L6 · Order of Operations with Rational Numbers; Estimation

Learning goal

“I can evaluate expressions involving rational numbers, brackets and powers in the conventional order, and I can estimate a result first so I can judge whether an exact answer is reasonable.”

Order of operations — the convention: brackets first, then powers, then multiplication and division (left to right), then addition and subtraction (left to right). It exists so an expression has ONE value, agreed by everyone.

Power — a number multiplied by itself: (2/3)^2 = 2/3 x 2/3 = 4/9. Powers apply to the number (or bracket) they sit on - including a negative sign inside the bracket.

Estimate — a deliberate approximation made BEFORE or INSTEAD of exact work, to predict size or to judge an answer.

Benchmark — a nearby easy value used for estimating: 7/8 is benchmarked to 1, -19.6 to -20.

Worked example

-1/2 + 3/4 x 2. Step 1: Scan for brackets and powers - none. Multiplication/division before addition: 3/4 x 2 = 3/2. Step 2: Now the addition: -1/2 + 3/2 = 1. So -1/2 + 3/4 x 2 = 1. (Adding first would give (-1/2 + 3/4) x 2 = 1/2 - a different value; the convention exists precisely to forbid that ambiguity.) Second example, with a power: (1/2)^2 + 3/4 = 1/4 + 3/4 = 1 - the power is evaluated before the addition, and the power applies to the bracketed 1/2 as a whole.

Check yourself — explain it

Why do we multiply before adding in -1/2 + 3/4 x 2?

Paired problem — example still visible

(2/5)^2 + 1/10 -> power first: 4/25 + 1/10 = 8/50 + 5/50 = 13/50

Faded completion — last step is yours

"4 - 3/2 / 1/2: division before subtraction - 3/2 / 1/2 = 3, so 4 - 3 = ___." -> 1

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10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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