Mathematics गणित

Quick summary

Roman numerals are an old way of writing numbers using capital letters. In this chapter we learn the symbols I, V and X and the rules to write numbers from 1 to 20 (and a little more with L, C, D, M).

Key points

• Roman numerals were used in Europe long ago. They use capital letters as symbols.
• I = 1, V = 5, X = 10. There is no symbol for zero.
• The value of a symbol does not change with its place.
• Rule 1: If I or X is repeated two or three times, add them: II = 2, XX = 20.
• Rule 2: I and X can be repeated at most three times. V is never repeated.
• Rule 3: A smaller symbol on the right of a bigger one is added: VI = 6, XV = 15.
• Rule 4: I written on the left of V or X is subtracted: IV = 4, IX = 9.
• More symbols: L = 50, C = 100, D = 500, M = 1000.

Important terms

• Roman numerals – number symbols using letters.
• International numerals – the digits 0 to 9 that we normally use.

Formulas

• 1 to 10: I, II, III, IV, V, VI, VII, VIII, IX, X
• 11 to 20: XI, XII, XIII, XIV, XV, XVI, XVII, XVIII, XIX, XX

5 important questions

Q1. Write 14 and 19 in Roman numerals.

14 = 10 + 4 = X + IV = XIV. 19 = 10 + 9 = X + IX = XIX.

Q2. Write the value of XVIII.

X + V + III = 10 + 5 + 3 = 18.

Q3. Is IIII correct for 4? Correct it.

No. I cannot be repeated four times. 4 = IV.

Q4. Is VVV correct for 15? Correct it.

No. V is never repeated. 15 = XV.

Q5. Write 8 and 17 in Roman numerals.

8 = V + III = VIII. 17 = X + V + II = XVII.

Exam tip

Remember: subtract only I before V or X. Never repeat V.

Quick summary

We already know numbers up to five digits. In this chapter we learn six-digit and seven-digit numbers (lakhs and ten lakhs), the crore, place value, expanded form and comparing numbers.

Key points

• 99,999 + 1 = 1,00,000 (one lakh), the smallest six-digit number.
• 1 lakh = 100 thousand. Half a lakh = 50,000.
• 9,99,999 + 1 = 10,00,000 (ten lakh), the smallest seven-digit number.
• 99,99,999 + 1 = 1,00,00,000 (one crore), the smallest eight-digit number.
• Indian places: units, tens, hundreds, thousands, ten thousands, lakhs, ten lakhs, crores.
• Commas in Indian system: after the first three digits from right, then after every two digits: 12,34,567.
• Place value = digit × value of its place.
• Expanded form shows a number as the sum of its place values.
• To compare, the number with more digits is bigger. If digits are equal, compare from the left.

Important terms

• Lakh – 1,00,000.
• Crore – 1,00,00,000 = 100 lakh.
• Place value – value of a digit due to its place.
• Expanded form – number written as the sum of place values.

Formulas

• 1 lakh = 100 thousand; 10 lakh = 1 million; 1 crore = 100 lakh.
• Smallest 6-digit = 1,00,000; greatest 6-digit = 9,99,999.
• Smallest 7-digit = 10,00,000; greatest 7-digit = 99,99,999.

5 important questions

Q1. Write 3,45,672 in words.

Three lakh, forty-five thousand, six hundred and seventy-two.

Q2. Write the expanded form of 7,08,315.

7,00,000 + 0 + 8,000 + 300 + 10 + 5.

Q3. What is the place value of 6 in 26,43,091?

6 is in the lakhs place. Place value = 6 × 1,00,000 = 6,00,000.

Q4. Which is greater: 4,56,789 or 4,65,123?

Lakhs are equal (4). Ten thousands: 5 < 6. So 4,65,123 is greater.

Q5. How many rupees is two and a half lakh?

2 lakh 50 thousand = ₹2,50,000.

Exam tip

Always put commas in the Indian way before reading a big number.

Quick summary

In this chapter we add and subtract six-digit and seven-digit numbers, with carrying and borrowing, and solve word problems.

