Ray's New Higher Arithmetic · Arts. 9–21 · Unit 1: Introduction
2. Propositions, rules and numbers
Goal: Your student can tell an axiom from a theorem and a problem, and knows what a rule, a formula, a unit and a number are.
You'll need
- The book
- A slate, small whiteboard or scrap paper
- A notebook and pencil
What's in the book
Definitions of proposition, axiom, theorem, problem, demonstration, solution, operation, rule and formula, then unit and number, abstract and concrete.
The lesson, step by step
Warm-up
Review last time: What is quantity? What is the difference between a science and an art?
Kinds of statements
Read:
A Proposition is the statement of a principle, or of something proposed to be done.
An Axiom is a self-evident truth.
A Theorem is a truth to be proved.
A Problem is a question proposed for solution.
Give an example of each: "The whole is greater than its part" (axiom); "The sum is the same in whatever order you add" (theorem, needs a reason); "What do 7 pencils cost at 3 cents each?" (problem).
Rules and formulas
Read:
A Rule is a general direction for solving all problems of a particular kind.
A Formula is the expression of a general rule or principle in algebraic language
Show a tiny formula: cost = price × number. Ask your student to say the same thing as a rule in words.
Units and numbers
Read:
A Unit is one thing, or one.
Number is the expression of a definite quantity.
The book divides numbers into abstract (just 5) and concrete (5 apples). Concrete numbers are also called denominate numbers; the word comes back in the chapter on measures.
Practice
Your student labels each as abstract or concrete: 12; 12 miles; 3 dozen eggs; 7; $40; 365 days. Then writes one axiom, one theorem and one problem of their own.
Wrap-up
Ask: Is "Find 15% of 80" an axiom, a theorem or a problem? (A problem.)