Algebra Lecture Notes

Key Concepts

Area vs. Volume

• Area: Measures 2D space.
  • Example: A square with sides 2 cm x 2 cm has an area of 4 cm².
  • Calculation involves multiplying length x width.
• Volume: Measures 3D space.
  • Example: A cube with dimensions 2 cm x 2 cm x 2 cm has a volume of 8 cm³ (Note: Initial calculation error corrected from 16 to 8).
  • Calculation involves multiplying length x width x height.

Understanding Axes in 3D Space

• In 2D: X-axis (horizontal) and Y-axis (vertical).
• In 3D: Adds Z-axis (appears diagonal, but represents depth).

Units and Exponents

• Area: cm x cm = cm².
• Volume: cm x cm x cm = cm³ (cubic centimeters).
• Exponents in Algebra: When multiplying like variables, add exponents (e.g., y³ from y² * y).*

Relation to Mass and Density

• Volume: Amount of space occupied.
• Mass: Amount of matter within the space.
• Density: Mass per unit volume (Density = Mass/Volume).
  • High density: More mass in the same volume (e.g., packed suitcase).
  • Low density: Less mass in the same volume (e.g., empty suitcase).

Conceptual Relationships

• Density Changes:
  • If volume remains constant and mass increases, density increases.
  • If volume remains constant and mass decreases, density decreases.
  • If mass remains constant and volume increases, density decreases.
  • If mass remains constant and volume decreases, density increases.
• Identifying Density Changes: If density increases, either mass increased or volume decreased (cannot determine which without additional information).

Units and Conversions

• Milliliters and Cubic Centimeters:
  • 1 milliliter (mL) = 1 cm³.
  • Used interchangeably, especially for liquids.

Problem-Solving Steps

1. Identify Given Information: List all known variables and their units.
2. Write Down the Formula: For density, use D = Mass/Volume.
3. Ensure Consistent Units: Convert units so they share the same prefix or have no prefix.
4. Plug in Values: Insert known values into the formula.
5. Solve: Calculate the unknown using the formula.

Example Problem

• Given: Mass = 3 kg, Density = 10.5 g/cm³
• Find Volume.
  • Convert 3 kg to 3,000 g.
  • Use formula Volume = Mass/Density.
  • Calculate: Volume = 3,000 g / 10.5 g/cm³ = 285.7 cm³.

Additional Notes

• Always ensure units are compatible before performing calculations.
• Understand the relationship between mass, volume, and density for accurate problem-solving.