Chapter 2 of 12

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ICSE Class 4 Mathematics★ ICSE Class 4 Mathematics · Full Year · Board: ICSE

Mathematics — Numbers, Operations, Geometry & Measurement

Numbers up to LAKHS. ADDITION, SUBTRACTION, MULTIPLICATION, and DIVISION with larger numbers. GEOMETRY — lines, angles, 2D and 3D shapes. And MEASUREMENT — length, weight, capacity, and time. ICSE Class 4 Mathematics.

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By the end of this chapter you'll be able to…

1Read, write, compare, and expand numbers up to lakhs (6 digits); identify place value
2Round numbers to the nearest 10, 100, or 1,000 using the 'look right' rule
3Read and write Roman numerals up to 1,000 using addition and subtraction rules
4Perform 5-digit addition and subtraction; multiply 3-digit × 2-digit; divide 4-digit ÷ 1-digit
5Find all factors of a number; list multiples; classify numbers as prime, composite, or neither
6Add and subtract like fractions; compare unlike fractions by converting to a common denominator
7Classify angles as acute, right, or obtuse; identify line segments, rays, and lines
8Find lines of symmetry for common shapes; read and draw bar graphs; extend number and shape patterns

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Why this chapter matters

Class 4 Mathematics is the bridge between basic arithmetic and algebraic thinking. Prime vs composite numbers is the foundation of HCF/LCM (Class 5–6) and all factorisation. Roman numerals appear in history, clocks, and exam questions throughout school. The introduction of unlike fractions (needing LCM) prepares students for Class 6 rational numbers. Symmetry and angle classification (acute/right/obtuse) appear constantly in geometry through Class 10. Getting these foundational facts right now is a multiplier for future learning.

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A 5-minute refresher here will save you 30 minutes of confusion below.

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ICSE Class 3 Mathematics

4-digit numbers, four operations, basic shapes, measurement conversions

Numbers, Operations, Geometry & Measurement

1. Numbers Up to Lakhs

Place Value

Ones → Tens → Hundreds → Thousands → Ten Thousands → LAKHS (1,00,000).

Example: 3,45,678 = Three lakh forty-five thousand six hundred seventy-eight.

Comparing Numbers

More digits = LARGER. Same number of digits → compare from the LEFT.

Rounding

Round to nearest 10, 100, 1000. 'Look at the digit to the RIGHT. If 5 or more → ROUND UP.'

Roman Numerals

I(1), V(5), X(10), L(50), C(100), D(500), M(1000).

2. The Four Operations

Addition and Subtraction

Up to 5-digit numbers. Align correctly by place value. Carry/borrow.

Multiplication

3-digit × 2-digit. 4-digit × 1-digit.
'Multiply each digit. Add partial products.'

Division

4-digit ÷ 1-digit. 3-digit ÷ 1-digit.
Dividend = Divisor × Quotient + Remainder. Remainder < Divisor.

3. Factors and Multiples

Factors

Numbers that divide EXACTLY. Factors of 12: 1, 2, 3, 4, 6, 12.

Multiples

Skip counting: Multiples of 3 = 3, 6, 9, 12, 15...

Prime and Composite

Prime: EXACTLY two factors (1 and itself). 2, 3, 5, 7, 11, 13...
Composite: MORE than two factors. 4, 6, 8, 9, 10...

4. Fractions

What Is a Fraction?

A PART of a WHOLE. ½ (half) . ¼ (quarter) . ¾ (three-quarters) .

Types

Like fractions: Same denominator. Compare numerators.
Unlike fractions: Different denominators. Convert to like fractions first.

Add/Subtract Like Fractions: Add numerators. Keep denominator.

5. Geometry

Lines

Line segment: Has TWO endpoints. Measure with a RULER.
Ray: Has ONE endpoint. Extends INFINITELY in one direction.
Line: Extends INFINITELY in BOTH directions.

Angles

Acute (<90°). Right (=90°). Obtuse (>90°).

