Introduction to Counting & Probability Solutions Manual
This is an outstanding math program for the math-gifted student. It is rigorous and oriented to the independent problem-solver. The texts are based on the premise that students learn math best by solving problems - lots of problems - and preferably difficult problems that they don't already know how to solve. Most sections, therefore, begin by presenting problems and letting students intuit solutions BEFORE explaining ways to solve them. Even if they find ways to answer the problems, they should read the rest of the section to see if their answer is correct and if theirs is the best or most efficient way to solve that type of problem. Textual instruction, then, is given in the context of these problems, explaining how to best approach and solve them.
Throughout the text there are also special, blue-shaded boxes highlighting key concepts, important things to retain (like formulas), warnings for potential problem-solving pitfalls, side notes, and bogus solutions (these demonstrate misapplications). There are exercises at the end of most sections to see if the student can apply what's been learned. Review problems at the end of each chapter test understanding for that chapter. If a student has trouble with these, he should go back and re-read the chapter. Each chapter ends with a set of Challenge Problems that go beyond the learned material. Successful completion of these sets demonstrates a high degree of mastery.
A unique feature in this series is the hints section at the back of the book. These are intended to give a little help to selected problems, usually the very difficult ones (marked with stars). In this way, students can get a little push in the right direction, but still have to figure out the solution for themselves. The solution manuals do contain complete solutions and explanations to all the exercises, review problems and challenge problems. It is best for students not to access these until they have made several attempts to solve the problems first. I particularly like one of the motivating boxes in the text that coaches, "If at first you don't know how to solve a problem, don't just stare at it. Experiment!". That pretty much sums up the philosophy of the course, encouraging children to take chances, become aggressive problem solvers, and attack problems with confidence. I wonder how far some children would go if they were encouraged this way instead of being spoon fed?
Though this course is used in classroom settings, the texts are student-directed, making them perfect for the independent learner or homeschooler. Students should start the introductory sequence with the Prealgebra book. Afterwards, begin the Introduction to Algebra. Students will be prepared for both the Introduction to Counting and Probability and Introduction to Number Theory courses after completing the first 11 chapters of Algebra. It won't matter whether they do these along with Algebra, put aside Algebra and complete the other two or finish Algebra first and then do them. All of them should be completed prior to the Introduction to Geometry book. If you are coming into this course from another curriculum, you will probably want to take a placement test to decide where to enter this program. Even if your student has finished Algebra 2 elsewhere, you will want to make sure that all of the material from this series has been covered before continuing on to the Intermediate series.
Taken together, these constitute a complete curriculum for outstanding math students in grades 6-10 and one that prepares them for competitions such as MATHCOUNTS and the American Mathematics Competitions. The material is challenging and in-depth; this is not a course for the mathematically faint of heart. If your child loves math, is genuinely math-gifted, or is interested in participating in math competitions, you definitely need to give this one serious consideration.
The Art of Problem Solving folks present: Beast Academy. Proving that any monster can do math, these comically monstrous beasts take you along with them as they face math challenges both inside and outside the Academy. Described as “rigorous, challenging, engaging, and fun” by the publisher, I would expect nothing less from a curriculum designed to ready your child for the demands of MATHCOUNTS and post-elementary AOPS curricula.
There are four parts for each grade level though the series is not all available as of this writing. Each part consists of two books: a Guide and a Practice book. Core instruction takes place in the Guide which is written in colorful, comic-book style and stars a cast of not-quite-cute monsters. Each book is divided topically into three chapters.
Grade 2 chapters are: 2A – Place Value, Comparing, Addition; 2B – Subtraction, Expressions, Problem Solving; 2C – Measurement, Strategies (+ & -), Odds & Evens; 2D – Big Numbers, Algorithms (+ & -); Problem Solving. [Currently, only 2A and 2B are available. Contact us for availability.]
Grade 3 chapters are: 3A – Shapes, Skip-Counting, Perimeter and Area; 3B – Multiplication, Perfect Squares, Distributive Property; 3C – Variables, Division, Measurement; 3D – Fractions, Estimation, Area. Grade 4 topics are: 4A - Shapes, Multiplication, Exponents; 4B - Counting, Division, Logic; 4C - Factors, Fractions and Integers; 4D - Fractions, Decimals and Probability. These chapters are further divided into sections (subtopics). For example, Chapter 1 of Grade 3A: Shapes contains sections labeled: Angles, Triangles, Quadrilaterals, and Polyominoes. 5A practices 3D solids, integers, expressions and equations. 5B covers statistics, factors & multiples and fractions. 5C will teach sequences, ratios & rates and decimals. 5D includes square roots and exponents.
Activity sheets, asides, notes, extensions, and games are often interspersed between the sections. Helpful, red rectangles at the end of some pages tell you when to break and do practice problems. Stop signs also appear in the corners of some panels. These contain questions for the student to answer before continuing to read. Guides are non-consumable. All written work is done in the corresponding Practice book which contains 300-400 practice problems and puzzles that reinforce the Guides’ instruction. Practice chapters each begin with a recommended sequence (like a lesson plan) correlating the reading (Guide) to practice pages. Problem sets generally begin with a recap of the concepts covered and a model problem with solution. Some problems have a pencil icon next to them signaling the student to write an explanation in addition to their answer. “Very challenging” problems are starred. Doubly-starred problems are, well, beastly. For students requiring a little help with these, each has a hint to get them started. These are found in a separate section in the back of the book so students won’t be tempted to use them unless necessary. Complete solutions to all problems in the book are also included. All pages in the Practice book are perforated, so you can optionally tear out either problem sets or solution keys if you desire.
So how does this program compare to other math curriculums? Compared to Saxon Grade 3, they cover basically the same topics, though each covers a few that the other doesn’t. Saxon follows a spiral approach and integrates review whereas this program is very topical and by the strand with neither tests nor chapter reviews. Saxon is a traditional math course whereas Beast Academy is not. It is innately more conceptual and interactive because of its unique lesson format. While Saxon is intended to be teacher-led, Beast Academy is written to the student. While manipulatives are an integral part of Saxon instruction, they would only be used here by choice as student problem-solving aids (though the characters use “real” objects in the context of the lessons). Students are encouraged to think and problem solve to a greater degree as you might expect in a curriculum from this publisher. There is no shortage of practice problems, a shortcoming often found in less traditional programs. The math-oriented student would get the most mileage (and challenge) out of this program. It is probably NOT for the student that needs a lot of hand-holding. That said, my child that most disliked math would have loved this program because of its highly-visual format and contextual presentation of concepts.
Beast Academy has its own support website (www.beastacademy.com) which contains free pre-assessments (with solutions), selected printable practice pages, coloring pages, and errata. When your little beast is finished with grade 5, he or she can move on to AOPS (or any other demanding) middle-school math program. Grade Packages include all four Guides and all four Practice Books for each grade level.