Secondary School Mathematics

Goals

1. Facility in basic problem solving
   1. Facility in solving problems using multiple modes
   2. Ability to focus attention on pertinent aspects of a problem situation
   3. Facility in dividing complex problems into smaller problems
   4. Facility in solving a problem in stages (as by using lemmas)
2. Facility in using real numbers
3. Working ability in logic
   1. Basic ability to recognize unexpected and counter-intuitive relationships
   2. Ability to use the concepts of proof and disproof
   3. Facility in using basic inductive logic
   4. Facility in using basic deductive logic
4. Awareness of axiomatic systems as virtual realities
5. Awareness of synergy in combining problem-solving methods

Previous: intermediate. The skills described here would be followed by vocational or college preparatory studies.

Objectives

• Concepts of inductive logic in the context of natural numbers
  • Apply inductive logic to a real or imagined situation
  • Explain how mathematical induction differs from less formal induction
  • Prove a simple formula about natural numbers using mathematical induction
• Concepts of deductive logic in the context of rational numbers
  • Explain how algebraic axioms define a virtual world
  • Express a numerical relationship as a formal conditional statement
  • Demonstrate that a conditional statement does not imply its converse [⇒ is not a symmetric relationship in algebra]
  • Express a correct syllogism for a simple algebraic proposition
  • Chain 3 or more syllogistic conclusions to prove an algebraic proposition
  • Express a deductive proof of a algebraic proposition formally
• Concepts of real numbers
  • Find factors of an integer
  • Perform complete factorization of an integer
  • Define prime numbers
  • Express division using negative integer exponents
  • ☆ Explain negative integer exponents in terms of consistency of computation
  • Express roots using non-integer exponents
  • ☆ Explain non-integer exponents in terms of consistency of computation
  • Define the difference between rational and irrational numbers
  • Use irrational numbers [such as √2] to solve an algebraic problem
  • Define the difference between algebraic and transcendental numbers
  • Use transcendental numbers [such as π] to solve a mathematical problem
  • Explain what mathematicians mean by saying "almost all" numbers are irrational
  • ☆ Explain how irrational numbers can be understood as simplifying or regularizing mathematics
• Basic concepts of geometrical thinking
  • Express an analogical understanding of a mathematical point
  • Express an analogical understanding of a mathematical line
  • Express a conceptual understanding of parts of a line
  • Express a conceptual understanding of distance
  • Express a conceptual understanding of angle
  • Express the difference between drawn figures and the mathematical figures represented
  • Demonstrate the idea of invariant, abstract length (the "sameness" of length in different places)
  • Demonstrate the idea of invariant, abstract angle (the "sameness" of angle measure in different places)
  • Express a conceptual understanding of congruence
  • Express a conceptual understanding of similarity
• Solving problems using geometrical thinking
  • Demonstrate the properties of equality geometrically
  • Express arithmetic operations geometrically
  • Demonstrate basic formulas for area geometrically
  • Demonstrate division of a line segment into an arbitrary number of equal parts
  • Demonstrate equality of area for rectangles with different shapes
  • ☆ Demonstrate the solution of proportionality problems geometrically
• Concepts of deductive logic in the context of planar figures
  • Explain how geometric axioms define a virtual world
  • Express a relationship between geometric figures as a formal conditional statement
  • Demonstrate that a conditional statement does not imply its converse [⇒ is not a symmetric relationship in geometry]
  • Express a correct syllogism for a simple geometric proposition
  • Chain 3 or more syllogistic conclusions to prove a geometric proposition
  • Express a deductive proof of a geometric proposition formally
• Concepts of constructive logic
  • Explain how postulates describe the constructions we are able (or allowed) to make
  • Express a real, physical relationship with a geometrical construction
  • Create a correct construction to demonstrate a simple proposition
  • Chain 3 or more constructions to prove a proposition
  • Express a constructive proof of a geometric proposition formally
• Simple analytics
  • Apply a coordinate system to planar figures
  • Define origin, ordinate, and abscissa in the coordinate plane
  • Solve geometrical problems using algebraic techniques
  • Solve problems by combining geometric and algebraic concepts
  • Describe the concept of a mathematical relation
  • Discriminate between functions and other relations
  • Graph simple relations in the coordinate plane
  • Rewrite equations into standard [canonical] forms
  • Classify the differences which result from varying the constants of a relation
• Concepts of disproof
  • Identify logical errors in a syllogism
  • Disprove a proposition with a counterexample
  • Disprove a proposition by proving a contrary proposition
  • Disprove a proposition by proving a contradiction
  • Identify logical errors in a deductive proof
  • Identify logical errors in a geometric construction

May, 2014