2D Shapes

Shape	Sides	Special Properties
Triangle	3	Sum of angles = 180°
Square	4 equal	All angles 90°
Rectangle	4	Opposite sides equal. All angles 90°
Circle	0 (curved)	Radius. Diameter = 2 × r

3D Shapes

Cube (6 square faces). Cuboid (6 rectangular faces). Sphere. Cylinder.

6. Measurement

Length: 1 km = 1,000 m. 1 m = 100 cm. 1 cm = 10 mm.

Weight: 1 kg = 1,000 g.

Capacity: 1 L = 1,000 mL.

Time

1 day = 24 hours. 1 hour = 60 min. 1 min = 60 s. Reading clocks: 'Quarter past' (15 min past). 'Half past' (30 min). 'Quarter to' (15 min before).

7. Data and Patterns

Bar Graphs

Bars of different heights. READ the scale. TALLER bar = MORE.

Number Patterns

Arithmetic: 3, 6, 9, 12, ___, ___ (15, 18 — add 3)
Geometric: 2, 4, 8, 16, ___, ___ (32, 64 — multiply by 2)

Symmetry

A shape has line symmetry if folding along a line = two EXACT halves. Square: 4. Equilateral triangle: 3. Circle: INFINITE.

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Place Value (Lakhs)

Lakhs | Ten-thousands | Thousands | Hundreds | Tens | Ones. E.g. 3,45,678 = 3L + 45T + 6H + 7T + 8O

Indian system groups as: ones/tens/hundreds, then thousands/ten-thousands, then lakhs.

Rounding Rule

Look at the digit ONE place RIGHT of where you're rounding. If ≥ 5 → round UP. If < 5 → keep same.

E.g. Round 4,567 to nearest 100: look at tens digit (6 ≥ 5) → 4,600.

Roman Numerals — Addition Rule

Smaller after larger = ADD: VI = 6, XV = 15, LX = 60

Max 3 of the same symbol in a row: III = 3, but IIII is wrong (use IV).

Roman Numerals — Subtraction Rule

Smaller before larger = SUBTRACT: IV = 4, IX = 9, XL = 40, XC = 90, CD = 400, CM = 900

Only I, X, C can precede larger numerals (not V, L, D). These 6 pairs must be memorised.

Division Algorithm

Dividend = Divisor × Quotient + Remainder; Remainder < Divisor

Always verify by substituting back.

Factor Pairs

Find factor pairs by testing: 1×N, 2×?, 3×?, up to √N

Once the two numbers in a pair cross (e.g. for 12: 4×3 already covered as 3×4), stop.

Prime Number Definition

Exactly 2 factors: 1 and the number itself. First primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37…

1 is NEITHER prime nor composite. 2 is the ONLY even prime.

Like Fractions (Add/Subtract)

Add/subtract numerators, keep denominator: a/n ± b/n = (a±b)/n

E.g. 3/8 + 2/8 = 5/8. Never add denominators!

Angle Types

Acute: 0° < angle < 90°. Right: exactly 90°. Obtuse: 90° < angle < 180°.

Mnemonic: A-cute angle is 'small and sharp'; O-btuse is 'big and blunt'.

Lines of Symmetry

Square: 4 lines. Rectangle: 2 lines. Equilateral triangle: 3 lines. Circle: infinite. Scalene triangle: 0.

A fold along a line of symmetry creates two perfectly matching halves.

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Calling 1 a prime number

✓ 1 is NEITHER prime nor composite. PRIME requires EXACTLY TWO factors (1 and itself). The number 1 has only ONE factor — itself. So it fails the definition. When listing primes, ALWAYS start from 2.

WATCH OUT

✗ Writing IIII for 4 in Roman numerals instead of IV

✓ You cannot repeat a symbol more than 3 times in a row. For 4, use the SUBTRACTION rule: IV (5−1). Similarly, 9 = IX, 40 = XL, 90 = XC, 400 = CD, 900 = CM. These six pairs must be memorised.

