**Fractions, Decimals & Advanced Place Value: **

**The Three Essential Elements of Algebra Readiness**

###### By David Berg, E.T.

Founder/Director of the Making Math Real Institute

Creator of the Making Math Real Multisensory Structured Methodologies in Mathematics, K-12

How many times have I heard exasperated students exclaim, “*I hate fractions!*” Even students who feel math is easy often claim, “*Yeah, that other stuff is pretty easy, but I’m really bad at fractions – oh, and decimals, too.*” Fractions are challenging, and are estimated to have an 80% failure rate nationwide.

The challenges students face in learning fractions are equal, if not exceeded by, the challenges math educators face in successfully teaching fractions to the full diversity of student populations. Furthermore, fractions’ fundamental connection to place value and decimals makes for a critical educational imperative, because within the K-12 architecture of mathematics, fractions, decimals, and advanced place value are the most crucial bridge connecting elementary mathematics to algebra through calculus. However, students are frequently taught fractions from purely procedural instructional methods, and this, combined with textbooks that may skip over significant content connections, compound the educational challenges students face in developing effective algebra readiness.

First, and foremost, significant numbers of students struggle with fractions because they have not developed automaticity with the multiplication and division facts. Additional common areas of challenge for students include limited conceptual understanding of:

- what fractions are
- numerator/denominator relationships
- equivalence and lowest terms
- mixed fractions and improper fractions
- multiplication and division with fractions
- cross simplifying for multiplication
- inverting and multiplying for division
- least common denominators
- ratios

In applying fractions, students also demonstrate significant challenges with:

- addition and subtraction with unlike denominators
- generating equivalent fractions and lowest terms
- conversions between mixed fractions and improper fractions
- multiplication and division with simple and mixed fractions
- comparing and ordering
- generating greatest common factors and divisors of two products

To be successful with fractions, students require comprehension of the 4 concepts of fractions: parts of a whole (2 and 3-dimensional), parts of a group (2 and 3-dimensional), parts of a line (segment) (1-dimensional: x-axis), and parts of a cup (both 3-dimensional: in the cup and 1-dimensional: y-axis). Students need to be fluent in decoding and encoding the symbolic structure of fractions based on the complete and specific understanding of what numerators and denominators signify, their functions, and what the interrelationships of numerators and denominators indicate. Furthermore, students need concept-procedure integration of lowest-terms and equivalent fractions, greatest common factors and greatest common divisors of two products, least common denominators (multiples) of two products, and fractions’ relationship to ratios. This integration provides the essential foundation for all future fraction processing including the 4 operations through mixed fractions; ratios and probability; and rational expressions and equations throughout algebra I and II, geometry, trigonometry, and calculus.

In working with thousands of diverse students over the last 34 years, one of the most exceptional benefits I have observed students experience while integrating fractions is the development of automaticity with the multiplication and division facts, especially for students who benefit from using the mental organizing structure embedded within the 9 Lines Multiplication Fact Acquisition and Application Strategy™. I expressly designed and created the 9 Lines™ mental organizing system for two principal purposes. First, to develop students’ automaticity with the multiplication and division facts, and second, to support fraction processing, specifically to mentally find products, quotients, missing factors and divisors, greatest common factors and greatest common divisors of two products, and least common denominators (multiples) of two products. Students’ continuous experience of using this mental organizing system to apply all components of multiplication/division fact families in application of fraction problem solving provides the culminating experience in developing automaticity with the multiplication and division facts.

To be developmentally and mathematically ready for success in algebra, fractions alone are insufficient. Students also require integration of advanced place value (through the billions) and decimals (at least through the thousandths). As with fractions, students require comprehensive integration of conceptual understanding with procedural fluency in both place value and decimals. Again, textbooks often leave out the majority of structure and content connecting fractions, place value, and decimals, thereby making it difficult for students and educators to effectively synthesize the interrelationship of these three essential elements of algebra readiness. For example, many, if not most textbooks, present the decimals unit prior to the fractions unit. Conceptually, this represents a mathematical and educational disconnection since decimals are fractions, but are merely recorded using a different code. The decimal and the fraction are both expressing the same part-of-a-whole fraction picture. The fraction uses a numerator/denominator relationship to express the relationship: 3 of the 10 parts are shaded. The decimal code specifically represents whole units and parts of a unit to express the concept-applications of place: 0.3 = 0 whole units (ones place) are fully shaded and 3 of the 10 parts (tenths place) are shaded. Furthermore, it is the decimal code of recording fractions as whole units and parts of a unit that connects fractions to the place value system. Therefore, if students are taught decimals prior to fractions it is possible they may not perceive the conceptual connection that decimals are fractions expressed in the code of place.

The developmental outcome of students’ successful integration of fractions, decimals, and advanced place value is the essential fundamental readiness for pre-algebra. Making Math Real: *Fractions, Decimals, and Advanced Place Value* provides the most intensive and comprehensive incrementation in connecting fractions, decimals, and advanced place value by presenting:

- the 4 concepts of fractions
- numerator/denominator definition and relationship
- hands-on, concrete connection of lowest-terms and equivalent fractions
- generating equivalent fractions and lowest terms
- number theory: factors and multiples, whole number factoring, greatest common factors and least common multiples
- hands-on, concrete connection of least common denominators
- addition and subtraction with like and unlike denominators
- hands-on, concrete connection of multiplying fractions
- multiplication and division with simple and mixed fractions
- structured explanation of using cross-simplifying for multiplication and inverting and multiplying for division
- hands-on, concrete connection of mixed fractions and improper fractions
- converting mixed fractions and improper fractions
- comparing and ordering fractions
- addition and subtraction with renaming for proper and mixed fractions
- advanced place value through the billions including standard form, word form, expanded form, and exponent form
- decoding the decimal code: connecting fractions decimals and place value
- decimals on the number line
- comparing and ordering decimals
- rounding and estimating with whole numbers, decimals, and fractions
- 4 operations with simple and mixed decimals

Most importantly, this course provides the entire multisensory structured conceptual and content incrementation spanning the most elemental concrete basis of fractions, decimals, and advanced place value all the way to their highest levels of abstract problem solving. The concrete to abstract structure scaffolds students’ direct experience of connecting the concrete reality of the math to its specific reconstruction in abstract symbolic form, thereby maximizing comprehension and successful decoding, encoding, and the interrelationship of fraction, decimal, and place value coding systems.

Providing students with the requisite incrementation helps students make and retain all connections in learning the mathematics with in-depth understanding, all of which fosters mathematical and cognitive development to the greatest extent. Being equipped with the content knowledge, incrementation and methods for teaching the math, empowers us as educators to be the primary source of instruction, not the textbook. The demands of teaching require the educator to adapt curriculum delivery to be aligned with the diverse processing needs of students. Therefore, textbooks cannot, and should not, be used as the primary source of instruction. The most appropriate usage of the textbook is as a resource for support, practice, and test preparation. Students need and deserve teachers that can successfully connect all aspects of mathematical instruction to maximize comprehension and integration. For this purpose, Making Math Real: *Fractions, Decimals, and Advanced Place Valu*e in combination with Making Math Real: *The 9 Lines Intensive*, have been designed expressly to provide educators with comprehensive content, structure, methods, and incrementation for maximizing student readiness for, and success in, algebra and advanced mathematics. Therefore, Making Math Real: *Fractions, Decimals, and Advanced Place Value* is the required prerequisite course for Making Math Real: *Pre-Algebra* because the content methods provided in the pre-algebra course, as with all Making Math Real courses, is developmental, and extends directly from the content methods from the fractions, decimals, and advanced place value course.