2.2 The Four Operations & Order of Operations

ICAS papers D to J include number sentences with all four operations, order-of-operations questions, and multi-step money problems — all without a personal calculator. The students who score Distinctions are not faster at arithmetic; they use smarter strategies and always estimate before they compute. This section gives you both.

Addition and Subtraction Strategies

Counting On and Bridging

For mental work, count on from the smaller number. To find 5 002 − 1 487, count up: 1 487 + 13 = 1 500, + 500 = 2 000, + 3 000 = 5 000, + 2 = 5 002. Total counted on: 13 + 500 + 3 000 + 2 = 3 515. Bridging through round numbers like 1 500 and 2 000 keeps the steps easy.

Partitioning

Split numbers by place value and recombine: 356 + 274 = (300 + 200) + (50 + 70) + (6 + 4) = 500 + 120 + 10 = 630. Partitioning is exactly what the written column method records, so understanding it prevents "carrying" errors.

Rearranging and Compensation

Add numbers in the most convenient order: 47 + 156 + 53 = (47 + 53) + 156 = 100 + 156 = 256. For compensation, adjust then fix: 456 − 198 = 456 − 200 + 2 = 258 (subtracting 198 is the same as subtracting 200 and giving back 2).

The Inverse Check

Addition and subtraction are inverse operations — each undoes the other. After any subtraction, add the answer back to the number you subtracted: 3 515 + 1 487 = 5 002 ✓. ICAS tests this directly with questions like "◻ − 236 = 418"; solve by adding 418 + 236 = 654.

Multiplication: Facts to Multi-Digit

Know the times tables to 12 × 12 cold — they feed every later topic. For larger numbers, use the area model (partitioning): 47 × 6 = (40 × 6) + (7 × 6) = 240 + 42 = 282.

For two-digit by two-digit, partition both: 34 × 26 = (30 × 26) + (4 × 26) = 780 + 104 = 884. Multiplying by 10, 100 or 1 000 shifts every digit left and appends zeros: 46 × 100 = 4 600. Multiplying by 5? Multiply by 10 and halve: 38 × 5 = 380 ÷ 2 = 190. Multiplying by 9? Multiply by 10 and subtract one lot: 27 × 9 = 270 − 27 = 243.

Division: Short, Long and Remainders

Division is the inverse of multiplication, so your tables do the heavy lifting. Short division handles divisors up to 12:

• 936 ÷ 4: 9 ÷ 4 = 2 remainder 1 → 13 ÷ 4 = 3 remainder 1 → 16 ÷ 4 = 4. Answer 234.
• Check: 234 × 4 = 936 ✓.

Long division is the same thinking with a two-digit divisor, written out in full. 1 750 ÷ 25: 25 × 70 = 1 750, so the answer is exactly 70 — estimating with friendly multiples (25 × 4 = 100) keeps long division short.

Interpreting Remainders in Word Problems

The remainder is not decoration — the context decides what to do with it:

ICAS sets all three styles. Read the final sentence of the problem and ask: does the leftover need another container, get discarded, or get shared as a fraction?

Order of Operations: BIDMAS

When a number sentence mixes operations, everyone must calculate in the same order — otherwise 36 + 4 × 5 could be 56 or 200. The agreed order is BIDMAS (also called BODMAS):

1. Brackets
2. Indices (powers and roots — "O" stands for Orders)
3. Division and Multiplication — equal rank, work left to right
4. Addition and Subtraction — equal rank, work left to right

Worked example 1: 36 + 4 × 5 − 2

• Multiply first: 4 × 5 = 20.
• Left to right: 36 + 20 − 2 = 54.
• The trap answer 198 comes from doing 36 + 4 first — multiplication outranks addition.

Worked example 2: (18 − 6) ÷ 3 + 2 × 7

• Brackets: 18 − 6 = 12.
• Division and multiplication: 12 ÷ 3 = 4 and 2 × 7 = 14.
• Addition: 4 + 14 = 18.

Worked example 3 (left-to-right trap): 24 ÷ 4 × 2. Division and multiplication share a rank, so go left to right: 24 ÷ 4 = 6, then 6 × 2 = 12 — not 24 ÷ 8 = 3.

Multi-Step Money Problems

ICAS embeds BIDMAS in shopping problems. Three notebooks cost $4.50 each and a pen costs $2.75. How much change from a $20 note?

• Cost of notebooks: 3 × $4.50 = $13.50.
• Total: $13.50 + $2.75 = $16.25.
• Change: $20.00 − $16.25 = $3.75 (count on: $16.25 → $17 → $20 is 75c + $3).

Write each step down; ICAS options include the results of common half-finished calculations like $16.25.

Estimate Before You Calculate

Estimation is your built-in error detector and a genuine ICAS skill in its own right.

• 49 × 21 ≈ 50 × 20 = 1 000. The exact answer 1 029 sits close by; an answer of 129 or 10 290 signals a place-value slip.
• 5 002 − 1 487 ≈ 5 000 − 1 500 = 3 500, matching our exact answer 3 515.

Round each number to one significant figure, compute mentally, and reject any option far from your estimate before you spend time on exact working.