Grade 10 Essential Mathematics Study Notes

Welcome to Numbers and Algebra

You use numbers every single day — checking change at the shop, weighing fruit at the market, reading the time, sending money on M-PESA. This strand takes the numbers you already know and shows you how to sort them, operate on them, and use letters in place of unknown numbers to solve problems faster.

What this strand covers

• Real Numbers — sorting numbers into odd, even, prime, composite, rational and irrational; doing combined operations on them; and finding reciprocals with a calculator.
• Indices — using powers as a short way of writing repeated multiplication, and the laws that go with them.
• Quadratic Equations — forming and solving equations of the form ax² + bx + c = 0, often from real-life problems on area and dimensions.

Lesson focus — Real Numbers

This sub-strand classifies all the numbers you have ever met — whole numbers, fractions, decimals, square roots — into a tidy system, and then shows you how to combine them in calculations.

Classifying real numbers as rational or irrational

Real numbers are all the numbers we use in everyday counting, measuring and calculating — whole numbers, negative numbers, fractions, decimals, recurring decimals and square roots all live inside this set.

Before we sort real numbers into rational and irrational, we first warm up with two simpler ways to sort whole numbers — by odd/even, and by prime/composite.

Three sortings to learn in this lesson

1. Odd vs even — by the last digit of a whole number.
2. Prime vs composite — by how many factors a whole number has.
3. Rational vs irrational — by whether the number can be written as a fraction.

Odd and even numbers

An even number is a whole number that can be divided exactly by 2, with nothing left over. An odd number is a whole number that, when divided by 2, leaves a remainder of 1.

The quick test — look at the ones digit

• Ends in 0, 2, 4, 6 or 8 → even.
• Ends in 1, 3, 5, 7 or 9 → odd.

Worked example

Classify 746, 1 253, 9 080, 4 517, 62 344 and 88 199 as odd or even.

Why this matters in real life

Even numbers can always be shared equally between two people. Odd numbers cannot — there will always be one left over. This is why we use it for grouping items into pairs and checking ticket or seat numbers.

Prime numbers and composite numbers

Step 1 — What is a factor?

A factor of a number is a whole number that divides into it exactly, leaving no remainder.

Example — find the factors of 12. Try dividing 12 by 1, 2, 3, 4, 5, 6 … and see which divide exactly:

12 ÷ 1 = 12 ✓   12 ÷ 2 = 6 ✓   12 ÷ 3 = 4 ✓   12 ÷ 4 = 3 ✓   12 ÷ 5 = 2 r 2 ✗   12 ÷ 6 = 2 ✓   12 ÷ 12 = 1 ✓

So the factors of 12 are 1, 2, 3, 4, 6 and 12 — six factors in total.

Step 2 — Sort whole numbers by counting factors

Once we know how to list factors, we can sort whole numbers (greater than 1) into two groups:

• A prime number has exactly two factors — 1 and itself.
• A composite number has more than two factors.
• The number 1 is neither prime nor composite — it has only one factor (itself).

Step 3 — Worked example: classify several numbers

Classify 13, 15, 17, 22 and 29 as prime or composite.

Solution
For each number, list its factors and count them.

Step 4 — Worked example: grouping 24 learners

A teacher intends to divide 24 learners into equal groups for a class activity. She wants to know whether 24 is prime or composite, so she can decide how to group them.

Solution
List the factors of 24:
24 ÷ 1 = 24,   24 ÷ 2 = 12,   24 ÷ 3 = 8,   24 ÷ 4 = 6,   24 ÷ 6 = 4,   24 ÷ 8 = 3,   24 ÷ 12 = 2,   24 ÷ 24 = 1.

Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24 — eight factors.

Since 24 has more than two factors, 24 is composite.

Because 24 is composite, the teacher has many ways to group the learners equally: 2 groups of 12, 3 groups of 8, 4 groups of 6, 6 groups of 4, 8 groups of 3, or 12 groups of 2.

Step 5 — Worked example: arranging 17 books

A librarian has 17 new books to arrange on shelves. He wants to put the same number of books on each shelf, using more than 1 shelf but fewer than 17 shelves. Can he do this?

Solution
List the factors of 17:
17 ÷ 1 = 17,   17 ÷ 17 = 1. No other whole number divides 17 exactly.

Factors of 17 = 1, 17 — only two factors.

