Numerical expressions: from a situation to its value — 11–13
Write a situation as an expression, then calculate it with parentheses, powers and the order of operations.
From a situation to an expression
A numerical expression writes a situation using only numbers and operation signs. “4 packs of 10 stickers and 6 loose ones” is 4 × 10 + 6, and its value is 46. You can also go the other way: 3 × (5 + 2) could count 3 boxes, each with 5 blue and 2 red pencils. The parentheses show what is calculated together: here, the pencils in one box.
One rule, one value
If everyone calculated in a different order, the same line would give different results: 3 + 4 × 5 would be 35 or 23. So everyone agreed on one rule: first what is inside parentheses, then powers, then × and ÷, and finally + and −. Within the same level you go from left to right. This way any person, calculator or computer reaches the same value.
Tiles, a power and parentheses
Situation: you buy 3 boxes of tiles, each with 4 rows of 4; 6 tiles break and the rest is shared between 2 equal jobs. How many tiles per job? Expression: (3 × 4² − 6) ÷ 2. Step 1, the power: 4² = 16. Step 2: 3 × 16 = 48. Step 3: 48 − 6 = 42. Step 4: 42 ÷ 2 = 21. Without the parentheses, 3 × 4² − 6 ÷ 2 = 48 − 3 = 45, which answers a different question.
Reading left to right
The reasonable mistake is to calculate in the order things are written, like reading a text: 2 + 3 × 4 → 5 × 4 = 20. But multiplication comes first: 3 × 4 = 12 and 2 + 12 = 14. Only when two operations have the same priority (× and ÷, or + and −) do you go from left to right: 24 ÷ 4 × 3 = 6 × 3 = 18, not 24 ÷ 12 = 2. If you want another order, use parentheses: (2 + 3) × 4 = 20.
Calculators, spreadsheets, code
Scientific calculators, spreadsheets and programming languages follow this same priority: typing 2+3*4 gives 14. Some simple calculators work in the order of the keys and give 20, so test yours with 2 + 3 × 4. In a spreadsheet, the formula =(3*4^2-6)/2 uses parentheses to say what is calculated first, and gives 21.
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