Key points

• Write numbers one below the other so that units come under units, tens under tens, and so on.
• Add from the units place. If a sum is 10 or more, carry 1 to the next place.
• Subtract from the units place. If the upper digit is smaller, borrow 1 from the next place.
• Check subtraction: difference + smaller number = bigger number.
• Order of adding does not change the sum: a + b = b + a.
• In word problems, words like 'total', 'altogether' mean add; 'how many more', 'left', 'difference' mean subtract.

Important terms

• Sum – answer of addition.
• Difference – answer of subtraction.
• Carry / Borrow – moving 1 to or from the next place.

Formulas

• Sum = first number + second number.
• Difference = bigger number − smaller number.
• Check: difference + smaller = bigger.

5 important questions

Q1. Add 3,48,659 and 2,75,486.

3,48,659 + 2,75,486 = 6,24,145.
Units 9+6=15 (write 5, carry 1); tens 5+8+1=14; hundreds 6+4+1=11; thousands 8+5+1=14; ten thousands 4+7+1=12; lakhs 3+2+1=6.

Q2. Subtract 1,87,594 from 5,02,310.

5,02,310 − 1,87,594 = 3,14,716. Check: 3,14,716 + 1,87,594 = 5,02,310 ✓

Q3. A factory made 4,25,600 bags in one year and 3,98,750 the next. How many in all?

4,25,600 + 3,98,750 = 8,24,350 bags.

Q4. A town had 12,45,000 people. 85,300 moved away. How many are left?

12,45,000 − 85,300 = 11,59,700 people.

Q5. What must be added to 6,58,000 to get 10,00,000?

10,00,000 − 6,58,000 = 3,42,000.

Exam tip

Always check subtraction by adding back. Write the answer with Indian commas.

Quick summary

This chapter covers multiplying bigger numbers, counting the number of ways of pairing things from two groups, and dividing numbers, with word problems.

Key points

• Multiply by the units digit first, then by the tens digit (shift one place left), then add.
• Pairing: if one group has 3 things and another has 2, the number of different pairs is 3 × 2 = 6.
• Division parts: dividend ÷ divisor = quotient, with a remainder.
• The remainder is always smaller than the divisor.
• Any number multiplied by 0 gives 0; multiplied by 1 gives the same number.
• Dividing by 10, 100 removes zeros when the number ends in zeros: 4500 ÷ 100 = 45.

Important terms

• Product – answer of multiplication.
• Dividend, Divisor, Quotient, Remainder – the parts of a division.

Formulas

• Dividend = Divisor × Quotient + Remainder.
• Number of pairs = (items in group 1) × (items in group 2).

5 important questions

Q1. Multiply 3,456 by 27.

3,456 × 7 = 24,192. 3,456 × 20 = 69,120. Add: 24,192 + 69,120 = 93,312.

Q2. Ajay has 3 shirts and 2 trousers. In how many ways can he pair them?

3 × 2 = 6 different pairs.

Q3. Divide 7,845 by 15.

7,845 ÷ 15 = 523, remainder 0. Check: 15 × 523 = 7,845 ✓

Q4. A school bought 48 benches at ₹1,250 each. Find the total cost.

48 × 1,250 = ₹60,000.

Q5. 1,000 sweets are packed 24 in a box. How many boxes are full and how many sweets are left?

1,000 ÷ 24 = 41, remainder 16. So 41 full boxes and 16 sweets left.

Exam tip

Check every division using Dividend = Divisor × Quotient + Remainder.

Quick summary

A fraction shows a part of a whole. In this chapter we learn equivalent fractions, like and unlike fractions, comparing fractions, adding and subtracting them, finding a fraction of a collection, and mixed fractions.

