Mathematics

A whole number is made of digits, and where a digit stands gives its place value. Reading from the right, the places are ones, tens, hundreds, thousands, ten-thousands, hundred-thousands and millions. Each place to the left is 10 times bigger. In 48 275, the digit 8 is in the thousands place, so its total value is 8 000.

We write numbers in words too: 48 275 is forty-eight thousand, two hundred and seventy-five. In expanded form, 143 = (1 × 100) + (4 × 10) + (3 × 1).

To round off, look at the digit just after the place you are rounding to. If it is 5 or more, round up; if it is 4 or less, round down. For example, 3 472 to the nearest hundred is 3 500 because the tens digit 7 is 5 or more. Rounding helps us estimate quickly, such as guessing that a matatu carrying 47 and 52 passengers moved about 50 + 50 = 100 people.

Addition means putting amounts together to find the sum (total). We line up numbers according to place value — ones under ones, tens under tens — and add each column from the right. When a column adds up to 10 or more, we regroup (carry) to the next column.

For example, 24 560 + 8 375: adding ones gives 5, tens give 13 (write 3, carry 1), and so on, giving 32 935. Always keep your columns straight so the carrying is correct.

Addition is common with money. If a shopkeeper sells goods worth KES 12 450 in the morning and KES 9 875 in the afternoon, the day's sales are 12 450 + 9 875 = KES 22 325.

We can estimate first to check our answer is sensible. Rounding KES 12 450 to 12 000 and KES 9 875 to 10 000 gives about KES 22 000, which is close to the exact answer, so 22 325 looks correct.

Subtraction means taking one amount away from another to find the difference. We arrange the numbers in place-value columns and subtract from the right. When the top digit is smaller than the bottom digit, we borrow (regroup) one unit from the next column.

For example, 50 000 − 23 475. Because we cannot take 5 from 0, we borrow across the columns, giving the answer 26 525.

We can always check a subtraction by adding back: 23 475 + 26 525 = 50 000, so the answer is correct. This is a good habit that catches mistakes.

Subtraction answers everyday questions. If a trader has KES 15 000 and spends KES 8 650 on stock, the money left is 15 000 − 8 650 = KES 6 350. If a tank held 4 500 ml of water and 1 750 ml was used, then 4 500 − 1 750 = 2 750 ml remains.

Multiplication is a fast way of doing repeated addition. 4 × 3 means 4 + 4 + 4 = 12. The answer is called the product.

To multiply by a two-digit number, we use long multiplication: multiply by the ones, then by the tens (shifting one column left because that digit stands for tens), then add the parts. For example, 342 × 24: 342 × 4 = 1 368 and 342 × 20 = 6 840. Adding, 1 368 + 6 840 = 8 208.

We can estimate a product by rounding first. 49 × 21 is about 50 × 20 = 1 000, so the exact answer 1 029 is reasonable.

Multiplication solves grouping problems. If one bag holds 36 mangoes, then 15 bags hold 36 × 15 = 540 mangoes. If a book costs KES 250, then 12 books cost 250 × 12 = KES 3 000.

Division shares or groups a number into equal parts. In 852 ÷ 4, 852 is the dividend, 4 is the divisor, and the answer 213 is the quotient. Sometimes there is a leftover called the remainder.

Division is the inverse of multiplication, so we can check answers: 213 × 4 = 852. This link also helps us estimate a quotient.

When a calculation mixes operations, we follow the order of operations: do division and multiplication before addition and subtraction. For example, 20 − 6 ÷ 3: first 6 ÷ 3 = 2, then 20 − 2 = 18. Doing it left to right would give a wrong answer.

Division solves sharing problems. If 90 sweets are shared equally among 6 children, each gets 90 ÷ 6 = 15 sweets. If KES 1 200 is shared equally among 4 pupils, each receives 1 200 ÷ 4 = KES 300.

A fraction shows part of a whole. The top number is the numerator (parts taken) and the bottom is the denominator (equal parts in the whole). Fractions that name the same amount are equivalent, for example 1/2 = 2/4 = 3/6.

To simplify, divide the numerator and denominator by their greatest common factor. 8/12 divides by 4 to give 2/3.

To add or subtract fractions with the same denominator, work with the numerators only: 2/5 + 1/5 = 3/5. When denominators are different, first make them the same using equivalent fractions. For 1/2 + 1/3, the common denominator is 6: 1/2 = 3/6 and 1/3 = 2/6, so 3/6 + 2/6 = 5/6.

To compare fractions, rewrite them with a common denominator. Comparing 3/4 and 2/3 over 12 gives 9/12 and 8/12, so 3/4 is larger. This helps when sharing food or measuring cloth fairly.

A decimal uses a point to separate whole numbers from parts of one. The places after the point are tenths (1/10), hundredths (1/100) and thousandths (1/1000). In 4.567, the 5 is tenths, 6 is hundredths and 7 is thousandths.