So 17 is prime. The only equal arrangements are 1 shelf of 17 or 17 shelves of 1 — neither of which fits his condition.

The librarian cannot arrange the 17 books equally under his rule.

Why this matters in real life

Composite numbers share evenly; prime numbers refuse to share. This is why prime numbers are used in passwords and computer security — they are hard to break into smaller pieces.

Classification of real numbers as rational or irrational

All real numbers fall into one of two groups. The test is whether the number can be written as a fraction.

• A rational number is any number that can be written in the form p/q, where p and q are integers and q ≠ 0.
• An irrational number is a number that cannot be written in the form p/q.

Examples of rational numbers

• Whole numbers — e.g. 9 = 9/1, –12 = –12/1.
• Simple fractions — e.g. 1/2, 5/4.
• Terminating decimals — e.g. 1.8 = 18/10, 0.5 = 1/2.
• Recurring decimals — e.g. 0.333… = 1/3.

Examples of irrational numbers

• π (pi) — the ratio of a circle's circumference to its diameter, 3.14159…
• √5, √2 — square roots of numbers that are not perfect squares.
• Decimals that go on forever without repeating, e.g. 2.874690156…

Worked example

Why this matters in real life: every shop price (Ksh 25.50), every weight on a scale (1.5 kg), every quantity in a recipe (¾ cup) and every distance you walk to school is a rational number. Irrational numbers show up when you measure the diagonal of a square, a circle's circumference, or use GPS coordinates that don't settle into a pattern.

Terminating, recurring and non-terminating decimals

Every real number can be written as a decimal — but the decimals come in three different shapes. The shape tells you instantly whether the number is rational or irrational.

• Terminating decimal — the digits stop. E.g. 1/4 = 0.25. Always rational.
• Recurring decimal — the digits go on forever, but a pattern repeats. E.g. 1/3 = 0.333…. Always rational.
• Non-terminating, non-repeating decimal — forever, no pattern. E.g. π = 3.14159…, √2 = 1.41421…. Always irrational.

Square roots — a quick way to spot irrational numbers

The square root of a perfect square (1, 4, 9, 16, 25, 36, …) is rational. For example, √36 = 6.

The square root of any other whole number is irrational. For example, √2, √3, √5, √7 — none of these can be written as a fraction.

Why this matters: the diagonal of a square with side 1 m is √2 m — irrational. The circumference of a circle with diameter 1 m is π m — irrational. We round in practice (π ≈ 3.14, √2 ≈ 1.41).

Performing combined operations on rational numbers

Part 1 — Addition and subtraction

A combined operation uses more than one operation in a single expression. To get the right answer, follow BODMAS:

• Brackets
• Orders (powers and roots)
• Division and Multiplication — left to right
• Addition and Subtraction — left to right

Worked example — shopping

Alex bought 3 exercise books at Ksh 45.50 each, 2 pens at Ksh 25.75 each, and 1 mathematical set at Ksh 120.25. Calculate the total amount he spent.

Solution
Cost of exercise books = 3 × Ksh 45.50
= Ksh 136.50

Cost of pens = 2 × Ksh 25.75
= Ksh 51.50

Cost of the mathematical set = 1 × Ksh 120.25
= Ksh 120.25

Total cost = Ksh 136.50 + Ksh 51.50 + Ksh 120.25
= Ksh 308.25

Alex spent a total of Ksh 308.25 on stationery.

Worked example — a simple budget

For a school trip, Mary received Ksh 1 200 from her parents and Ksh 500 from her aunt. She spent Ksh 650 on snacks and Ksh 425 on bus fare. How much was left?

Solution
Total received = Ksh 1 200 + Ksh 500
= Ksh 1 700

Total spent = Ksh 650 + Ksh 425
= Ksh 1 075

Balance = Ksh 1 700 – Ksh 1 075
= Ksh 625

Mary was left with Ksh 625.

How to attack a word problem

1. Read it twice. Underline the numbers and circle the operations ("more", "less", "total", "remained" tell you which).
2. Label each step in words first, then write the calculation.
3. Apply BODMAS step by step.
4. Check the answer makes sense in the story. This is exactly how a budget is built — list income, list expenditure, then add and subtract to see what is left.

