Grade 10 core mathematics Numbers and Algebra – Real Numbers Notes

1. Classify whole numbers as odd, even, prime and composite in different situations.
2. Classify real numbers as rational and irrational in different situations.
3. Determine the reciprocal of real numbers by division.
4. Determine the reciprocal of real numbers by use of mathematical tables and calculators.
5. Apply reciprocals of real numbers in mathematical computations.
6. Promote the use of real numbers in day‑to‑day activities.

Key definitions and concepts

• Whole numbers: 0, 1, 2, 3, ... — each is either even (divisible by 2) or odd (not divisible by 2).
• Prime numbers: whole numbers greater than 1 whose only positive divisors are 1 and itself (e.g., 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...).
• Composite numbers: whole numbers greater than 1 that have more than two positive divisors (e.g., 4, 6, 8, 9, 12, 15...).
• Real numbers: include all rational and irrational numbers.
• Rational numbers: can be written as a fraction a/b, where a and b are integers and b ≠ 0. Decimal forms either terminate (0.75) or repeat (0.333...).
• Irrational numbers: cannot be written as such fractions. Decimal expansion is non‑terminating and non‑repeating (e.g., √2 ≈ 1.4142135..., π ≈ 3.14159...).
• Reciprocal (multiplicative inverse): for any nonzero real x, reciprocal is 1/x. Multiplying a number by its reciprocal gives 1.

Visual: Simple number line (whole numbers & integers)

Classification examples

• Odd / Even: 54 (even), 73 (odd). Example: if a farmer packs maize in 54 kg bags each containing 2 sacks, the total sacks per bag is even.
• Prime / Composite: 47 (prime), 48 (composite; 48 = 6 × 8). Note: 2 is the only even prime.
• Rational: 3/4, 0.25, 0.333... (1/3). Irrational: √2, π. For example, the diagonal of a square of side 1 m is √2 m — an irrational length.

Reciprocal — what it is and how to find it

Applying reciprocals in computations (practical Kenyan examples)

• Unit price: If a 50 kg bag of maize costs KSh 3,000, the cost per kg = 3,000 × (1/50) = 3,000 ÷ 50 = KSh 60. Here we used the reciprocal of 50.
• Sharing: Divide KSh 1,200 equally among 8 people: each gets 1,200 × (1/8) = 150.
• Speed/time: If distance = 120 km and speed = 60 km/h, time = distance × (1/speed) = 120 × (1/60) = 2 hours.
• Converting rates: If a bus fare is KSh 50 for 5 km, fare per km = 50 × (1/5) = 10 KSh/km.
• Algebra: Solve 5x = 35 → x = 35 × (1/5) = 7. We multiply by the reciprocal of 5.

Suggested learning experiences (classroom & community)

1. Number hunt (group activity): In small groups, students list 10 examples of numbers they encounter at home or in the market (prices, weights, lengths, bus fares). Classify each as whole, rational, irrational (if possible) and mark odd/even where appropriate.
2. Reciprocal practice with calculators and tables: Provide a worksheet with integers, fractions and decimals. Students find reciprocals by:
   • a) division on calculators (1 ÷ x),
   • b) if available, using printed reciprocal tables,
   • c) by flipping fractions (e.g., 3/7 → 7/3).
3. Real‑life problem solving: Give real Kenyan contexts — splitting harvests, computing unit prices at a local kiosk, converting quantities for recipes — and ask learners to use reciprocals to find unit rates and shares.
4. Whiteboard race (classification): Teams race to classify a rapid list of numbers called out by the teacher (odd/even, prime/composite, rational/irrational). Points for correct classification and quick explanation.
5. Number line construction: Each learner draws a number line and places examples of rational (fractions/decimals) and marks an approximation for an irrational number like √2 or π.
6. Extension: Show how reciprocals are used in solving linear equations and ratios; link to everyday budgeting and measurement tasks.

Classroom assessment tasks (formative)

• Give 12 numbers: classify each as odd/even, prime/composite and rational/irrational (where applicable). Mark and discuss mistakes.
• Worksheet: find reciprocals of mixed list — integers, simple fractions, and decimals. Check using calculators.
• Apply: A shopkeeper sells 3 identical tins of cooking fat for KSh 270. What is the price of one tin? (Show solution using reciprocal.)
• Practical homework: at home, record a list of five different quantities (e.g., petrol volume, sugar weight, bus fare, phone credit). Classify and compute unit rates using reciprocals.

Sample exercises (with answers)

1. Classify 17, 18, 0, 1, 25. (Answers: 17 prime, odd; 18 composite, even; 0 whole, even by definition; 1 neither prime nor composite; 25 composite, odd.)
2. Identify rational or irrational: 0.125, 2/7, √3, π. (Answers: rational, rational, irrational, irrational.)
3. Find reciprocals: (a) 5 → (b) 2/5 → (c) -0.25. (Answers: (a) 1/5 or 0.2; (b) 5/2 or 2.5; (c) -4.)
4. Market problem: 3 kg of beans cost KSh 330. Find cost per kg. (Solution: 330 × (1/3) = KSh 110.)

Resources & teacher tips

• Use simple calculators and printed reciprocal tables where available (some national exam practice uses tables).
• Relate examples to local contexts (market stalls, school stores, farm yields, bus fares) to increase relevance for Kenyan learners.
• Encourage students to practise flipping fractions for reciprocals and checking on calculators.
• Address common misconceptions: 0 has no reciprocal; 1 is its own reciprocal (1 × 1 = 1); negative reciprocals keep the sign.