Module 1 Study Guide: Proportional Relationships, Scale Factors, and Dimensional Scaling

Proportional Relationships and Direct Variation

  • Core Concept:

    • Two quantities form a proportional relationship when they change together at a constant rate and vary directly.

    • Governing equation: y=kxy = kx, where yy is the dependent variable, xx is the independent variable, and kk represents the constant of proportionality.

  • Constant of Proportionality (kk):

    • Calculated by dividing the dependent value by the matching independent value:     k=yxk = \frac{y}{x}

    • Remains constant across all data pairs, excluding the origin (0,0)(0, 0).

  • Graphical Criteria:

    • The graph must form a straight line (linear).

    • The line must pass directly through the origin (0,0)(0, 0).

Geometric Scale and Scale Factors

  • Key Definitions:

    • Scale: A ratio comparing dimensions in a drawing to actual original dimensions.

    • Scale Factor (SF\text{SF}): The constant multiplier applied to the side lengths of an original figure to produce a similar copy:     Scale Factor=CopyOld\text{Scale Factor} = \frac{\text{Copy}}{\text{Old}}

    • Enlargement: The figure expands to a larger size when Scale Factor>1\text{Scale Factor} > 1.

    • Reduction: The figure shrinks to a smaller size when 0<Scale Factor<10 < \text{Scale Factor} < 1.

    • Scale factors must always be positive (Scale Factor>0\text{Scale Factor} > 0).

Solving for Missing Dimensions

  • Step-by-Step Procedure:

    • Step 1 (Match Sides): Identify and match corresponding sides between the original figure and the scaled copy.

    • Step 2 (Find Scale Factor): Divide a known side length of the copy by the corresponding original side length:     Scale Factor=CopyOld\text{Scale Factor} = \frac{\text{Copy}}{\text{Old}}

    • Step 3 (Verify Magnitude): Confirm that the scale factor exceeds 11 for an enlargement or is between 00 and 11 for a reduction.

    • Step 4 (Calculate Missing Side): Multiply the known original side by the scale factor:     Unknown Copy Side=Old Side×Scale Factor\text{Unknown Copy Side} = \text{Old Side} \times \text{Scale Factor}

Multi-Dimensional Area Scaling

  • Area Scaling Principle:

    • Area is two-dimensional (length×width\text{length} \times \text{width}).

    • When side lengths scale by a scale factor, the area scales by the scale factor multiplied twice:     New Area=Old Area×SF×SF\text{New Area} = \text{Old Area} \times \text{SF} \times \text{SF}     New Area=Old Area×(SF)2\text{New Area} = \text{Old Area} \times (\text{SF})^2