Grade 10 Math Overview and Techniques

Oct 11, 2024

Grade 10 Math Lecture Notes

Introduction

  • One hour video covers the entire grade 10 math curriculum.
  • Aimed to reduce learning time from 110 hours to one hour.
  • Useful for review or introduction to grade-level math topics.

Solving Linear Systems

Methods

  1. Graphing

    • Rearrange equations to y = mx + b form.
    • Plot y-intercept and use slope to draw lines.
    • Identify intersection point as the solution.

  2. Elimination

    • Align equations to x + y = constant format.
    • Make coefficients of one variable equal in absolute value.
    • Add or subtract equations to eliminate a variable.
    • Solve for remaining variable.

  3. Substitution

    • Isolate one variable in an equation.
    • Substitute expression into the other equation.
    • Solve for one variable, then substitute back to find the other.

Example Problem

  • Solve the system: x - y = 5 and 3x + y = 3 using all three methods.
  • Solution: x = 2, y = -3.

Application of Linear Systems

  • Example: Determine the number of shoes sold.
    • Define variables for each type of shoe.
    • Create and solve system using given totals and prices.
    • Use elimination to solve for variables.

Midpoint and Distance

Midpoint

  • Formula: ((x1 + x2) / 2, (y1 + y2) / 2).
  • Example: Find midpoint between points A(5, -3) and B(-1, 5). Result: A midpoint of (2, 1).

Distance

  • Formula derived from Pythagorean Theorem.
  • Example: Find distance between points A and B. Result: 10 units.

Geometry

Triangles

  • Median: Line segment from a vertex to the midpoint of the opposite side.
  • Right Bisector: Line that is perpendicular to a line segment and passes through its midpoint.
  • Classifying Triangles: Determine if scalene, isosceles, or has a right angle.

Circles

  • Equation: x^2 + y^2 = r^2.
  • Example: Circle centered at origin, radius 6.
  • Determine if a point lies on, inside, or outside the circle by substituting into the equation.

Quadratic Functions

Forms

  1. Standard Form: Useful for y-intercept.
  2. Vertex Form: Useful for finding the vertex.
  3. Factored Form: Useful for x-intercepts.

Solving Quadratics

  • Methods include factoring, completing the square, and quadratic formula.
  • Example: Solve by factoring x^2 - 36 = 0. Solution: x = 6, -6.

Graphing Quadratics

  • Identify key properties: intercepts, vertex, and axis of symmetry.
  • Example: Graph y = x^2 + 8x + 12.

Trigonometry

Right Triangle Trigonometry

  • SOHCAHTOA: Ratios for right triangles.
  • Sin, Cos, Tan to find sides or angles using given side lengths.

Non-Right Triangle Trigonometry

  • Sine Law: Use when given two angles and a side or two sides and an angle not between them.
  • Cosine Law: Use when given three sides or two sides and the angle between them.
  • Examples provided for both sine and cosine laws.

This summarizes the techniques and examples for solving linear systems, geometric calculations involving midpoints, distances, and quadratics, as well as right and non-right triangle trigonometry covered in the lecture.