Key points

• In a/b, a is the numerator and b is the denominator.
• Equivalent fractions have the same value: 1/2 = 2/4 = 3/6. Multiply or divide numerator and denominator by the same number.
• Like fractions have the same denominator (2/7, 5/7). Unlike fractions have different denominators.
• Unlike fractions are made like by finding equivalent fractions with a common denominator.
• Like fractions: the one with the bigger numerator is bigger (5/9 > 2/9).
• Same numerator: the one with the smaller denominator is bigger (3/4 > 3/8).
• Add or subtract like fractions by adding or subtracting numerators only.
• A mixed fraction has a whole number and a fraction, like 2 1/3 (= 7/3).

Important terms

• Numerator / Denominator – top / bottom number of a fraction.
• Equivalent fractions (सममूल्य अपूर्णांक) – fractions with equal value.
• Like / Unlike fractions – same / different denominators.
• Mixed fraction – a whole number plus a proper fraction.

Formulas

• a/b = (a × k)/(b × k)
• a/c + b/c = (a + b)/c; a/c − b/c = (a − b)/c
• Fraction of a collection: (a/b) of N = (N ÷ b) × a

5 important questions

Q1. Write two equivalent fractions of 3/5.

3/5 = 6/10 = 9/15.

Q2. Add 1/3 and 1/4.

Make like fractions: 1/3 = 4/12, 1/4 = 3/12. 4/12 + 3/12 = 7/12.

Q3. Subtract 2/9 from 7/9.

7/9 − 2/9 = 5/9.

Q4. Find 3/4 of 20 mangoes.

20 ÷ 4 = 5; 5 × 3 = 15 mangoes.

Q5. Write 11/4 as a mixed fraction.

11 ÷ 4 = 2 remainder 3. So 11/4 = 2 3/4.

Exam tip

Before adding unlike fractions, always make the denominators equal.

Quick summary

An angle is formed when two rays start from the same point. In this chapter we learn to name angles, measure and draw them with a protractor, the types of angles, and parallel and perpendicular lines.

Key points

• An angle has a vertex (common point) and two arms.
• ∠ABC means the angle with vertex B; it can also be written ∠CBA. The vertex letter is always in the middle.
• Angles are measured in degrees (°) using a protractor.
• To measure: put the protractor's centre on the vertex, its base line on one arm, and read the scale starting from 0 on that arm.
• Right angle = 90°. Acute angle: less than 90°. Obtuse angle: more than 90°.
• A right angle can be made by folding a paper twice.
• Parallel lines never meet, however far they are extended. The distance between them stays the same.
• Perpendicular lines meet at a right angle (90°).

Important terms

• Vertex (शिरोबिंदू) – the common point of the arms.
• Arms – the two rays forming the angle.
• Protractor – instrument to measure angles.
• Parallel lines – lines that never meet.
• Perpendicular lines – lines meeting at 90°.

Formulas

5 important questions

Q1. Name the vertex and arms of ∠PQR.

Vertex Q; arms QP and QR.

Q2. Classify: 45°, 90°, 120°.

45° acute, 90° right, 120° obtuse.

Q3. What angle do the clock hands make at 3 o'clock?

A right angle (90°).

Q4. How will you draw an angle of 70°?

Draw ray BC. Put the protractor centre at B along BC. Mark a point at 70°. Join B to it and name it A. ∠ABC = 70°.

Q5. What are parallel lines? Give an example.

Lines that never meet; e.g. the two rails of a railway track.

Exam tip

Read the protractor scale that starts from 0 on your arm, or you will get 180° minus the answer.

Quick summary

A circle is a round closed figure in which every point is at the same distance from the centre. In this chapter we learn radius, chord, diameter, drawing a circle with a compass, and interior and exterior of a circle.

Key points

• The fixed middle point is the centre.
• A line joining the centre to any point on the circle is a radius. All radii of a circle are equal.
• A line joining any two points on the circle is a chord.
• A chord passing through the centre is a diameter. It is the longest chord.
• Diameter = 2 × radius. All diameters of a circle are equal.
• A circle is drawn with a compass: set the radius, keep the metal tip fixed at the centre, turn the pencil once around.
• Points inside the circle are in its interior; outside are in its exterior; points on the circle are on the circle.
• A part of a circle is called an arc.

