9th Grade Math: Whole Numbers

• Whole Numbers*

Properties of Whole Numbers

• When two or more whole numbers are added, the result is always a whole number.
• When two or more whole numbers are multiplied, the result is always a whole number.
• The system of whole numbers is closed under addition and multiplication.
• 0 is the identity element for addition, meaning that when 0 is added to a number, the result is the same number.

Types of Numbers

• Natural numbers: counting numbers, excluding 0.
• Whole numbers: natural numbers, including 0.
• Integers: whole numbers, including negative numbers.
• Rational numbers: numbers that can be expressed as a fraction of two integers.
• Irrational numbers: numbers that cannot be expressed as a fraction of two integers.
• Real numbers: numbers that include both rational and irrational numbers.
• Calculations with Whole Numbers*

Estimating and Rounding

• Estimating: getting close to an answer without performing precise calculations.
• Rounding: simplifying numbers to make calculations easier.

Addition and Subtraction in Columns

• Addition and subtraction can be performed by writing numbers in columns and adding or subtracting each digit starting from the rightmost digit.

Multiplication and Division

• Multiplication can be performed using expanded notation or a shorter method.
• Long division: dividing a number step-by-step, subtracting multiples of the divisor from the dividend, and continuing until the remainder is less than the divisor.
• Multiples and Factors*

Lowest Common Multiple (LCM) and Highest Common Factor (HCF)

• Consecutive multiples: multiples of a number, obtained by multiplying the number by 1, 2, 3, etc.
• Common multiples: multiples of two or more numbers.
• LCM: the smallest common multiple of two or more numbers.
• HCF: the largest number that divides each of the numbers without leaving a remainder.
• Solving Problems*

Ratio and Rate

• Ratio: a comparison of two quantities.
• Rate: a ratio of two quantities, often expressed as a unit rate.

Financial Contexts

• Discount: a percentage reduction in the original price.
• Profit: the difference between the selling price and the cost price.
• Loss: the difference between the cost price and the selling price.

Powers and Roots

• Powers: repeated multiplication of a number.
• Roots: the opposite of powers, finding the original number.
• Integers*

Adding and Subtracting Integers

• Adding a negative number is equivalent to subtracting a positive number.
• Subtracting a negative number is equivalent to adding a positive number.
• The sum of two negative numbers is negative.
• The sum of a positive and a negative number is positive if the positive number is larger, and negative if the negative number is larger.

Multiplying and Dividing Integers

• The product of two negative numbers is positive.
• The product of a positive and a negative number is negative.
• The quotient of a positive and a negative number is negative.
• The quotient of two negative numbers is positive.

Properties of Integers

• The additive inverse of a number is the number that, when added to the original number, results in 0.
• The sum of a number and its additive inverse is 0.
• The product of a number and its additive inverse is 1.
• Exponents*

Laws of Exponents

• The product of two numbers with the same base is equal to the base raised to the sum of the exponents.
• The quotient of two numbers with the same base is equal to the base raised to the difference of the exponents.
• The power of a product is equal to the product of the powers.
• The power of a quotient is equal to the quotient of the powers.

Scientific Notation

• A shorthand method for writing very large or very small numbers.
• Written in the form ±a × 10^n, where a is a decimal number between 1 and 10, and n is an integer.
• Patterns*

Geometric Patterns

• Patterns formed by combining shapes, such as squares, triangles, and circles.
• Identifying constants and variables in patterns.

Number Patterns

• Identifying and extending patterns in sequences.
• Using formulae to describe patterns.

Algebraic Expressions

Terminology

• Monomial: an expression with one term.
• Binomial: an expression with two terms.
• Trinomial: an expression with three terms.
• Variable: the symbol representing the unknown value.
• Coefficient: the number multiplying the variable.
• Constant: a fixed value in the expression.

Equivalent Expressions

• Expressions with different sequences of operations but the same numerical value.

Conventions for Writing Algebraic Expressions

• The multiplication sign is often omitted.
• Known numbers are written first in a product.
• Addition and subtraction are performed from left to right.
• Multiplication and division are performed before addition and subtraction.

Simplifying Expressions

• Combining like terms.
• Applying the distributive property.

Multiplying and Dividing Algebraic Expressions

• Multiplying polynomials by monomials.

• Simplifying expressions using division.

• Evaluating squares, cubes, and roots of monomials.### Distributive Property of Division

• The distributive property of division can be expressed as: (x + y)/z = x/z + y/z and (x - y)/z = x/z - y/z

• This property is used to divide a sum or difference by a number

Factorizing Algebraic Expressions

Expansion of Binomials

• To expand the product of two binomials, apply the distributive property
• (x + a)(x + b) = x^2 + (a + b)x + ab
• (x - y)(x + z) = x^2 + (z - y)x - yz

Expansion of Binomials and Squares

• The square of a binomial (a + b) or (a - b) can be expressed as:
  • (a + b)^2 = a^2 + 2ab + b^2
  • (a - b)^2 = a^2 - 2ab + b^2

Substitution into Algebraic Expressions

• Evaluate expressions by substituting specific values for variables to check for equivalence and simplification
• Examples of algebraic expressions for substitution:
  • (a + b)(c + d)
  • ax^2 + bx + c + dx^2 + ex + f
  • g(ax^2 - bx + c) - h(dx - e)
  • jx^2 + kx - l
  • mx^2 - nx - o
  • px + q

Functions and Relations

Input and Output Values

• Set A: Natural numbers smaller than 10: 1, 2, 3, 4, 5, 6, 7, 8, 9
• Set B: Multiples of 10 between 20 and 90: 20, 30, 40, 50, 60, 70, 80, 90

Calculating Outputs from Inputs

• When using a rule, multiply a number by 5 and subtract the result from 50
• Apply the rule to different sets and determine the output numbers

Application to Even Numbers

• Consider even numbers (e.g., 2, 4, 6, 8, 10) and apply different rules (e.g., 2n + 1, 2n - 1, 2n + 5, 3n + 1)

Equivalent Forms

• Determine the output numbers for each set when the rule is applied
• Analyze the output types:
  • Applying the rule x - 1000 to natural numbers smaller than 1000 results in negative numbers
  • Applying the rule x/10 + 10 to natural numbers smaller than 10 results in positive numbers between 10 and 11
  • Applying the rule 30x + 2 to positive fractions with denominators 2, 3, and 5 results in positive numbers greater than 2

Representation of Functions

• A relationship between two variables in which there is only one output number for each input number is called a function
• Functions can be represented in different ways:
  • Table: Shows some values of the two variables and clearly indicates which output value corresponds to each input value
  • Flow Diagram: Illustrates the calculations needed to determine the output number for a given input variable
  • Formula: Describes the calculations to be done to determine the output number for a given input variable
  • Graph: Provides a visual representation of the relationship