WATCH OUT

✗ Adding denominators when adding fractions: ½ + ½ = 2/4

✓ NEVER add denominators. ½ + ½ = 2/2 = 1 (a whole). The denominator stays: add ONLY numerators when denominators are the same. ½ + ½: numerators are 1+1=2, denominator stays 2, so 2/2 = 1.

WATCH OUT

✗ Using only divisor-1 as the test limit when finding factors

✓ Find factor PAIRS. For factors of 36: test 1×36, 2×18, 3×12, 4×9, 6×6. Stop at 6×6 because the next pair (9×4) is already listed in reverse. Systematic pairs ensure you find ALL factors.

WATCH OUT

✗ Confusing lines of symmetry in a rectangle — thinking it has 4 like a square

✓ A rectangle has ONLY 2 lines of symmetry (horizontal and vertical through the middle). The diagonal is NOT a line of symmetry — folding along a diagonal doesn't create matching halves. A square has 4 (including both diagonals) because all sides are equal.

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· roman-numerals

Write in Roman numerals: (a) 49 (b) 94 (c) 450.

▶ Show solution

Step 1 — Break each number into Roman numeral components.

(a) 49 = 40 + 9 = XL + IX = XLIX
(b) 94 = 90 + 4 = XC + IV = XCIV
(c) 450 = 400 + 50 = CD + L = CDL

✦ Answer: (a) XLIX (b) XCIV (c) CDL.

Q2EASY· prime-composite

List all prime numbers between 20 and 40.

▶ Show solution

Step 1 — Test each number from 21 to 39.
  21 = 3 × 7 (composite). 22 = 2 × 11 (composite). 23 = factors: 1, 23 only → PRIME.
  24, 25, 26, 27, 28, 29, 30: 29 = factors: 1, 29 only → PRIME.
  31 = factors: 1, 31 only → PRIME.
  32, 33, 34, 35, 36, 37: 37 = factors: 1, 37 only → PRIME.
  38, 39 = composite.

✦ Answer: 23, 29, 31, 37.

Q3EASY· factor-pairs

Find all factors of 48.

▶ Show solution

Step 1 — List factor pairs (both numbers that multiply to 48):
  1 × 48, 2 × 24, 3 × 16, 4 × 12, 6 × 8. Next would be 8 × 6 — already seen, stop.

Step 2 — Collect all unique factors in order:
  1, 2, 3, 4, 6, 8, 12, 16, 24, 48. (10 factors total.)

✦ Answer: Factors of 48 = {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}.

Q4EASY· rounding

Round each number: (a) 4,532 to nearest 100. (b) 7,865 to nearest 1,000.

▶ Show solution

(a) Nearest 100: look at tens digit of 4,532 → 3. Since 3 < 5 → round DOWN → 4,500.
(b) Nearest 1,000: look at hundreds digit of 7,865 → 8. Since 8 ≥ 5 → round UP → 8,000.

✦ Answer: (a) 4,500. (b) 8,000.

Q5MEDIUM· like-fractions

A strip of cloth is 7/9 m long. A piece of 2/9 m is cut off. What length remains? Then find 4/11 + 5/11.

▶ Show solution

Strip remaining: 7/9 − 2/9 = (7−2)/9 = 5/9 m.
(Like fractions: subtract numerators, keep denominator 9.)

4/11 + 5/11 = (4+5)/11 = 9/11.
(Like fractions: add numerators, keep denominator 11.)

✦ Answer: Remaining cloth = 5/9 m. 4/11 + 5/11 = 9/11.

Q6MEDIUM· symmetry-angles

Draw and name a shape with (a) exactly 2 lines of symmetry, (b) 0 lines of symmetry. Also: name the angle type for 45°, 90°, 130°.

▶ Show solution

(a) 2 lines of symmetry: RECTANGLE (horizontal and vertical midlines are lines of symmetry; diagonals are NOT).
(b) 0 lines of symmetry: SCALENE TRIANGLE (all sides different, no fold creates matching halves).