Important terms

• Centre (केंद्र) – fixed middle point.
• Radius (त्रिज्या) – centre to circle.
• Diameter (व्यास) – chord through the centre.
• Chord (जीवा) – line joining two points on the circle.
• Arc (कंस) – part of a circle.

Formulas

• Diameter = 2 × Radius
• Radius = Diameter ÷ 2

5 important questions

Q1. The radius of a circle is 4 cm. Find its diameter.

Diameter = 2 × 4 = 8 cm.

Q2. The diameter of a circle is 22 cm. Find its radius.

Radius = 22 ÷ 2 = 11 cm.

Q3. Which is the longest chord of a circle?

The diameter.

Q4. How do you draw a circle of radius 3 cm?

Open the compass to 3 cm on a ruler. Fix the metal tip at a point O and turn the pencil one full round.

Q5. What is an arc?

A part of the circle between two points on it.

Exam tip

Remember: every diameter is a chord, but not every chord is a diameter.

Quick summary

This chapter teaches multiples and factors of numbers, simple tests of divisibility by 2, 5 and 10, prime and composite numbers, co-prime numbers and Eratosthenes' method of finding primes.

Key points

• If a number divides another exactly, the first is a factor and the second is a multiple. 3 is a factor of 12; 12 is a multiple of 3.
• Every number is a multiple of 1 and of itself.
• Divisible by 2: units digit is 0, 2, 4, 6 or 8.
• Divisible by 5: units digit is 0 or 5.
• Divisible by 10: units digit is 0.
• A prime number has only two factors: 1 and itself (2, 3, 5, 7, 11…).
• A composite number has more than two factors (4, 6, 8, 9…).
• 1 is neither prime nor composite. 2 is the only even prime number.
• Two numbers with only 1 as a common factor are co-prime (8 and 15).
• Eratosthenes' sieve: list numbers 1–100, cross out 1, then cross out multiples of 2, 3, 5, 7 (not the numbers themselves). The numbers left are primes.

Important terms

• Factor (विभाजक) – a number that divides another exactly.
• Multiple (पट) – result of multiplying a number by 1, 2, 3…
• Prime number (मूळ संख्या) – exactly two factors.
• Composite number – more than two factors.
• Co-prime numbers – only common factor is 1.

Formulas

• Primes between 1 and 100: 25 numbers (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97).

5 important questions

Q1. Write all factors of 24.

1, 2, 3, 4, 6, 8, 12, 24.

Q2. Write the first five multiples of 7.

7, 14, 21, 28, 35.

Q3. Is 390 divisible by 2, 5 and 10?

Units digit is 0, so it is divisible by 2, 5 and 10.

Q4. Is 21 prime or composite?

Factors are 1, 3, 7, 21. More than two, so it is composite.

Q5. Are 9 and 16 co-prime?

Factors of 9: 1, 3, 9. Factors of 16: 1, 2, 4, 8, 16. Only common factor is 1, so yes.

Exam tip

Remember 1 is neither prime nor composite, and 2 is the only even prime.

Quick summary

Decimal fractions are another way of writing fractions with denominators 10 and 100. In this chapter we learn tenths, hundredths, the decimal point, place value, adding and subtracting decimals, and their use in money and measurement.

Key points

• 1/10 is written 0.1 (one tenth). 1/100 is written 0.01 (one hundredth).
• The dot is the decimal point. Digits to its left are the whole number; to its right are tenths and hundredths.
• 3.47 = 3 units + 4 tenths + 7 hundredths.
• 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75, 1 1/4 = 1.25.
• To add or subtract decimals, write the decimal points one below the other. Fill empty places with 0.
• Decimals are used in money (₹12.50) and measurement (2.5 kg, 1.75 m).
• 1 rupee = 100 paise, so 50 paise = ₹0.50.
• 1 m = 100 cm, so 35 cm = 0.35 m.

Important terms

• Decimal fraction (दशांश अपूर्णांक) – fraction written with a decimal point.
• Tenths / Hundredths – first / second place after the point.

