The four operations, fractions and decimals

What this heading covers

What you do with numbers once you have them. Four operations, the two families of number that give children the most trouble — fractions and decimals — and the convention that decides the order of a calculation.

Place value, the families of numbers, integers, factors and multiples have a lesson of their own.

The thread through this heading, and the reason the pedagogy paper cares about it: most errors here are whole-number rules carried into places they do not hold. Adding denominators, assuming multiplication makes things bigger, judging a decimal by its length — each is a rule that worked before and does not work now.

The four operations

Operation	Words a question uses for it	Its inverse, and its identity
Addition	sum, total, altogether	Inverse: subtraction. Identity: 0
Subtraction	difference, how many more, how many left	Inverse: addition. No identity
Multiplication	product, times, repeated addition	Inverse: division. Identity: 1
Division	quotient, shared equally, how many groups	Inverse: multiplication. No identity

Which properties hold for which

This is asked directly, and the pattern is worth memorising as a block:

Operation	Commutative?	Associative?
Addition	Yes	Yes
Subtraction	No	No
Multiplication	Yes	Yes
Division	No	No

Commutative means the order may change without changing the result; associative means the grouping may. Addition and multiplication allow both; subtraction and division allow neither.

Distribution runs one way only: multiplication distributes over addition and subtraction, so a × (b + c) = ab + ac. Addition does not distribute over multiplication.

Two consequences questions use: 8 − 3 ≠ 3 − 8, and 12 ÷ 4 ≠ 4 ÷ 12. If an option claims subtraction or division is commutative, it is wrong.

The two meanings of division

Worth separating, because the pedagogy paper asks and the content paper assumes it:

• Equal sharing — 12 sweets shared among 4 children; how many each?
• Equal grouping (measurement) — 12 sweets in packets of 4; how many packets?

Both are 12 ÷ 4, but they are different questions, and a child who has only met one does not recognise the other.

The division algorithm

A named heading on the primary track, and simply the statement that ties the four parts of a division together:

Dividend = (Divisor × Quotient) + Remainder,     with 0 ≤ Remainder < Divisor

The condition matters as much as the formula: the remainder is always less than the divisor. A remainder equal to or larger than the divisor means the quotient was taken too small.

Worked: divide 47 by 5. Quotient 9, remainder 2. Check: (5 × 9) + 2 = 47, and 2 < 5.

Used backwards, which is the commoner question: a number divided by 7 gives quotient 12 and remainder 5. The number is (7 × 12) + 5 = 89.

Two standard item types:

• The least number to be subtracted from 1000 so that it is exactly divisible by 7. Divide: 1000 ÷ 7 leaves remainder 6, so subtract 6.
• The least number to be added. Divisor minus remainder: 7 − 6 = 1.

Operations on integers

The sign rules, which whole-number practice does not supply.

Addition and subtraction

• Same signs — add the values, keep the sign: (−4) + (−3) = −7.
• Different signs — subtract the smaller value from the larger, take the sign of the larger: (−9) + 4 = −5.
• Subtracting is adding the opposite: 5 − (−3) = 5 + 3 = 8. Two minuses meeting make a plus, and this is where most errors are.

Multiplication and division

	Result
positive × positive	positive
negative × negative	positive
positive × negative	negative

The same signs rule applies to division. A quick check: an even number of negative factors gives a positive product; an odd number gives a negative one.

Fractions

A proper fraction is less than 1, an improper fraction is 1 or more, and a mixed number pairs a whole with a proper fraction. Like fractions share a denominator; unlike fractions do not; equivalent fractions have the same value in different terms; a unit fraction has numerator 1.

Converting: an improper fraction to a mixed number is a division — 17/5 = 3 remainder 2 = 3⅖. Back again: (whole × denominator) + numerator over the denominator.

Comparing

• Bring to a common denominator, or cross-multiply.
• With the same denominator, the larger numerator is the larger fraction.
• With the same numerator, the larger denominator makes the smaller fraction — ⅓ is less than ½. This is the single most common misconception in primary mathematics, because it reverses the whole-number rule.

The four operations on fractions

• Adding and subtracting: a common denominator first, then operate on the numerators only. Adding numerators and denominators together is the classic error, and it is a whole-number rule imported.
• Multiplying: numerators together, denominators together. Simplify before multiplying where you can.
• Dividing: multiply by the reciprocal — ⅔ ÷ ⅘ = ⅔ × 5/4 = 10/12 = ⅚.

Two results that defeat whole-number intuition, and the paper uses both:

• Multiplying by a proper fraction makes a number smaller. 12 × ½ = 6.
• Dividing by a proper fraction makes it larger. 12 ÷ ½ = 24 — because the question is "how many halves are in 12?"

Decimals

Decimals are fractions with denominators of 10, 100, 1000, written in place-value form. The places after the point are tenths, hundredths, thousandths.