To compare decimals, line up the decimal points and compare digit by digit from the left, filling gaps with zeros. So 0.709, 0.700 and 0.680 compare as thousandths 709 > 700 > 680, meaning 0.709 is the largest.

To add or subtract, write the numbers with their decimal points in a line, then work column by column. For example, 2.5 + 1.35 = 3.85, and 6.4 − 2.15 = 4.25 (check: 2.15 + 4.25 = 6.40).

Decimals are everywhere in money: KES 4.50 means 4 shillings and 50 cents. A tailor may buy 2.5 m and 1.35 m of cloth, using 2.5 + 1.35 = 3.85 m in total.

Length tells us how long or far something is. The units, from small to large, are the millimetre (mm), centimetre (cm), metre (m) and kilometre (km). The key relationships are: 1 cm = 10 mm, 1 m = 100 cm, and 1 km = 1 000 m.

We use the kilometre for long distances such as the road from home to town. To change metres into kilometres, divide by 1 000; to change kilometres into metres, multiply by 1 000. For example, 4 500 m = 4 500 ÷ 1 000 = 4.5 km, and 3 km = 3 × 1 000 = 3 000 m.

We can add and subtract lengths that use the same unit. If a learner walks 1 250 m to school and back, the total is 1 250 + 1 250 = 2 500 m, which is 2.5 km. A 250 cm ribbon is the same as 250 ÷ 100 = 2.5 m of ribbon.

Area is the amount of flat surface a shape covers. We measure it in square units, such as the square centimetre (cm²) — a square whose sides are each 1 cm. We can find area by counting how many 1 cm squares fit inside a shape.

For a rectangle, the squares form rows and columns, so area is a product: Area = length × width. A rectangle 5 cm long and 3 cm wide covers 5 × 3 = 15 cm². A 3 cm by 3 cm square holds 3 × 3 = 9 unit squares.

A square has equal sides, so Area = side × side. A square of side 4 cm has area 4 × 4 = 16 cm².

We can also work backwards. If a rectangular garden has area 20 cm² on a plan and a length of 5 cm, its width is 20 ÷ 5 = 4 cm. Area helps us plan tiles, farms and playgrounds.

Volume is the amount of space a solid object takes up. We measure it in cubic units, such as the cubic centimetre (cm³) — a cube whose edges are each 1 cm. We can find the volume of a box by counting how many 1 cm cubes fill it.

For a cuboid (a box shape), the cubes fill layers, so: Volume = length × width × height. A cuboid 4 cm long, 3 cm wide and 2 cm high holds 4 × 3 × 2 = 24 cm³, which is the same as 24 one-centimetre cubes.

A cube has equal edges, so a cube of side 2 cm has volume 2 × 2 × 2 = 8 cm³.

Volume helps us know how much a container can hold. A carton 10 cm by 5 cm by 4 cm has volume 10 × 5 × 4 = 200 cm³. Later we learn that 1 000 cm³ of water is exactly 1 litre.

Capacity is how much liquid a container can hold. The main units are the litre (L) and the millilitre (ml). The relationship is: 1 L = 1 000 ml.

We use millilitres for small amounts, like a 500 ml soda bottle or a 250 ml cup of milk, and litres for larger amounts, like a 20 L jerrican of water.

To change litres into millilitres, multiply by 1 000: 2 L = 2 × 1 000 = 2 000 ml. To change millilitres into litres, divide by 1 000: 2 500 ml = 2 500 ÷ 1 000 = 2.5 L.

We can add, subtract and divide capacities. If a 2 L jug of juice is poured equally into 500 ml cups, it fills 2 000 ÷ 500 = 4 cups. If a tank holds 5 L and 1 500 ml is used, 5 000 − 1 500 = 3 500 ml remains.

Mass tells us how heavy an object is. We measure small masses in grams (g) and larger masses in kilograms (kg). A paperclip is about 1 g, while a 2 kg packet of maize flour is much heavier. The relationship is: 1 kg = 1 000 g.

We use a weighing balance or scale to find mass. In a shop, sugar, rice and beans are weighed in grams and kilograms.

To change kilograms into grams, multiply by 1 000: 2 kg = 2 × 1 000 = 2 000 g. To change grams into kilograms, divide by 1 000: 4 000 g = 4 000 ÷ 1 000 = 4 kg.

We can add and subtract masses in the same unit. A mixture of 2 kg 500 g is the same as 2 500 g. If a bag of 5 kg sugar is packed into 500 g packets, it makes 5 000 ÷ 500 = 10 packets.

Time is measured in seconds, minutes and hours. The second is a small unit — about one heartbeat — while the minute and hour are longer. The key relationships are: 1 minute = 60 seconds and 1 hour = 60 minutes.

