Lesson 2: Fractions — The Singapore Way

Fractions defeat more students than any other primary-school topic — yet the Singapore approach makes them intuitive. The secret is CPA: fold paper and cut fraction discs (concrete), draw bar models and pie diagrams (pictorial), and only then compute with numerators and denominators (abstract). This lesson builds fractions from the ground up, the Singapore way, using examples from hawker centres, pizza nights, and MRT journeys.

What Is a Fraction? — Concrete Beginnings

A fraction represents equal parts of a whole: 3/4 means 3 parts out of 4 equal parts. The bottom number (denominator) names the size of the parts — fourths; the top number (numerator) counts how many — three. Singaporean children first meet fractions by folding paper strips: fold a strip into 4 equal parts, shade 3, and you have built 3/4 with your hands before ever writing the symbol.

The "equal parts" condition is crucial and commonly violated. A "half" of a cake cut into unequal pieces is not 1/2 — it is just a piece. Singaporean teachers stress this with deliberate non-examples: showing unequal divisions and asking "is this a half?" trains the eye to demand equality. This concrete discipline prevents years of later confusion.

Fractions appear constantly in Singapore life: half a dozen eggs, a quarter tank of petrol, three-quarters of an MRT journey completed, a recipe calling for 1/2 cup of coconut milk. Each is a part of a whole — the dozen, the tank, the journey, the cup. Naming the whole is the first step in every fraction problem.

Equivalent Fractions — Same Value, Different Look

Equivalent fractions name the same quantity differently: 1/2 = 2/4 = 3/6 = 4/8. Pictorially, shading half a bar, two quarters of the same bar, or three sixths all cover the same area — the bar model makes equivalence visible. Abstractly, multiplying or dividing numerator and denominator by the same number preserves value: 3/4 = (3×2)/(4×2) = 6/8.

Finding equivalents is the key that unlocks comparing, adding, and subtracting fractions. To compare 3/4 and 5/6, convert to common denominators: 3/4 = 9/12, 5/6 = 10/12, so 5/6 is larger. The pictorial version — two bars of 12 parts, one with 9 shaded, one with 10 — shows why the abstract rule works.

A hawker example: Ahmad drank 3/4 of his teh tarik; Mei drank 5/6 of hers. Who drank more? Convert: 3/4 = 9/12, 5/6 = 10/12. Mei drank more — by 1/12 of a cup. The common denominator is just a common language for comparing.

Adding and Subtracting Fractions

Addition requires a common denominator — you can only add like parts. 1/4 + 2/4 = 3/4 is easy (same parts); 1/3 + 1/6 needs conversion: 1/3 = 2/6, so 2/6 + 1/6 = 3/6 = 1/2. The pictorial bar model shows why: you cannot combine thirds and sixths until you cut the thirds into sixths. The abstract rule "find the lowest common denominator" is just efficient cutting.

Mixed numbers (like 2 3/4) combine wholes and parts. To add 2 3/4 + 1 1/2: convert 1 1/2 to 1 2/4, then add wholes (2 + 1 = 3) and parts (3/4 + 2/4 = 5/4 = 1 1/4), giving 4 1/4. Alternatively, convert everything to improper fractions: 11/4 + 6/4 = 17/4 = 4 1/4. Both routes work; the bar model shows both.

Subtraction follows the same logic. A cake recipe needs 3/4 cup of sugar; you have 1/2 cup. How much more do you need? 3/4 − 1/2 = 3/4 − 2/4 = 1/4 cup. Always ask: are the parts the same? If not, make them the same first.

Multiplying and Dividing Fractions

Multiplication of fractions is beautifully pictorial: 1/2 × 1/3 means "half of one-third." Draw a bar, shade 1/3, then take half of the shaded part — the result is 1/6 of the whole bar. Abstractly: multiply numerators, multiply denominators: (1×1)/(2×3) = 1/6. "Of" means multiply — a translation worth memorizing.

A hawker-centre problem: a recipe for 4 people uses 3/4 kg of rice flour. For 6 people (1.5 times as much), you need 3/4 × 3/2 = 9/8 = 1 1/8 kg. The abstract computation is quick; the pictorial check — a bit more than the original 3/4 — confirms sense.

Division is the inverse: 3/4 ÷ 1/2 asks "how many halves fit into three-quarters?" Pictorially, 3/4 contains one full 1/2 with 1/4 left over — and that leftover 1/4 is half of a half, so the answer is 1 1/2. Abstractly: invert and multiply: 3/4 × 2/1 = 6/4 = 1 1/2. The "invert and multiply" rule is not magic — it is the shortcut for the pictorial reasoning.

Fractions, Decimals and Percentages Together

Fractions, decimals, and percentages are three languages for the same ideas: 1/2 = 0.5 = 50%, 3/4 = 0.75 = 75%, 1/4 = 0.25 = 25%. Singaporean pupils learn to translate fluently, because each form suits different problems: fractions for exact reasoning, decimals for measurement, percentages for comparisons like discounts and GST.

Conversion is mechanical: fraction to decimal, divide numerator by denominator (3/8 = 0.375); decimal to percentage, multiply by 100 (0.375 = 37.5%); percentage to fraction, write over 100 and simplify (37.5% = 37.5/100 = 3/8). The common benchmarks — halves, quarters, fifths, eighths, tenths — should be known by heart.