Formulas

5 important questions

Q1. Write 6/10 and 45/100 as decimals.

6/10 = 0.6; 45/100 = 0.45.

Q2. Write the place value of each digit in 5.38.

5 units = 5; 3 tenths = 0.3; 8 hundredths = 0.08.

Q3. Add 12.75 and 3.5.

12.75 + 3.50 = 16.25.

Q4. Subtract 2.68 from 10.

10.00 − 2.68 = 7.32.

Q5. Write 3 metres 40 cm in metres.

40 cm = 40/100 m = 0.40 m. So 3.40 m.

Exam tip

Always line up the decimal points and add zeros to fill empty places.

Quick summary

This chapter teaches seconds, a.m. and p.m., the 24-hour clock and how to find time intervals.

Key points

• 1 minute = 60 seconds; 1 hour = 60 minutes; 1 day = 24 hours.
• The thin, fast-moving hand of a clock is the seconds hand.
• a.m. (ante meridiem) means before noon: 12 midnight to 12 noon.
• p.m. (post meridiem) means after noon: 12 noon to 12 midnight.
• In the 24-hour clock, time is counted from 0 to 24 hours starting at midnight. Railways, airlines and offices use it.
• For p.m. times, add 12 to get 24-hour time: 3:15 p.m. = 15:15.
• For 24-hour times greater than 12, subtract 12 and write p.m.: 19:40 = 7:40 p.m.
• Midnight is 00:00 hours; noon is 12:00 hours.

Important terms

• a.m. – time before noon.
• p.m. – time after noon.
• 24-hour clock – time shown from 00:00 to 23:59.

Formulas

• 1 min = 60 s; 1 h = 60 min = 3600 s.
• p.m. time → 24-hour: add 12 hours.

5 important questions

Q1. How many seconds are there in 5 minutes?

5 × 60 = 300 seconds.

Q2. Write 8:30 p.m. in the 24-hour clock.

8 + 12 = 20. So 20:30 hours.

Q3. Write 17:45 in a.m./p.m. form.

17 − 12 = 5. So 5:45 p.m.

Q4. A train leaves at 10:20 a.m. and reaches at 2:50 p.m. How long is the journey?

10:20 a.m. to 12:20 p.m. = 2 h; 12:20 to 2:50 = 2 h 30 min. Total 4 h 30 min.

Q5. How many minutes are there in 3 hours 15 minutes?

3 × 60 + 15 = 195 minutes.

Exam tip

Count time intervals in steps up to the next full hour or noon.

Quick summary

This chapter uses units of length, weight (mass) and capacity to solve everyday word problems involving adding, subtracting, multiplying and dividing measurements.

Key points

• Length: 1 km = 1000 m; 1 m = 100 cm; 1 cm = 10 mm.
• Weight: 1 kg = 1000 g; 1 quintal = 100 kg.
• Capacity: 1 litre = 1000 ml.
• Before adding or subtracting, change quantities to the same unit.
• To change a bigger unit into a smaller one, multiply. To change a smaller unit into a bigger one, divide.
• Activities like measuring shadows or a pen refill help us estimate lengths.

Important terms

• Length – how long something is.
• Weight (mass) – how heavy something is.
• Capacity – how much liquid a container holds.

Formulas

5 important questions

Q1. Convert 3 km 250 m into metres.

3 × 1000 + 250 = 3250 m.

Q2. A bag holds 2 kg 750 g of rice. 1 kg 900 g is used. How much is left?

2750 g − 1900 g = 850 g.

Q3. One bottle holds 750 ml. How much do 8 bottles hold?

8 × 750 = 6000 ml = 6 litres.

Q4. A 3 m ribbon is cut into pieces of 50 cm. How many pieces?

3 m = 300 cm. 300 ÷ 50 = 6 pieces, nothing left over.

Q5. Write 4500 g in kilograms.

4500 ÷ 1000 = 4.5 kg (4 kg 500 g).

Exam tip

Change everything to the smaller unit first; it avoids mistakes.

