Grade 9 Math Culminating Notes PDF

1 Grade 9 math culminating notes By: Karam Al-Sakaji  2 Numbers - Natural numbers : Counting Numbers (N) - Whole numbers : Counting numbers and zero (W) - Integers : positive and negative...

1 Grade 9 math culminating notes By: Karam Al-Sakaji  2 Numbers - Natural numbers : Counting Numbers (N) - Whole numbers : Counting numbers and zero (W) - Integers : positive and negative whole numbers ( i ) - Rational numbers : numbers that can be expressed as a fraction where the numerator and denominator are integers (Q) - Real numbers : all the 4 numbers and numbers that non-terminating (never ending) and non-repeating Fractions - Compare Fractions with Fractions: Find the LCD, adjust fractions, then compare numerators. - Compare Fractions with Decimals: Convert fractions to decimals, then compare. - Convert Mixed/Improper Fractions: Mixed to improper: Multiply and add. Improper to mixed: Divide, quotient is the whole number. - Add/Subtract Fractions: Same denominator: Add/subtract numerators. Different denominators: Find LCD, adjust, then add/subtract. - Multiplying Fractions: Multiply the numerators and denominators, then simplify if needed. - Dividing Fractions: Flip the second fraction, multiply as usual, and simplify if needed. Powers - Working with Powers: Rewrite powers as repeated multiplication before evaluating. - Positive vs. Negative Solutions: Use the sign of the base and whether the exponent is odd/even to determine the result. - Multiplying Powers: Add exponents when multiplying powers with the same base  3 - Dividing Powers: Subtract exponents when dividing powers with the same base. - Power of a Power Rule: Multiply the exponents together. - Simplification: Combine like terms and apply exponent rules before evaluating. Power Rules -(Multiply) Ex: A^m x A^n= A^m+n to multiply powers with the same base. Keep the base and add the exponents -(Divide) Ex: A^m ÷ A^n = A^m+n -To divide powers with the same base, Keep the base and subtract the exponents Scientific Notation - A method to write very large or very small numbers using powers of 10. - Format: a × 10^n ○ a = Coefficient (1 ≤ aa < 10). ○ n = Exponent (shows decimal movement). - Positive nn: For large numbers, move decimal right. - Negative nn: For small numbers, move decimal left. Order of operations (BEDMAS) B = bracket, E = exponent, D or M = Division or multiplication (whichever comes first) A or S = Addition or Subtraction (whichever comes first)  4 Terms Coefficient: tells us how many of a term there are Monomial: has 1 term 5y Binomial: has 2 terms 7f + 2 Trinomial: has 3 terms ex: 4x + 3 + 9c Polynomial: the sum of 1 or more terms Distributive properties When a sum is multiplied by a number, each value in the sum is multiplied by the number separately and the products are added Ex: a ( b+c ) = ab+ ac -Distributive property : when a sum is multiplied by a number, each value in the sum is multiplied by the number separately and the products are added What is an Equation and Solving A statement showing two expressions are equal, using an (=) sign. Used to find the value of a variable Use opposite operations to isolate the variable. The goal is to make both sides equal. Linear Relations - A linear relation is a relationship between two variables that forms a straight line when graphed. It can be represented by a linear equation, in the form (y=ax+b), where a is the slope, and b is the y-intercept.  5 Rate of change (or slope) Measures the rate at which one variable is changing compared to the second variable (The change in y compared to the change in Slope = Special cases of lines and inequalities -A horizontal line has a slope of 0 -A Vertical line has no slope Linear inequalities Linear Systems The Point of Intersection is the solution to a system of linear equations and is where those lines cross. The point of intersection will be on both lines. Collection of data -Primary data:original data collected for a specific purpose.  6 -Secondary data:pre-existing data collected by others, used for a new purpose. Scatter plots: a chart used to display the relationship between two variables, with each variable's values represented by dots. The relationship can be positive (both variables increase together) or negative (one variable increases while the other decreases). It can also be linear (a straight-line pattern) or non-linear (a curved or irregular pattern), and the correlation can be strong (dots closely follow a trend) or weak (dots are spread out with little discernible pattern). Line of best fit To draw a line of best fit: - must be a straight line - must follow the trend of the data - should go through as many points as possible - should have (approximately) the same amount of data points above and below the line -Interpolation is estimating values within the range of existing data points. -Extrapolation is estimating values outside the range of existing data points. Correlation coefficient -Positive correlation means both variables move in the same direction, with values between 0 and +1. -Negative correlation means variables move in opposite directions, with values between 0 and -1.  7 Central tendency Mean is the average of a set of numbers, calculated by adding them all together and dividing by the number of numbers. Median is the middle value in a sorted list of numbers. Mode is the number that appears most in a dataset. Range is the difference between the highest and lowest numbers in a dataset. Boxplots -(The 5 number summary) Minimum: Smallest value. Q1 (First Quartile): is the median of the first half of the data set. Median (Q2): Middle value. Q3 (Third Quartile): the median of the second half of the data set. Maximum: Largest value. Interquartile Range (IQR): Difference between Q3 and Q1. Income Active Income: money earned in exchange for your time ex: salary Passive Income: money earned that continues to make money without any additional effort ex: dividends, interest, asset appreciation, affiliate marketing, Discretionary/Disposable Income: This is what you have left after you've paid all of your expenses. Budget: A budget is a plan that manages income and expenses to achieve financial goals. Unit Price, Better Buy, Taxes, and Discounts  8 Appreciation and Depreciation -Appreciation: When an object or investment increases in value by a percentage. -Depreciation: the term used to describe the loss of value of an item over time. Interest -Simple Interest Formula Formula: I = Prt I: amount of simple interest earned ($) P: principal (original) amount ($) r: interest rate (% as a decimal) t: time (months), years, or days Total Amount Formula Formula: A = P (1 + rt) A: final amount ($) P: principal (original) amount ($) r: annual interest rate (% as a decimal) t: time (months) -Compound interest Formula: A = P(1 + i)^n A: Final amount ($) (balance) P: Principal (original) amount ($) i: Interest rate (% as a decimal) (For credit cards, monthly interest rate) n: Number of compounding periods (For credit cards, months) n is the number of times interest is calculated Interest Paid Formula: Formula: I(total interest) = A (final amount) - P (Original amount)  9 Numbers P2 Fractions P2 Powers P2 Power Rules P3 Scientific Notation P3 Order of operations P3 Terms P4 Distributive Property P4 Equations P4 Linear Relations P4 Special cases of lines and inequalities P5 Lines and Inequalities P5 Linear Systems P5 Collection of Data P5 Scatter plot P6 Line of best fit P6 Correlation Coefficient P6 Central Tendency P7 Box plots P7 Income P7 Unit Price, Better Buy, P7 Appreciation and Depreciation P8 Interest P8