications presented in their correct scope and sequence across the grades including:

- Concepts of “decimal-fractions”
- Decoding and encoding mixed and simple decimals through the 0.001s place
- Decimal equivalence
- Comparing and ordering decimals
- Rounding and estimating decimals through the 0.001s place
- Decimals on the number line
- Addition and subtraction with decimals
- Increasing and decreasing decimals by powers of 10
- Multiplication and division with decimals
- Decimal long division

**ADVANCED PLACE VALUE**

The development of number sense is directly related to the development of place and place value. This development is essential as it provides the foundation upon which all number-based mathematics will be derived, and therefore, the concepts and applications of place and place value *must* be the first unit in every grade, K-5. “Advanced” place value refers to the decoding and encoding of numbers from the 1,000s period through the billions period (100 billions) in standard and alternate number forms, whereas baseline place value, the essential fundamental for advanced place value, covers the decoding and encoding of standard and alternate number forms from 1 through 999 and is the domain of K-2nd grade. Baseline place value is presented in Making Math Real: *The 4 Operations and the 400 Math Facts* course.

In addition to decoding and encoding numbers in standard and alternate number forms, the advanced place value unit also includes place value-related applications, such as rounding and estimating and the development of sequencing and numeration by crossing place value landmarks counting forward and backward by any and all values, ones through 100 billions.

Course emphasis will be on developing number sense through all place value concept-applications presented in their correct scope and sequence across the grades 3 – 5 including:

- Decoding and encoding numbers in standard form for the periods of thousands through billions
- Decoding and encoding standard form to word form, and word form to standard form for the periods of thousands through billions
- Decoding and encoding standard form to expanded form, and expanded form to standard form for the periods of thousands through billions
- Decoding and encoding standard form to exponent form form, and exponent form to standard form for the periods of thousands through billions
- Rounding to the nearest 0.01 through rounding to the nearest 100 billion
- Estimation and the four operations with estimation
- Development of number sense through the applications of “Base 10 Puzzles”

**NUMBER THEORY**

The number theory unit covers 4th and 5th grades and will present the required fundamentals for all future applications of factoring in algebra through calculus. All of the number theory unit is based on the applications and extensions of **the 9 Lines Multiplication Facts Acquisition and Application Strategy™**. Course emphasis will be on developing requisite factoring applications presented in their correct scope and sequence across the grades including:

- Definitions and derivations of factors, products and multiples
- Factoring whole numbers
- Definition and derivation of Greatest Common Factor (GCF)
- Mentally finding the GCF of two products for GCFs ≤ 9
- Definition and derivation of Least Common Multiple (LCM)
- Mentally finding the LCM of two products for GCFs ≤ 9
- Definition and derivation of Least Common Denominators
- Primes and composites
- Prime factorization
- GCF games and factoring puzzles

**Extension and enrichment activities for each unit are presented.** Extensive color-coding is a critical element of the structure. Please bring 4 colored pens or pencils in blue, green, red and black.

**RELATED ARTICLE:** Fractions, Decimals & Advanced Place Value – The Three Essential Elements of Algebra Readiness

**FROM THE DESK OF DAVID BERG, ET:**

*Published on: August 19th, 2017
*The 4 Operations and The 400 Math Facts: The Essential Building Blocks for All Future Mathematics

** Optional Academic Credit Costs & Registration: **Optional units of credit are available for select courses at the low cost of $134 per unit, paid directly to CSUEB Continuing Education, by the last day of every course. CSUEB Online credit registration and payment instructions, along with the MMR grading policy, will be provided on the first day of every class. All credit registrations and payments must be submitted online through CSUEB’s website by the end of each course. No exceptions.

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