• Comparing: compare digit by digit from the left, not by length. 0.5 > 0.25, though 0.25 has more digits. Adding zeros on the right changes nothing: 0.5 = 0.50 = 0.500.
• Adding and subtracting: line up the decimal points, not the right-hand ends.
• Multiplying: multiply as whole numbers, then place the point so the answer has as many decimal places as both factors together. 0.3 × 0.02 has three decimal places: 0.006.
• Dividing by a decimal: shift the point in both numbers until the divisor is whole — 4.5 ÷ 0.5 becomes 45 ÷ 5 = 9.

Converting

• Fraction to decimal: divide the numerator by the denominator.
• Decimal to fraction: write over 10, 100 or 1000 by the number of decimal places, then simplify — 0.75 = 75/100 = ¾.
• A fraction gives a terminating decimal when, in lowest terms, its denominator has only 2s and 5s as prime factors. Otherwise it is a recurring (repeating) decimal: ⅓ = 0.333…, written with a bar over the repeating part. So ⅜ terminates because 8 = 2³, while ⅙ recurs because 6 = 2 × 3.

Percentage is a fraction out of 100 — to convert, multiply by 100 for a percentage and divide by 100 to go back. Ratio compares two quantities of the same kind, has no units, and is written in lowest terms.

The order of operations

BODMAS — Brackets, Of, Division, Multiplication, Addition, Subtraction.

Division and multiplication rank equally and are worked left to right; so do addition and subtraction. That detail decides questions:

• 12 ÷ 3 × 2 is 8, not 2 — the division comes first because it is further left.
• 10 − 4 + 3 is 9, not 3.
• Brackets are resolved innermost first; "of" means multiplication, so ½ of 8 is 4.

Questions of the kind the paper sets

A number divided by 7 gives quotient 12 and remainder 5. What is the number? (a) 84 (b) 89 (c) 96 (d) 77

(7 × 12) + 5.

What is the least number that must be added to 1000 to make it exactly divisible by 7? (a) 6 (b) 1 (c) 7 (d) 3

1000 ÷ 7 leaves 6, so 7 − 6 = 1. Option (a) is the answer to subtracted.

5 − (−3) equals — (a) 2 (b) −8 (c) 8 (d) −2

Subtracting is adding the opposite.

(−4) × (−5) × (−2) equals — (a) 40 (b) −40 (c) −9 (d) 11

Three negative factors — an odd number — so the product is negative.

Which is greater, ⅖ or ⅗? (a) ⅖ (b) ⅗ (c) equal (d) cannot be compared

Same denominator, so the larger numerator wins.

Which is greater, ⅓ or ¼? (a) ¼ (b) ⅓ (c) equal (d) cannot be compared

Same numerator, so the smaller denominator is the larger fraction — the reverse of the whole-number rule.

12 ÷ ½ equals — (a) 6 (b) 24 (c) 12 (d) ¼

How many halves are there in 12?

Which is greatest: 0.7, 0.25, 0.108? (a) 0.108 (b) 0.25 (c) 0.7 (d) they are equal

Compare from the left; length is irrelevant.

12 ÷ 3 × 2 equals — (a) 2 (b) 8 (c) 12 (d) 6

Equal rank, so left to right.

Which property is shown by 7 × (4 + 3) = 7 × 4 + 7 × 3? (a) Commutative (b) Associative (c) Distributive (d) Closure

Mistakes that cost marks here

• Saying subtraction or division is commutative or associative.
• Forgetting the condition on the division algorithm — the remainder is always less than the divisor.
• Answering least number to be added with the remainder instead of divisor minus remainder.
• Getting two minuses wrong: 5 − (−3) is 8.
• Counting the sign of a product by guesswork rather than by whether the count of negatives is odd or even.
• Adding numerators and denominators when adding fractions.
• Thinking the larger denominator makes the larger fraction.
• Assuming multiplication always makes bigger and division always smaller. With proper fractions both reverse.
• Comparing decimals by how many digits they have.
• Lining up decimals by their right-hand ends rather than the point.
• Working BODMAS strictly by the letters instead of left to right within each rank.

In one line

The operations are easy and their properties are the examinable part; the division algorithm is one identity with one condition attached; and almost every fraction and decimal error in the paper is a whole-number habit applied where it no longer holds — so the question to ask of any answer here is whether it would still be true if the numbers were whole.

For the classroom

Fractions go wrong because they are taught as symbols before they are met as quantities. Paper strips folded into halves, thirds and quarters settle ⅓ against ¼ in a way no rule does, and a child who has laid the strips side by side does not later write that ¼ is bigger.

The same holds for the two meanings of division. Share twelve counters among four children, then make packets of four from twelve counters, and ask what was different — both are 12 ÷ 4, and a child who has done only the first will not recognise the second in a word problem.