Quick summary

Perimeter is the length of the boundary of a figure. Area is the space a figure covers. This chapter gives the formulas for perimeter and area of rectangles and squares, with word problems.

Key points

• Perimeter is measured in units of length (cm, m).
• Area is measured in square units (sq cm, sq m).
• A rectangle has two lengths and two breadths. Opposite sides are equal.
• A square has all four sides equal.
• Area can be found by counting unit squares, or by formula.
• Use the same unit for length and breadth before applying a formula.

Important terms

• Perimeter (परिमिती) – total length of the boundary.
• Area (क्षेत्रफळ) – space enclosed by a figure.
• Square centimetre – area of a square of side 1 cm.

Formulas

• Perimeter of rectangle = 2 × (length + breadth)
• Perimeter of square = 4 × side
• Area of rectangle = length × breadth
• Area of square = side × side

5 important questions

Q1. Find the perimeter of a rectangle 12 cm long and 7 cm broad.

2 × (12 + 7) = 2 × 19 = 38 cm.

Q2. Find the perimeter of a square of side 9 m.

4 × 9 = 36 m.

Q3. Find the area of a rectangular room 6 m by 4 m.

6 × 4 = 24 sq m.

Q4. The side of a square tile is 25 cm. Find its area.

25 × 25 = 625 sq cm.

Q5. A rectangular field is 50 m long and 30 m wide. How much wire is needed for fencing it once?

Perimeter = 2 × (50 + 30) = 160 m of wire.

Exam tip

Write the unit in every answer: cm for perimeter, sq cm for area.

Quick summary

Objects like boxes, balls and dice have length, breadth and height, so they are three-dimensional (3D). This chapter teaches how to draw 3D objects on paper and how a flat net can be folded to make a 3D shape.

Key points

• Flat shapes like squares and triangles have only length and breadth; they are two-dimensional (2D).
• Objects with length, breadth and height are three-dimensional (3D): cube, cuboid, cylinder, cone, sphere.
• A 3D object can be drawn on paper by showing its hidden edges with dotted lines.
• A cube has 6 square faces, 12 edges and 8 vertices.
• A cuboid has 6 rectangular faces, 12 edges and 8 vertices.
• A net is a flat pattern which can be folded to make a 3D object.
• A cube's net has 6 equal squares joined edge to edge.
• A shape made by joining five squares edge to edge is a pentomino. Some pentominoes fold into an open box (a cube without a lid).

Important terms

• Face – a flat surface of a solid.
• Edge – a line where two faces meet.
• Vertex – a corner where edges meet.
• Net – a flat shape that folds into a solid.
• Pentomino – a shape of five squares joined edge to edge.

Formulas

5 important questions

Q1. How many faces, edges and vertices does a cube have?

6 faces, 12 edges and 8 vertices.

Q2. What is a net?

A flat figure that can be folded to make a 3D object.

Q3. How many squares are there in the net of a cube?

6 equal squares.

Q4. What is the difference between 2D and 3D shapes?

2D shapes have length and breadth; 3D objects also have height (thickness).

Q5. What is a pentomino?

A figure made by joining five equal squares edge to edge.

Exam tip

Cut out nets from paper and fold them; it helps you see which nets make a cube.

Quick summary

A pictograph shows numerical information (data) using pictures. Each picture stands for a fixed number. This chapter teaches how to read and draw pictographs.

Key points

• Information in the form of numbers is called data.
• In a pictograph, a picture or symbol represents a certain number of things.
• The scale (key) tells how much one picture stands for, e.g. 1 picture = 10 trees.
• Half a picture shows half that number.
• To read: count pictures and multiply by the scale.
• To draw: choose a scale, then divide each number by it to know how many pictures to draw.
• Pictographs make comparison easy at a glance.

Important terms

• Data – information in numbers.
• Pictograph (चित्रालेख) – data shown with pictures.
• Scale / Key – the value of one picture.

Formulas

• Value shown = number of pictures × value of one picture.
• Number of pictures = quantity ÷ value of one picture.

