Ray's New Higher Arithmetic · Arts. 82–86 · Unit 6: Division
20. Arithmetical signs and the order of operations
Goal: Your student evaluates a string of numbers joined by +, −, × and ÷, doing multiplications and divisions before additions and subtractions.
You'll need
- The book
- A slate, small whiteboard or scrap paper
- A notebook and pencil
What's in the book
A short section on signs: positive and negative numbers, how + and − mark off the parts of an expression, how × and ÷ act in order, and examples to work. Example 1 has its answer (14); the later examples are hard to read in the scan.
The lesson, step by step
Warm-up
Ask: What is 5 + 3 × 2? Let your student argue for 16 or 11.
Positive and negative
Read:
Positive numbers are distinguished by the sign
plus, and negative numbers by minus. A number with no sign is positive.
What + and − control
The book says a + or − governs the whole result of the operations that follow it, up to the next + or −. So in 5 + 7 × 2 × 9 − 2 × 6, the + adds 7 × 2 × 9 = 126 and the − takes away 2 × 6 = 12. Result: 5 + 126 − 12 = 119.
× and ÷ in order
Read:
are to have their particular effects in the exact order of their occurrence.
So 96 ÷ 12 × 4 = 8 × 4 = 32, but 96 ÷ (12 × 4) = 2.
Guided practice
Work the book's Example 1:
4X3 + 7X2 — 9X3+6X4 — 3X3
12 + 14 − 27 + 24 − 9 = 14.
Independent practice
The later examples are blurred in the scan. Your student works whatever is readable, then makes up three strings of their own for you to solve.
Wrap-up
Ask: In 20 − 6 ÷ 2 × 3, what do you do first? (6 ÷ 2 = 3, then × 3 = 9, then 20 − 9 = 11.)