Ray's New Higher Arithmetic · Arts. 22–39 · Unit 1: Introduction
3. The signs of arithmetic
Goal: Your student reads and uses the signs +, −, ×, ÷, √, =, :, ::, parentheses and ∴, and knows the order of the fundamental operations.
You'll need
- The book
- A slate, small whiteboard or scrap paper
- A notebook and pencil
What's in the book
Integer, fraction and mixed number; the signs of addition, subtraction, multiplication, division, root, equality, ratio and proportion, with short notes on who first used some of them; signs of aggregation; the primary proposition; and the fundamental operations. A topical outline of the Introduction follows.
The lesson, step by step
Warm-up
Ask your student to write, with signs: seven plus five equals twelve; twenty divided by four equals five.
Kinds of number
Read:
An Integral number, or Integer, is a whole number
The book also names fractional numbers and mixed numbers (a whole number and a fraction together). Ask for one example of each.
The signs
Read:
A Sign is a character used to show a relation among numbers, or that an operation is to be performed.
The sign of Addition is [+], and is called plus.
Plus means more; minus means less.
Go through ×, ÷ and the radical sign on the page. The book notes that the radical's small index figure tells which root is wanted: √36 = 6, and the cube root of 125 is 5.
Equality, ratio and proportion
Read:
Ratio is the relation which one number bears to another of the same kind.
Proportion is an equality of ratios.
The ratio sign is the colon (a division sign with its line left out). Look at the book's example 3 : 6 :: 4 : 8 and ask: why are those equal? (3 is half of 6, and 4 is half of 8.)
Grouping signs
Parentheses and the vinculum (a bar over numbers) make several numbers count as one. Work 24 − (8 + 7) = 9 together, then 24 − 8 + 7 = 23, and talk about why they differ.
The primary proposition
Read:
Arithmetic depends upon this primary proposition: that any number may be increased or diminished.
The fundamental operations follow in this order:
Numeration and Notation, Addition, Subtraction, Multiplication, and Division.
Wrap-up
Look over the book's topical outline of the Introduction together. Your student points to each heading and says one sentence about it.
Tip
Ray's historical notes (Steifel, Oughtred, Pell) are fun but optional. Don't quiz on dates.