5 important questions

Q1. What is a pictograph?

A way of showing data using pictures, each picture standing for a fixed number.

Q2. If 1 picture = 5 students, how many students do 7 pictures show?

7 × 5 = 35 students.

Q3. If 1 picture = 10 trees, how many pictures show 45 trees?

45 ÷ 10 = 4.5, so 4 full pictures and half a picture.

Q4. Why is a scale needed?

When numbers are big, one picture can stand for many things, so the chart stays small and clear.

Q5. In a pictograph, village A has 6 pictures and B has 4 (1 picture = 100 people). How many more people live in A?

(6 − 4) × 100 = 200 more people.

Exam tip

Always write the scale (key) when you draw a pictograph.

Quick summary

Patterns are designs that repeat in a regular way. This chapter looks at patterns in floor tiles, saris and shawls, and patterns made with shapes and numbers.

Key points

• A pattern is an arrangement that repeats according to a rule.
• Floor tiles fit together without gaps. Squares, rectangles, equilateral triangles and regular hexagons can cover a floor fully.
• Patterns can be made by repeating, turning or flipping a shape.
• Borders of saris and shawls show repeating patterns.
• Number patterns follow a rule, like adding or multiplying the same number each time.
• To continue a pattern, first find its rule.

Important terms

• Pattern (आकृतिबंध) – a design that repeats by a rule.
• Tiling – covering a surface with shapes without gaps or overlaps.
• Rule – what changes from one step to the next.

Formulas

• Examples of rules: 3, 6, 9, 12 … (add 3); 2, 4, 8, 16 … (multiply by 2).

5 important questions

Q1. What is a pattern?

An arrangement of shapes or numbers that repeats according to a rule.

Q2. Name shapes that can cover a floor without gaps.

Squares, rectangles, equilateral triangles and regular hexagons.

Q3. Continue: 5, 10, 15, 20, __, __.

Rule: add 5. Next: 25, 30.

Q4. Continue: 1, 3, 9, 27, __.

Rule: multiply by 3. Next: 81.

Q5. Continue: 100, 90, 80, 70, __, __.

Rule: subtract 10. Next: 60, 50.

Exam tip

Find the difference or ratio between consecutive terms to discover the rule.

Quick summary

Algebra uses letters along with numbers to write rules in short. In this chapter we learn equality, inequality with the signs =, ≠, <, >, and using letters to stand for numbers.

Key points

• Two expressions with the same value are equal: (9 − 5) = (2 × 2).
• If they are not equal we use ≠: (9 − 5) ≠ (15 ÷ 3).
• < means 'is less than'; > means 'is greater than'. 8 < 12, 20 > 7.
• Brackets show which operation to do first.
• Letters like a, b, x can stand for any number.
• Sum of two numbers a and b is a + b. Order does not matter: a + b = b + a.
• a × 1 = a; a × 0 = 0; a + 0 = a.
• Letters let us write rules that are true for all numbers.

Important terms

• Expression – numbers combined with operations, e.g. (4 × 7).
• Equality – two sides having the same value.
• Inequality – two sides not equal.
• Variable – a letter that stands for a number.

Formulas

• a + b = b + a; a × b = b × a
• a × 1 = a; a + 0 = a; a × 0 = 0

5 important questions

Q1. Fill <, > or =: (4 × 7) ___ (30 − 2).

4 × 7 = 28 and 30 − 2 = 28. So (4 × 7) = (30 − 2).

Q2. Fill <, > or =: (18 ÷ 3) ___ (2 + 5).

18 ÷ 3 = 6 and 2 + 5 = 7. 6 < 7, so (18 ÷ 3) < (2 + 5).

Q3. Write 'the sum of x and 7' using letters.

x + 7.

Q4. If a = 5 and b = 3, find a + b and a × b.

a + b = 8; a × b = 15.

Q5. Write the rule 'any number multiplied by 1 gives the same number' using a letter.

a × 1 = a.

Exam tip

Solve the brackets on both sides first, then compare.