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Mathematical Reasoning™

Bridging the Gap Between Computation and Math Reasoning

Grades: Toddler-9


Full Curriculum

Mathematical Reasoning™

Grades: Toddler-9


Full Curriculum

  •  Multiple Award Winner

Forget boring math lessons and dreaded drill sheets.  These fun, colorful books use engaging lessons with easy-to-follow explanations, examples, and charts to make mathematical concepts easy to understand.  They can be used as textbooks or comprehensive workbooks with your textbooks to teach the math skills and concepts that students are expected to know in each grade—and several concepts normally taught in the next grade.

Every lesson is followed with a variety of fun, colorful activities to ensure concept mastery.  The lessons and activities spiral slowly, allowing students to become comfortable with concepts, but also challenging them to continue building their problem-solving skills.  These books teach more than mathematical concepts; they teach mathematical reasoning, so students learn to devise different strategies to solve a wide variety of math problems.  All books are written to the standards of the National Council of Teachers of Mathematics.

Beginning 1, Beginning 2, and Level A Contents

     •  Counting
     •  Identifying
     •  Matching
     •  Ordering
     •  Position
     •  Comparing and Estimating
     •  Addition
     •  Subtraction
     •  Locating
     •  Writing
     •  Grouping
     •  Patterns
     •  Geometric Shapes
     •  Measurement
     •  Data/Probability
     •  Fractions

Level B, Level C, Level D, Level E, Level F, and Level G Contents
Mathematical Reasoning Content Chart

Understanding Pre-Algebra
This book teaches and develops the math concepts and critical thinking skills necessary for success in Algebra I and future mathematics courses at the high school level.  It was written with the premise that students cannot problem solve or take leaps of reasoning without understanding the concepts and elements that lead to discovery.  The author—with 35 years of experience teaching mathematics—is a firm believer that understanding leads to confidence and confidence gives students the resolve to succeed in higher level mathematics rather than fear it.  

It is standards-based, but what makes it different from other pre-algebra books is that it organizes concepts in a logical fashion, stressing practice and critical thinking. It avoids the mistakes—found in many other math books—of trying to teach new concepts before students receive the prerequisite skills and practice necessary for success. The concepts are presented clearly and in connection to other concepts. Math vocabulary is very important to success in higher mathematics, so this book includes easy-to-follow explanations and a user-friendly glossary.

Free Detailed Solutions are available!

Understanding Pre-Algebra Contents

     •  Family of Numbers
     •  Working With Integers
     •  Working With Rational Numbers
     •  Ratio, Proportion, and Percent
     •  Percent Applications
     •  Algebraic Expressions
     •  Equations and Solving Word Problems
     •  Inequalities and Applications
     •  Understanding Square Roots and Irrational Numbers
     •  Two Dimensional Geometry
     •  Understanding Volume and Surface Area
     •  Graphing on the Coordinate Plane
     •  Transformations and Congruency
     •  Understanding Functions
     •  Probability and Statistics

Understanding Geometry
The successful completion of this colorful 272-page book will prepare middle schoolers for high school geometry. It covers more than 50% of the concepts taught in high school geometry using a step-by-step approach and teaches the reasoning behind the properties taught in geometry–instead of merely asking them to memorize them. Students are also taught the basics of geometric proofs and coordinate geometry in a way middle school students can understand. Students who struggle with high school geometry usually have lower standardized test scores because it is a fundamental subject in high school standardized testing.

A glossary of terms that every student should master is included. This book can be used as a classroom textbook in Grades 7, 8, or 9 (usually over a two-year period) or as a reference for high school students. This book covers more than the National Math Standards for middle school mathematics.

Free Detailed Solutions are available!

Understanding Geometry Contents

     •  Geometry Notation
     •  Lines
     •  Planes
     •  Angles—Types and Properties
     •  Pythagorean Theorem
     •  Polygons
     •  Quadrilateral Properties
     •  Parallelogram Properties
     •  Perimeter
     •  Circumference
     •  Pi
     •  Trapezoids
     •  Geometric Constructions
     •  3D Shapes—Prism, Cylinders, Pyramids, and Cones
     •  Symmetry
     •  Transformations—Reflection, Translations, Rotations, and Dilations
     •  Tessellations
     •  Proofs
     •  Congruency
     •  Slopes

NOTE:  It is our recommendation that students complete Understanding Pre-Algebra (see description above) before attempting Understanding Geometry.

Understanding Algebra I
This is a one-year Algebra I course for Grades 7-9. Students who have a solid algebra background will have no trouble with the algebra problems from SAT and even the GRE.  This 384-page book highlights vocabulary and notation, and has examples from the history of math.

