Math Final Exam Tips

Overview

This lecture walks through a Grade 9 math final exam, providing step-by-step solutions and tips on exponent laws, algebra, linear relations, geometry, and data analysis.

Exponents & Algebra Basics

Any number (except 0) to the power of 0 equals 1.
When multiplying powers with the same base, add exponents: ( a^{10} \times a^{5} = a^{15} ).
For a power of a product, apply the exponent to both factors: ( (3x^4)^2 = 9x^8 ).
Dividing powers with the same base, subtract exponents: ( 7^6 / 7^3 = 7^3 ).
Negative exponents indicate reciprocals: ( 5^{-2} = 1/25 ).
Like terms have identical variables and exponents; coefficients do not matter.

Polynomials

Terms are separated by + or −; three terms = trinomial.
Degree of a polynomial is highest degree among its terms.
To combine like terms, add/subtract coefficients.

Simplifying Expressions & Equations

Expand by distributing multiplication over addition/subtraction.
Collect and simplify like terms after expansion.
To solve equations, use inverse operations to isolate the variable.
Cross-multiply to solve equations with fractions.
Always check solutions by substituting back into the original equation.

Linear Relations & Functions

A direct variation has the form ( y = mx ); partial variation includes a constant, ( y = mx + b ).
Slope (m) is "rise over run" or ( (y_2 - y_1) / (x_2 - x_1) ).
Horizontal lines: ( y = c ); vertical lines: ( x = c ).
Parallel lines have equal slopes; perpendicular lines have negative reciprocal slopes.
Use first differences to test for linearity in tables: constant difference signals linearity.

Geometry: Area, Perimeter, Volume, and Angles

Perimeter: add all side lengths.
Area of rectangle: length × width; area of triangle: ( 0.5 \times \text{base} \times \text{height} ).
Volume of prism: area of base × height; volume units are cubed.
Surface area of sphere: ( 4\pi r^2 ); cone: ( \pi r s + \pi r^2 ).
Pythagorean Theorem: ( a^2 + b^2 = c^2 ) for right triangles.
Angles on a straight line add to 180°; vertically opposite angles are equal.
Co-interior angles (C-pattern) add to 180°; alternate angles (Z-pattern) are equal.

Data & Graphs

Primary data: collected yourself; secondary data: collected by someone else.
Extrapolation: estimating beyond known data; interpolation: within known data.
Line of best fit shows trends; interpolation/extrapolation use this line for prediction.
Types of sampling: simple random, systematic, stratified, non-random.

Key Terms & Definitions

Exponent Rule — Laws for simplifying expressions with exponents (add for multiplication, subtract for division).
Like Terms — Terms with exactly the same variable parts.
Direct Variation — Linear relationship with no constant term (( y = mx )).
Partial Variation — Linear relationship with constant term (( y = mx + b )).
Slope (m) — Measures steepness of a line, calculated as rise/run.
Intercept — Where the line crosses an axis (x-intercept or y-intercept).
Perimeter — Distance around a shape.
Area — Space within a boundary.
Volume — Space within a 3D object.
Surface Area — Total area of all surfaces of a 3D object.
Pythagorean Theorem — Relation of sides in a right triangle.
Primary Data — Data collected firsthand.
Extrapolation/Interpolation — Predicting data outside/inside known values.

Action Items / Next Steps

Review exponent laws and practice simplifying expressions.
Practice graphing linear relationships and solving for slope/intercepts.
Complete practice problems on perimeter, area, volume, and angle rules.
Review types of data sampling and graph interpretation.
Check and correct any wrong answers by substituting into the original equations.