A shopping example: a S$80 shirt at 25% off. 25% = 1/4, so the discount is S$80 ÷ 4 = S$20, and the price is S$60. Or with GST: S$60 × 1.09 = S$65.40. Fluency across the three forms turns everyday arithmetic into mental reflex.

Conclusion — Why This Matters

Fractions are the gateway to all higher mathematics: algebra's rational expressions, calculus's limits, and probability's ratios all generalize fraction ideas. Students who truly understand fractions — concretely, pictorially, abstractly — rarely struggle later; students who only memorized rules do.

The Singapore lesson is general: when mathematics feels hard, go back down the CPA ladder. Confused by algebraic fractions? Draw a bar model. Lost in symbols? Fold paper. Understanding is built, not wished for.

Practice until the translations (1/2 = 0.5 = 50%) are instant and the operations are reliable. Fractions reward the diligent: they are finite, rule-governed, and completely masterable.

A Brief History of Fractions

Fractions are ancient. The Egyptians (c. 1800 BCE) used unit fractions — 1/2, 1/3, 1/4 — writing other fractions as sums (2/5 = 1/3 + 1/15, in their notation). The Rhind Papyrus contains tables for doubling unit fractions, the computational backbone of Egyptian engineering. Their restriction to unit fractions made arithmetic cumbersome — a cautionary tale about notation shaping thought.

India gave the world the modern fraction bar. Brahmagupta (7th century CE) and later Bhaskara formalized fraction arithmetic — including the rules for adding, multiplying, and dividing that Singaporean pupils learn today. The horizontal bar separating numerator from denominator is an Indian invention, transmitted through Arab mathematicians to Europe.

The decimal fraction — writing 1/2 as 0.5 — emerged much later, championed by the Flemish engineer Simon Stevin in 1585. Decimals made measurement and commerce vastly easier, which is why Singapore's metric, money-based economy runs on them. The fraction-to-decimal journey took three thousand years; pupils retrace it in three school years.

The moral for learners: today's "easy" notation is humanity's hard-won achievement. When fractions feel difficult, remember that the greatest civilizations struggled with them for millennia — and that the bar model you draw is part of a very long story of making parts of wholes thinkable.

Debate — Should Pupils Drill Fractions or Discover Them?

Traditionalists argue fractions demand drill: the operations (common denominators, invert-and-multiply) are procedures, and procedures become fluent only through repetition. A pupil who hesitates over 1/3 + 1/6 will drown in algebra's rational expressions. Drill builds the automaticity that frees working memory for harder thinking — just as times-table fluency underpins all arithmetic.

Discovery advocates counter that drilled procedures without understanding are brittle: pupils who memorized "invert and multiply" cannot explain why it works and collapse on unfamiliar problems. The Singapore CPA sequence — fold paper, draw bars, then compute — builds procedures on understanding, so the rules are reconstructable even when forgotten. Time spent discovering is repaid in retention.

The synthesis, and Singapore's actual practice, is both: concrete discovery first, then deliberate practice to fluency. Understanding without drill is slow; drill without understanding is fragile. The debate's real lesson is sequencing — discover, then drill — not choosing sides.

Worked Example — Bar Model or Algebra?

Problem: "After spending 3/8 of her money on books, Siti had S$45 left. How much money did she have at first?" Bar model: draw the whole as 8 equal units. She spent 3 units; 5 units remain, and 5 units = S$45. So 1 unit = S$9, and the whole (8 units) = S$72. Algebra: let the amount be x dollars. Then x − (3/8)x = 45, so (5/8)x = 45 and x = 45 × 8/5 = 72. Both routes give S$72 — check: 3/8 of 72 is 27, and 72 − 27 = 45 ✓.

The bar model turns the fraction 5/8 into something you can see and count; the algebra compresses the same reasoning into one line. Learn the bar first — the equation is its abbreviation.

Common Mistakes

• Adding denominators. 1/3 + 1/6 is not 2/9 — parts must be made equal first: 2/6 + 1/6 = 3/6 = 1/2.
• Cancelling wrongly in multiplication. 2/3 × 9/10: cancel 3 with 9 and 2 with 10 *before* multiplying to get 3/5; multiplying first (18/30) works but invites slips.
• Dividing without inverting. 3/4 ÷ 1/2 means "how many halves in three-quarters" — invert the *second* fraction only: 3/4 × 2/1 = 1 1/2.

Exam Experience

Question 1 [2 marks]. Express 42/56 in its simplest form, and write your answer as a decimal.

Question 2 [3 marks]. Calculate 2 3/4 + 1 5/6 − 1 1/3, giving your answer as a mixed number in its simplest form.

Question 3 [6 marks]. Mei has some ribbons. She gives 2/7 of them to her sister. She then gives 1/4 of the remainder to her friend. She has 45 ribbons left. (a) What fraction of the original ribbons did she give to her friend? (b) What fraction of the original ribbons does she have left? (c) How many ribbons did she have at first?

Marking scheme. Q1: B1 for 3/4; B1 for 0.75. Q2: M1 for converting to twelfths (33/12 + 22/12 − 16/12 or equivalent); M1 for correct addition and subtraction; A1 for 3 1/4. Q3: (a) M1 for finding the remainder 5/7 and taking 1/4 of it; A1 for 5/28. (b) M1 for multiplying 5/7 by 3/4 (the part of the remainder she keeps); A1 for 15/28. (c) M1 for the unit method (15 units = 45, so 1 unit = 3, or equivalent); A1 for 84 ribbons.