 Performing combined operations (multiplication and division)

BODMAS still rules, but now we add multiplication and division of fractions and decimals. Two reminders:

• Multiply fractions: tops × tops, bottoms × bottoms — e.g. (2/3) × (4/5) = 8/15.
• Divide by a fraction: multiply by its reciprocal (flip it) — e.g. (3/4) ÷ (1/2) = (3/4) × (2/1) = 3/2.

Worked example — repackaging flour

Amina bought 4½ bags of maize flour. Each bag had a mass of 10.5 kg. She repacked the flour into 1.75 kg packets. How many packets did she get?

Solution
Total mass = 4½ × 10.5 kg
= 4.5 × 10.5
= 47.25 kg

Number of packets obtained = 47.25 ÷ 1.75
= 27 packets

Amina obtained 27 packets.

Worked example — temperature change

The room temperature at 6:00 a.m. was 19.2°C. By 2:00 p.m. it had risen by 5.7°C. By 8:00 p.m. it had dropped by half of the morning rise. Determine the temperature at 8:00 p.m.

Solution
Temperature at 2:00 p.m. = 19.2°C + 5.7°C
= 24.9°C

Drop in temperature by 8:00 p.m. = ½ × 5.7°C
= 2.85°C

Temperature at 8:00 p.m. = 24.9°C – 2.85°C
= 22.05°C

The temperature at 8:00 p.m. was 22.05°C.

Worked example — all four operations in one

A trader has Ksh 2 000. She buys 3 packets of flour at Ksh 180 each, then sells half of one packet for Ksh 120. How much money does she have left?

Solution
Money left = 2 000 – 3 × 180 + 120 ÷ 2
= 2 000 – 540 + 60 (× and ÷ first)
= 1 460 + 60
= Ksh 1 520

The trader has Ksh 1 520 left.

 Reciprocal of real numbers using calculators

The reciprocal of a number is what you multiply that number by to get 1. The reciprocal of x is 1/x.

• Reciprocal of 4 = 1/4 = 0.25.
• Reciprocal of 2/5 = 5/2 = 2.5 (just flip the fraction).
• Reciprocal of 0.25 = 1/0.25 = 4.
• Reciprocal of –1.5 = –2/3 ≈ –0.6667.

Numbers and their reciprocals always multiply to give 1: 4 × 1/4 = 1; 2/5 × 5/2 = 1.

The book shows two calculator methods. The next two blocks walk through each one in order.

 Method A: Using a calculator with a "1/x" or x⁻¹ button

Most scientific calculators have a dedicated reciprocal key, marked 1/x or x⁻¹. On many models this is the SHIFT (or 2nd) function — press SHIFT first, then the key.

The five steps

1. Turn the calculator on.
2. Enter the number — for example, 4.
3. Press the x⁻¹ key (press SHIFT first if needed).
4. Press =. The display shows 0.25.
5. To see the answer as a fraction, press the a b/c key.

Worked example — find the reciprocal of 7

Solution
Enter the number 7 in the calculator.
Press the x⁻¹ button, then press =.
The reciprocal of 7 is 0.142857142…

Worked example — find the reciprocal of 2/5

Solution
Enter the number 2/5 in the calculator.
Press the x⁻¹ button, then press =.
The reciprocal of 2/5 is 2.5 (or 5/2 as a fraction).

Worked example — find the reciprocal of –1.5

Solution
Enter the number –1.5 in the calculator.
Press the x⁻¹ button, then press =.
The reciprocal of –1.5 is –0.6666…
Press the a b/c key to display it as –2/3.

 Method B: Using a calculator without the special button

If your calculator has no x⁻¹ key (a basic calculator), just calculate 1 ÷ the number directly.

The four steps

1. Turn the calculator on. Press 1.
2. Press ÷.
3. Enter the number — for example, 2.
4. Press =. The reciprocal of 2 appears as 0.5.

Worked example — find the reciprocal of 0.25

Solution
Press 1 followed by the ÷ sign.
Enter the number 0.25.
Press the = button.
The reciprocal of 0.25 is 4.

Why both methods give the same answer: the reciprocal of x is just 1/x. Method A uses the key; Method B types the division by hand.

Why reciprocals matter in real life: if a car covers 60 km in 1 hour, the time for 1 km is the reciprocal: 1/60 hour = 1 minute. The reciprocal also gives distance per unit time: if a runner takes 6 minutes to cover 1 km, her speed is 1/6 km per minute. So reciprocals connect time, speed and distance — turning a "per X" rate into a "per 1" rate, and back again.