To change minutes into seconds, multiply by 60: 2 min = 2 × 60 = 120 s. To change seconds into minutes, divide by 60: 180 s = 180 ÷ 60 = 3 min.

We add and subtract time to find durations. If a lesson starts at 8:00 and ends at 8:40, it lasts 40 minutes. 1 hour 30 minutes is the same as 60 + 30 = 90 minutes.

We can also divide time. If a 60-minute games period is shared equally among 4 activities, each activity gets 60 ÷ 4 = 15 minutes. Knowing time helps us keep a timetable, catch a matatu and run races fairly.

Kenyan money is counted in shillings (KES), using coins and notes. Handling money well means spending wisely and saving.

A budget is a plan for how to use money. We list our income (money received) and our expenditure (money spent), then check that spending does not go beyond income. For example, if a pupil receives KES 500 and plans KES 200 for lunch, KES 150 for a book and KES 50 for fare, the total spent is 200 + 150 + 50 = KES 400, leaving 500 − 400 = KES 100 to save. Budgeting stops us from running short.

Tax is money citizens and businesses pay to the government, for example on goods bought in shops. The government uses taxes to build roads, schools and hospitals, pay teachers and provide services for everyone. Paying tax honestly is the duty of a good citizen, and it helps develop the whole country.

A line is a straight path that has no thickness and continues in both directions. We describe lines by how they lie and how they meet.

A horizontal line runs level from side to side, like the edge where the sky meets the land (the horizon). A vertical line runs straight up and down, like a standing flagpole or the corner of a wall.

Two lines are perpendicular when they meet or cross at a right angle (90°). The corner of a book, a window frame or a floor tile shows perpendicular lines. We mark the right angle with a small square.

Lines are parallel when they are always the same distance apart and never meet, no matter how far they go — like the two rails of a railway line or the long edges of a ruler. Learning to spot and draw these lines helps in drawing shapes, reading maps and building neat structures.

An angle is the amount of turn between two lines (arms) that meet at a point called the vertex. We measure angles in degrees (°) using a protractor. A full turn is 360° and a half turn is 180°.

Angles are named by their size:

• A right angle is exactly 90° (a quarter turn), like the corner of a book.
• An acute angle is less than 90°, like the hands of a clock at 1 o'clock.
• An obtuse angle is more than 90° but less than 180°.
• A straight angle is exactly 180°, a straight line.

We see angles all around us: in the corners of a door, the hands of a clock, the letter V, and the roof of a house. When we open a door a little way it makes an acute angle; opening it wide makes an obtuse angle. Being able to measure angles helps in drawing shapes and reading directions.

3-D objects are solids that take up space; they have length, width and height. Common ones are the cube, cuboid, cylinder, cone, sphere and pyramid. A dice is a cube, a matchbox is a cuboid, a tin is a cylinder and a ball is a sphere.

Every flat surface of a solid is a face, and each face is a 2-D shape. A cube has 6 faces, and each one is a square. A cuboid also has 6 faces, but they are rectangles. A cylinder has 2 circular faces and one curved surface; a cone has 1 circular face and a curved surface narrowing to a point.

We also count edges (where two faces meet) and vertices (corners where edges meet). A cube has 6 faces, 12 edges and 8 vertices. Describing solids by their 2-D shapes helps us make boxes, tins and models.

Data is information we collect, such as the favourite fruits or number of pupils in each class. After collecting data, we can record it using tally marks and organise it in a table with clear headings, which makes it easy to read and compare.

A pictograph (picture graph) shows data using pictures or symbols. A key tells us how many items each picture stands for. For example, if 1 picture = 10 mangoes, then 3 pictures stand for 3 × 10 = 30 mangoes. A half picture stands for half the key value, so with 1 apple = 20, then 2 apples + 1 half = (2 × 20) + 10 = 50 apples.

To read a table, we look across the row and down the column. If a shop's sales table shows Mon 12, Tue 8 and Wed 15, the total sold is 12 + 8 + 15 = 35. Representing data clearly helps us make good decisions in real life.

An equation is a number sentence with an equals sign (=) showing that both sides are balanced. When a value is unknown, we use a letter such as x to stand for it. For example, a number plus 5 equals 12 is written x + 5 = 12.

To solve an equation, we keep it balanced: whatever we do to one side, we must do to the other. To remove the + 5, we subtract 5 from both sides: x + 5 − 5 = 12 − 5, so x = 7.

We do the opposite operation to undo. To solve x − 3 = 7, add 3 to both sides: x = 10. To solve 2x = 14 (two times x), divide both sides by 2: x = 7.

Always check by putting the answer back: for x + 5 = 12, 7 + 5 = 12 ✓. Equations model real life, such as "I had some money, spent KES 30, and KES 20 remained": x − 30 = 20, so x = 50.