Angle types:
  45° → ACUTE (less than 90°, sharp angle)
  90° → RIGHT (exactly 90°, like a corner of a square)
  130° → OBTUSE (more than 90° but less than 180°, 'wide' angle)

✦ Answer: Rectangle (2), Scalene triangle (0). Angles: 45°=acute, 90°=right, 130°=obtuse.

Q7MEDIUM· number-patterns

Find the next two terms: (a) 4, 8, 16, 32, ___, ___. (b) 100, 91, 82, 73, ___, ___.

▶ Show solution

(a) Each term is DOUBLED (multiplied by 2).
  32 × 2 = 64. 64 × 2 = 128.
  Pattern: 4, 8, 16, 32, 64, 128.

(b) Each term DECREASES by 9 (arithmetic pattern, common difference = −9).
  73 − 9 = 64. 64 − 9 = 55.
  Pattern: 100, 91, 82, 73, 64, 55.

✦ Answer: (a) 64, 128. (b) 64, 55.

Q8HARD· multi-step-division

A school buys 1,248 notebooks to distribute equally among 6 classes. Each class then shares the notebooks equally among 8 students. How many notebooks does each student get? How many are left over, if any?

▶ Show solution

Step 1 — Divide 1,248 by 6 (one per class).
  1,248 ÷ 6:
  12 ÷ 6 = 2. 04 ÷ 6 = 0 remainder 4. 48 ÷ 6 = 8.
  So 1,248 ÷ 6 = 208 notebooks per class. Remainder = 0.

Step 2 — Divide 208 by 8 (per student).
  208 ÷ 8:
  20 ÷ 8 = 2 remainder 4. 48 ÷ 8 = 6.
  So 208 ÷ 8 = 26 notebooks per student. Remainder = 0.

Verify: 26 × 8 × 6 = 26 × 48 = 1,248 ✓

✦ Answer: Each student gets 26 notebooks. No notebooks are left over.

Q9HARD· prime-composite-reasoning

Is every odd number prime? Justify your answer with examples and a counter-example. Also: between 50 and 60, find all prime numbers.

▶ Show solution

CLAIM: 'Every odd number is prime.' This is FALSE.

Counter-examples of odd COMPOSITE numbers:
  9 = 3 × 3 (factors: 1, 3, 9 — three factors, so composite)
  15 = 3 × 5 (factors: 1, 3, 5, 15 — four factors, so composite)
  21 = 3 × 7 (factors: 1, 3, 7, 21 — four factors, so composite)
  25 = 5 × 5 (factors: 1, 5, 25 — three factors, so composite)

Conclusion: Some odd numbers are prime (3, 5, 7, 11, 13…) but many are NOT.
Note: 2 is even and prime — the ONLY even prime.

PRIMES between 50 and 60:
  51 = 3 × 17 (composite). 52, 54, 56, 58, 60 (all even). 53: test ÷2, ÷3, ÷5, ÷7 — none divide evenly → PRIME. 55 = 5 × 11 (composite). 57 = 3 × 19 (composite). 59: test ÷2, ÷3, ÷5, ÷7 — none divide evenly → PRIME.

✦ Answer: NOT all odd numbers are prime (e.g. 9, 15, 25 are odd composites). Primes between 50 and 60: 53 and 59.

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Place value up to LAKHS: 3,45,678 = 3 lakhs 45 thousands 6 hundreds 7 tens 8 ones.
•Rounding: look ONE place RIGHT; ≥ 5 → round up, < 5 → round down.
•Roman numerals: 6 subtraction pairs to memorise: IV(4), IX(9), XL(40), XC(90), CD(400), CM(900).
•1 is NEITHER prime NOR composite. 2 is the ONLY even prime. Primes from 1–30: 2,3,5,7,11,13,17,19,23,29.
•Factors come in PAIRS. Find pairs systematically from 1×N up to √N.
•Like fractions: add/subtract NUMERATORS only. NEVER add denominators.
•Angles: Acute (0–90°), Right (exactly 90°), Obtuse (90–180°).
•Lines of symmetry: Square=4, Rectangle=2, Equilateral triangle=3, Circle=infinite, Scalene triangle=0.
•Division check: Dividend = Divisor × Quotient + Remainder. Remainder MUST be < Divisor.
•Bar graphs: always read the scale on the y-axis first; multiply the bar height by the scale value.

