maths paper 1
Whole Numbers
1. Definition of Whole Numbers
Non-negative integers including zero: (0, 1, 2, 3, ...).
2. Key Terminology
Place Value: Value of a digit based on its position in a number (e.g., in 345, 3 is in the hundreds place, 4 in the tens, and 5 in the units).
Face Value: The actual value of the digit itself (e.g., in 345, the face value of 4 is 4).
Number Value: Combined value of a number based on place and face values.
3. Prime and Composite Numbers
Prime Numbers: Numbers greater than 1 that have no divisors other than 1 and themselves (e.g., 2, 3, 5, 7).
Composite Numbers: Numbers greater than 1 that have more than two divisors (e.g., 4, 6, 8, 9).
4. Rounding Off
Rules for Rounding Numbers: Round to nearest whole number, tenths, or hundredths.
Example: Round 3.67 to the nearest whole number by looking at the digit after the decimal (6) and rounding up to 4.
5. Exponents and Roots
Exponents: Indicate how many times a number (base) is multiplied by itself (e.g., 2^3 = 2 × 2 × 2 = 8).
Square Roots: The number that, when multiplied by itself, gives the original number (e.g., square root of 9 = 3).
6. Exercises
Practice Exercises: Ex. 1.1, 1.3, 1.4, 1.5 on prime and composite numbers, place values, rounding numbers, and exponents.
BODMAS and Patterns
1. BODMAS
Order of Operations: Brackets, Orders (exponents), Division and Multiplication (left to right), Addition and Subtraction (left to right).
Example: In 2 + 3 × (4 - 1), first solve brackets: 4 - 1 = 3, then multiplication: 3 × 3 = 9, and finally addition: 2 + 9 = 11.
2. Recursive and Function Rules
Recursive Rule: A sequence where each term is derived from the previous terms (e.g., Fibonacci sequence).
Function Rule: A mathematical expression that relates input to output.
3. Geometric Patterns
Patterns involving shapes, where each term follows a specific rule (e.g., triangles, squares).
4. Flow Diagrams
Visual representations to show the steps in a process or the flow of operations.
5. Exercises
Practice Problems: Ex. 2.1, 2.2, 2.4 on BODMAS, recursive rules, and geometric patterns.
Addition and Subtraction
1. Basic Operations
Combination and separation of whole numbers using addition and subtraction.
Strategies for Addition: Grouping and using number lines.
2. Exercises
Practice Exercises: Ex. 3.1, 3.3, 3.6 on addition and subtraction problems.
Word Problems: Ex. 19.4, 19.6, 19.7, 19.8, 19.10 on complex addition/subtraction.
Common Fractions
1. Comparing Fractions
Use <, >, or = to compare fractions by finding a common denominator or converting to decimals.
2. ‘Of’ Sums
Finding a fraction of a whole (e.g., 1/4 of 20 = 5).
Formula: (1/2) × 50 = x; divide diagonally.
3. Equivalent Fractions
Different fractions representing the same value (e.g., 1/2 = 2/4).
4. Simplification of Fractions
Reducing fractions to their simplest form (e.g., 4/8 = 1/2 by dividing by the largest common factor).
5. Conversions Between Fractions
Changing between improper fractions and mixed numbers.
6. Add/Subtract Fractions
Finding a common denominator before using operations.
7. Exercises
Practice Problems: Ex. 4.1, 4.5, 4.6, 4.7, 4.8, 4.9, 4.10 on fractions.
Multiplication and Division
1. Multiples and Factors
Multiples: Numbers obtained by multiplying a given number by integers (e.g., multiples of 3: 3, 6, 9, ...).
Factors: Numbers that divide another number without leaving a remainder (e.g., factors of 12: 1, 2, 3, 4, 6, 12).
2. Vertical Multiplication
Arranging numbers vertically for easier multiplication.
Steps:
Write numbers vertically, aligning by the right.
Multiply each digit in the bottom number by the top number.
Add results, shifting left as necessary.
Example: 23 × 4 = 92.
3. Divisibility Rules
Rules to Determine Divisibility:
Divisible by 2: Last digit even.
Divisible by 3: Sum of digits divisible by 3.
Divisible by 4: Last two digits divisible by 4.
Divisible by 5: Last digit is 0 or 5.
Divisible by 6: Divisible by both 2 and 3.
Divisible by 7: Double last digit, subtract from rest; divisible by 7 if the result is valid.
Divisible by 8: Last three digits divisible by 8.
Divisible by 9: Sum of digits divisible by 9.
Divisible by 10: Last digit is 0.
Divisible by 11: Difference between sums of odd and even positions’ digits divisible by 11.
4. Long and Short Division
Long division for larger numbers; short division for smaller numbers.
5. Exercises
Practice Problems: Ex. 10.3, 10.4, 10.6, 10.7 on multiplication.
Division Exercises: Ex. 14.1 to 14.7.
Decimals and Percentages
1. Percentages Overview
Converting Fractions to Percentages
Multiply the fraction by 100, simplify if necessary, add the % symbol.
Example: (3/4) × 100 = 75%.
2. Converting Percentages to Fractions
Write percentage over 100 and simplify.
Example: 25% = 25/100 = 1/4.
3. Converting Percentages to Decimals
Remove % symbol and divide by 100.
Example: 35% = 0.35.
4. Converting Decimals to Percentages
Multiply by 100 and add % symbol.
Example: 0.72 = 72%.
5. Finding the Percentage of a Whole Number
Convert percentage to decimal, multiply by the whole number.
Example: 30% of 80 = 0.30 × 80 = 24.
6. Profit and Loss Overview
Key Concepts
Cost Price (CP): Price at which an item is bought.
Selling Price (SP): Price at which an item is sold.
Profit: When SP > CP.
Loss: When SP < CP.
Formulas for Profit and Loss
Profit = SP - CP
Loss = CP - SP
Profit Percentage = (Profit ÷ CP) × 100%
Loss Percentage = (Loss ÷ CP) × 100%.
7. Calculating Profit and Loss
Examples illustrating calculations for both profits and losses.
8. Practice Problems
Solve given problems showing calculations for profits and losses.
9. Tips for Solving Profit and Loss Problems
Identify CP and SP, determine profit or loss, use formulas, and double-check calculations.
Number Patterns
1. Definition of Number Patterns
A sequence of numbers that follows a specific rule.
Use operations: addition, subtraction, multiplication, and division.
2. Types of Rules for Number Patterns
Recursive Rule: Find next number based on previous numbers.
Example: Start with 5, rule is add 8: 5, 13, 21, 29, ...
Function Rule: Predict numbers based on their position.
Example: For sequence (4, 7, 10, 13, 16), find 20th number using rule (× 3 + 1).
3. Fibonacci Rule
Each number is the sum of the two before it (e.g., (0, 1, 1, 2, ...)).
Finding the 20th number explained through additions of earlier numbers.
4. Conclusion
Tips to identify rules and understand patterns without writing all numbers.