What makes this book unique and different from other algebra textbooks is that it is built from the experiences of an award-winning algebra teacher with more than 30 years of teaching experience. Many textbooks are written by a committee of authors, and many of those authors have little experience teaching beginning algebra students in middle school or high school.

Understanding Algebra I presents the most essential concepts and skills needed to fully understand and gain confidence in algebra in a step-by-step fashion, teaching students that algebra is generalized arithmetic. It helps students see the connection between mathematics that they already know and algebra, so that learning algebra becomes easier and less abstract. This book provides students with real strategies to succeed in solving word problems by using charts and translating strategies that guarantee success.

Understanding Algebra I Contents

     •  Set and Set Notation
     •  Number Lines
     •  Graphing
     •  Rationals
     •  Operations
     •  Expressions
     •  Equations
     •  Inequalities
     •  Word Problem
     •  Percent Problems
     •  Ratio Problems
     •  Motion Problems
     •  Work Problems
     •  Absolute Value
     •  Polynomials
     •  Factoring
     •  Radicals
     •  Linear Functions
     •  Slope
     •  Elimination
     •  Graphing
     •  Substitution
     •  Absolut Value
     •  Algebraic Fractions

Essential Algebra for Advanced High School and SAT
This 241-page math book is a new, ‘essential’ resource for building the foundation of ongoing excellence in mathematics. This book offers an in-depth approach to teaching math skills focusing on important algebraic concepts that are vital to a student’s success in advanced high school mathematics.

Algebra is essentially the language of all advanced mathematic courses. It is generalized arithmetic, so the rules that apply when we are working with number and fraction operations—including the knowledge of factors and multiples—also apply when we are working with algebra.

Without this knowledge and these skills, students often struggle or even fail in advanced mathematics courses. Imagine a good high school student who sees a problem like 3•x•y•4 and hesitates to write 12xy because he or she is unsure of the rules that govern multiplication. Or they don't understand how to add 2x to 1/4y to combine it into one entire fraction. Or why –62 is different than (–6)2. It is easy to see that not having a strong understanding of the foundational rules of algebra can stop even the smartest students from succeeding in advanced high school math courses.

The lessons within this book teach fundamental arithmetic principles that govern algebra so that students can apply them, as needed, in advanced mathematical courses and in life. The book begins with the definition of a "term" and the facts of when you can cancel and when you cannot. It discusses the importance of factors and multiples and integer rules. It teaches addition, subtraction, multiplication, and division of fractions – not by doing the problems up and down like they are mostly done in elementary school, but by doing them sideways just like they are done later in algebra and in higher-level math. This helps make everything so much easier down the road!

The explanations are clear and followed with basic step-by-step procedures to show students how to master and apply those fraction concepts, negative number operations, exponent rules, work with square roots, and doing real life percent problems with charts and tables, just like those on the SAT’s. Concepts are purposefully repeated in different settings, so students will keep re-applying what they learn throughout the book. The language is friendly to put students at ease and increase their confidence while learning concepts that are often confusing.

This book reviews the necessary reading strategies needed to translate word problems, but it is not the primary focus of the book. The book Understanding Algebra I (see above) devotes large sections to many types of word problems and many useful strategies if deeper study of this topic is desired. Essential Algebra is a supplemental book that can be used as an accompanying book to Algebra I, Algebra II and higher math courses. Also recommended as an Algebra II prep course!

In the last two chapters of the book, students are asked to apply the algebra they have just learned to an overview of problems in geometry. Also included is review of some important techniques that students frequently struggle with involving both linear and quadratic equations.

This book also includes an appendix of topics used as a review, but also includes new information that can be considered enrichment for students who may want to learn more. Teaching support and answers are also included.

Mathematical Reasoning™ Supplements
These supplemental books reinforce grade math concepts and skills by asking students to apply these skills and concepts to non-routine problems. Applying mathematical knowledge to new problems is the ultimate test of concept mastery and mathematical reasoning. These user-friendly, engaging books are made up of 50 theme-based collections of problems, conveniently grouped in self-contained, double-sided activity sheets that provide space for student work. Each collection contains relevant math facts at the end of the worksheet in case students need hints to solve the problems. Calculators are allowed on activity sets that have a calculator icon at the top of the front side of the set. Each activity set is accompanied by a single-sided answer sheet containing strategy tips and detailed solutions. Teachers and parents will appreciate the easy-to-understand, comprehensive solutions. These books are a wonderful enrichment tool, but also can be used to assess how well students have learned their grade level's math concepts.

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