ICSE marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Roman numerals in everyday life

Clock faces (VIII = 8), movie sequel names (Rocky IV), chapter numbers in books (Chapter XII), and centuries (21st century = XXI) all use Roman numerals. Knowing them is a genuine literacy skill.

Prime numbers in internet security

The encryption that keeps your bank account safe (RSA cryptography) relies on the fact that multiplying two large primes is easy, but figuring out which two primes made a huge number is nearly impossible. Primes are the foundation of digital security.

Symmetry in art and design

Logos, mandalas, architecture, and Islamic geometric patterns all use lines of symmetry. A Mughal architect designing the Taj Mahal used the same symmetry concepts you learn in Class 4.

Rounding in daily estimates

When you hear 'the crowd was about 50,000 people' or 'the project costs roughly ₹2 lakh' — that is rounding in action. Rounding is how humans communicate approximate quantities without getting lost in exact digits.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Roman numeral questions: first convert to groups (thousands, hundreds, tens, ones), then translate each group using addition/subtraction rules.
For factors, ALWAYS list in pairs (1×N, 2×?, …) to ensure no factor is missed — a common error is missing one.
When classifying numbers as prime/composite, always check whether any number from 2 to √n divides it exactly.
For fraction addition/subtraction, first check if denominators are the same (like fractions). If yes, operate on numerators only. If no, find LCM first.
Symmetry diagrams: if unsure, physically fold the shape in your mind — if one half exactly covers the other, it is a line of symmetry.

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Sieve of Eratosthenes: a method to find all primes up to any number — cross out multiples of 2, then 3, then 5, and so on. Remaining numbers are prime.
Twin primes: pairs of primes that differ by 2 — (3,5), (5,7), (11,13), (17,19), (29,31). Are there infinitely many? This is an unsolved mathematical problem.
Number base systems: we use base-10 (place values are powers of 10). Computers use base-2 (binary), where digits are only 0 and 1. 4 in binary = 100; 7 = 111.
Figurate numbers: triangular numbers (1, 3, 6, 10, 15…), square numbers (1, 4, 9, 16, 25…) — numbers that form geometric patterns. Used in advanced combinatorics.

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

ICSE Class 4 Annual ExaminationAll topics; Roman numerals, primes, and fractions each carry 5–15 marks

IMO (International Mathematics Olympiad) Class 4Number theory, patterns, and basic geometry are commonly tested

ASSET Mathematics Class 4Conceptual focus on why rules work; prime/composite reasoning is common

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

Every even number is divisible by 2. So any even number greater than 2 has at least three factors: 1, 2, and itself. That means it's composite. Only 2 itself is even and has exactly two factors (1 and 2) — making it the unique even prime.

Only the six pairs: IV(4), IX(9), XL(40), XC(90), CD(400), CM(900). Whenever you see a smaller numeral BEFORE a larger one, subtract. In all other cases, add.

Think of it physically. If you eat ½ of a pizza and your friend eats ½, you've eaten 1 whole pizza (2/2 = 1). If you wrongly added numerators AND denominators: (1+1)/(2+2) = 2/4 = ½ — that would mean you only ate half a pizza together, which is wrong.

Test divisibility by all primes up to its square root. For a number below 100, you only need to test 2, 3, 5, and 7. If none of these divide it exactly, it's prime.

Yes — infinitely many! Any line passing through the centre of a circle is a line of symmetry. This is why circles are used for wheels, rings, and compasses — they're perfectly balanced in every direction.

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