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Sample AlgebraVictory! Final Exam
Here is a sample AlgebraVictory! Final Examination. It covers many of the kinds of problems students will learn to solve in AlgebraVictory! But it does not cover every kind problem—for example, it only contains seven word problems, but students are required to know how to solve 15 different kinds of word problems. Most adults will be intimidated by this exam. Yet, by the time AlgebraVictory! students have completed Lesson 75, they will not be intimidated by it—they will sit down and do it, and likely get a passing score
The Six Chapters and the Subjects They Cover
- Chapter 1. The natural numbers; variables and when clauses; the integers; integer arithmetic (i.e., signed arithmetic); the rational numbers; fraction arithmetic; solving linear equations in one variable (including with parentheses, fractions and when clauses); simple syntax diagramming (factor, term, phrase, sentence); and word problems involving direct translation (in one variable).
- Chapter 2. Graphing linear equations in two variables; solving pairs of linear equations in two variables (including with parentheses, fractions and when clauses); decimals and decimal arithmetic (including repeating decimals); conversion between decimals, fractions, mixed numbers and percents; the real numbers; and two kinds of word problems, involving: (a) direct translation (two variables), (b) rectangles (linear).
- Chapter 3. Introduction to functions; syntax diagramming with powers, fractions and radicals; monomials; clearing parentheses involving monomials and polynomials; and three kinds of word problems, involving: (a) consecutive integers, (b) items in a ratio, and (c) percent increase/decrease.
- Chapter 4. Factoring polynomials (by five methods); simplifying algebraic fractions, including complex fractions.
- Chapter 5. Solving quadratic equations in one variable that have rational solutions; solving equations containing algebraic fractions; solving equations containing parameters; and five kinds of word problems: (a) area of rectangles and right triangles, (b) motion of an object with and against a moving medium, (c) independent motion of two objects, (d) discrete mixtures, and (e) continuous mixtures.
- Chapter 6. Radicals, the quadratic formula, solving equations with variables under radicals; solving and graphing inequalities in one variable and in two variables; and word problems involving power and work.
AlgebraVictory! is taught cumulatively. This means students are required to maintain and build on the skills they learn throughout the course. Although quizzes are just on newly learned skills, tests and the final exam include problems from the entire course. Before each test, there are two or three review lessons, and before the final there are four review lessons. Moreover, to a large extent, the material is inherently cumulative, because skills taught later in the course depend upon skills previously learned.
Algebra Is Holistic: Hold the Algebra in Your Right Hand.
AlgebraVictory! is taught holistically. Students learn a complex diversity of concepts, facts and skills, yet, at the same time, they learn how it all fits beautifully together into a unified whole. By the end of the course, students can (metaphorically) hold the whole of the algebra in the palm of their right hand. And it is not separate from them. The algebra is an integral and vibrant part of this universe, and hence an integral and vibrant part of who we are as living, breathing Beings with the capacity to comprehend our existence and to think rationally about it. Quite obviously, the complex concepts, facts and skills taught in AlgebraVictory! are crucial for the STEM fields. But what few people understand is that the holistic lessons (that can only be learned by mastering the algebra) are important for ALL students to develop intellectual maturity and spiritual awareness. This is because God is in the math, and some students are able to experience this.
Mechanical Problems Only; No Creative Problems Until GeometryPower
However, AlgebraVictory! students are NOT responsible for solving “creative problems.” In an educational context, every math problem requires either a mechanical or creative solution:
- Mechanical Problems require the student to mechanically apply concepts, facts and skills that have already been taught.
- Creative Problems require, in addition to mechanical skills, that the student create a new way to solve the problem—a way that has not yet been taught.
Creative Problems are harder (and can be vastly harder) than Mechanical Problems. Mark’s strong position is that Creative Problems should be deferred until later in the math curriculum, beginning with the GeometryPower course that will follow AlgebraVictory!.
He believes that the easiest way to learn Creative Problem-solving skills is by doing geometry proofs (Mark has 100% success teaching all geometry students to do proofs and to fully understand them). The left brain does linear thinking, while the right brain does non-linear thinking. Integrating the left and right brain is crucial for Creative Problem solving, and in geometry proofs the process of examining the diagram (right brain) and writing the steps of the proof (left brain) presents a magnificent learning space for students to become mathematically creative. In contrast, algebra is almost entirely done by linear, left brain thinking, and although we do graph equations in two variables in first-year algebra they are almost all straight lines, and hence are still linear.
Mark believes it is enough for first-year algebra students to master solving a complex diversity of Mechanical Problems. In doing so, they develop confidence and self-esteem: “Wow! Look at that Final Exam! I understand it, and I can do it!” There is no nagging feeling of what might happen if there is a problem on the test that they have not been taught how to do. Everything falls within the Master Flow Chart of the entire AlgebraVictory! course—in the palm of their right hand.
And that is another reason for excluding Creative Problems from AlgebraVictory!—if they were included, then it would not be possible for students to hold the whole algebra in their right hand! To achieve this important holistic goal, it is necessary to place strict (yet broad) curricular limits on what problems students must face.
Mark will teach AlgebraVictory! students to hold the complex stuff they learn in the metaphoric palm of their right hand, and they are responsible for doing Mechanical Problems based on this stuff. (The student’s metaphoric left hand is discussed in the next section.)
In today’s first-year algebra classrooms, only the most intelligent math students can ever solve math problems creatively. Mark teaches first-year algebra to all students who can pass the Prerequisite Quiz, including those of below-average intelligence. Mechanical Problems are more than sufficient.
Algebra is Magical: Touch the Mystery with Your Left Hand.
Throughout the AlgebraVictory! course Mark will explain many amazing, seemingly magical concepts and facts as students just sit back and listen. He does this by presenting “Mystery Questions” of which there are a total of 17 during the Course. Mark wants students to wonder about these things, be amazed by them, and touch them with their left hand.
For example, Mystery Question #3 asks, What happens when you divide by zero? Well… you get infinity! And what, precisely, is infinity? And what’s wrong with infinity? Well… infinity destroys arithmetic! Why? Well… because if we allow division by zero, then we can easily prove that 5=7! Or that 8=1000. Or that any number x, equals any other number y. That is NOT okay. We must therefore prohibit division by zero. This is amazing stuff that students LOVE to learn about. (In striking contrast, in algebra classes nationwide, students are simply taught that division by zero is prohibited without ever knowing or touching the incredible mystery that makes that rule necessary.)
In Lesson 6, Mark will present Mystery Question #5: Are there any numbers that are not rational? However, he will not answer it until near the end of the course in Lesson 65, when he teaches teach the fascinating, mind-boggling, and indeed magical distinction between (a) the rational numbers that can be written as fractions, and (b) the irrational numbers that cannot. You might be wondering: Can’t ALL numbers be written as fractions? Mark will prove in a way students can understand that the answer is NO, and hence the irrational numbers do indeed exist. Students will be (and should be) astounded by this! The profound relationship between the rational numbers and the irrational numbers is a transcendent reality of the universe that touches on the nature of the human mind and our very existence. This is stuff that all educated and spiritually awake people should know, yet only a tiny minority of the population has a clue about it!
Mark also wants students to grasp that many mathematical problems have never been solved. Some have consumed centuries of effort by top mathematicians. That is why, with Mystery Question #9 he will teach students about Goldbach’s Conjecture from the year 1742, which states that every even natural number greater than 2 equals the sum of two primes (e.g., 18=5+13; 100=3+97). It has neither been proven nor disproven, despite massive efforts spanning more than 275 years! That is amazing, and Mark wants his students to experience that amazement.
Feeling the wonder causes us to transcend from the surface chatter of the mind into our depth as Beings, where our hearts open and we can experience our Oneness with the universe, or with God. Indeed, God is in the mystery and magic!
Middle 80% Students AND Math-Smart 10% Students Will LOVE AlgebraVictory!
AlgebraVictory! provides a learning trail up Algebra Mountain that enables the Middle 80% to master the subject. But the Math-Smart 10% also love the learning trail. Indeed, they learn the algebra better than when they are forced to climb up Algebra Mountain.
Mark inspires students in the Middle 80% and students in the Math-Smart 10% to go further in their math education. AlgebraVictory! Trail gives them the solid foundation they need for success in more advanced mathematical studies.
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