Scaled Copies and Scale Factors

Overview

This lecture covers key concepts from seventh grade mathematics Unit 1, Lessons 1–12, focusing on scaled copies, scale factors, area, perimeter, and interpreting and creating scale drawings. The notes include definitions, step-by-step examples, visuals (described and illustrated), and strategies for solving problems.

Scaled Copies and Scale Factors

Definition:
A scaled copy is a figure where all lengths are changed by the same scale factor, but the shape remains the same (the figures are similar).
Scale Factor:
The constant number used to multiply all lengths in the original figure to create the scaled copy.
- If the scale factor is greater than 1, the figure is enlarged.
- If the scale factor is less than 1, the figure is reduced.
Angles:
Angles remain unchanged in scaled copies; only the lengths change.
Visual Example:

Original "A" shape: 2 units tall, 2 units wide.
Scaled copy with scale factor 2: 4 units tall, 4 units wide (all sides doubled).
Not a scaled copy: If only the height or width changes, or if the shape is distorted.
Step-by-Step: How to Make a Scaled Copy
1. Identify the scale factor.
2. Multiply each side length of the original figure by the scale factor.
3. Draw the new figure with the calculated side lengths.
4. Check that all angles remain the same.

Identifying and Creating Scaled Copies

How to Check for a Scaled Copy:
1. Compare corresponding side lengths. All must be multiplied by the same number.
2. Check that angles remain the same.
3. The shape must remain similar (no stretching or skewing).
Perimeter and Area:
- Perimeter of scaled copy = original perimeter × scale factor.
- Area of scaled copy = original area × (scale factor)².
Example:
- Original quadrilateral: sides 6, 9, 9, 12.
- Scaled copy with shortest side 2:
  - Scale factor = 2/6 = 1/3.
  - New side lengths: 2, 3, 3, 4.
  - Perimeter = 2 + 3 + 3 + 4 = 12 units.
Drawing Scaled Copies:
- Multiply each side length by the scale factor.
- For a scale factor of ½, halve each side; for 3, triple each side.
Visual Example:

Original polygon: 3 units by 2 units.
Scale factor 2: New polygon is 6 units by 4 units.
Step-by-Step: Drawing a Scaled Copy
1. Write down the side lengths of the original figure.
2. Multiply each side by the scale factor.
3. Draw the new figure using the new side lengths.
4. Label corresponding points to check similarity.

Corresponding Parts and Ratios

Corresponding Parts:
Points, sides, and angles in the original and scaled figure that match in position.
- Example: In polygons, label matching vertices (A with P, B with Q, etc.).
Ratios:
- Equivalent ratios are found by multiplying or dividing both terms by the same number.
- Example: A 6:5 ratio (pink to blue paint) is equivalent to 12:10 or 18:15 for 2 or 3 batches.
Visuals:

Use colored pencils or labels to highlight corresponding sides and points.
Draw lines between matching points on original and scaled figures.
Step-by-Step: Finding Equivalent Ratios
1. Write the original ratio.
2. Multiply or divide both numbers by the same value.
3. Check if the new ratio matches the context (e.g., number of batches).

Scale Drawings and Maps

Using Scales:
- A scale drawing uses a ratio (e.g., 1 cm : 10 m) to represent real objects.
- To find actual measurements, multiply the drawing measurement by the scale factor.
- To find drawing size for a new scale, divide the real measurement by the new scale.
Changing Scales:
- If the scale changes (e.g., from 1 cm : 1 m to 1 cm : 2 m), adjust drawing dimensions proportionally.
- New drawing will be smaller if each unit represents more real-world distance.
Area in Scale Drawings:
- Area changes by the square of the scale factor.
- Example: If the scale factor is 2, area increases by 4 times.
Examples:
- Map distance: If 1 inch = 60 miles, and the distance is 4 inches, actual distance = 240 miles.
- Blueprint: 3 inches × 5.5 inches at 1 inch : 10 feet = 30 feet × 55 feet room.
Visual Example:

Draw a rectangle on graph paper to represent a room. Use a ruler to measure and label the scaled dimensions.
Step-by-Step: Using a Scale Drawing
1. Identify the scale (e.g., 1 cm : 10 m).
2. Measure the length on the drawing.
3. Multiply the drawing length by the scale factor to get the real length.
4. Repeat for width or other dimensions.

Solving Equations and Ratios

Solving for Unknowns:
- For equations like “a × ? = b”, solve by dividing b by a.
- Example: 5 × ? = 15 → ? = 15 ÷ 5 = 3.
Proportions:
- To solve proportions, find an equivalent ratio or multiply/divide both sides by the same number.
- Example: If 1 batch uses 6 cups of water, 2 batches use 12 cups, 3 batches use 18 cups, etc.
Visuals:

Use tables to organize proportional relationships.
Draw arrows to show how numbers increase by the same factor.
Step-by-Step: Solving a Proportion
1. Write the proportion (e.g., 1 batch : 6 cups = x batches : y cups).
2. Set up the equation (1/6 = x/y).
3. Cross-multiply and solve for the unknown.

Applications with Units

Monetary Problems:
- Multiply the unit price by the number of items.
- Example: $1.25 per bottle × 10 bottles = $12.50.
Mixture/Recipe Scaling:
- Multiply each ingredient by the number of batches.
- Example: 1 cup detergent per batch × 4 batches = 4 cups.
Tables:
- Complete tables by recognizing and applying proportional increases.
Visual Example:

Draw a table with columns for batches, water, and detergent. Fill in each row by multiplying.
Step-by-Step: Scaling a Recipe
1. Write the amount needed for one batch.
2. Multiply each ingredient by the number of batches.
3. Fill in the table for each batch.

Key Terms & Definitions

Scaled Copy: A figure with all lengths changed by the same scale factor; shape remains similar.
Scale Factor: The constant used to multiply all lengths when making a scaled copy.
Corresponding Parts: Points, sides, or angles in similar figures that match in position.
Area: The space inside a two-dimensional shape, measured in square units.
Perimeter: The total length around a figure.
Scale Drawing: A drawing where all dimensions are proportional to the actual object using a given scale.
Ratio: A comparison of two quantities by division.
Visuals:

Draw and label figures to show corresponding parts.
Use arrows and color to connect matching sides and angles.

Examples and Visuals

Scaled Copy Identification:
- Figure A: 2 × 2 units.
- Figure B: 3 × 2 units (not a scaled copy).
- Figure C: 4 × 4 units (scaled copy, scale factor 2).
Area Scaling:
- Original area: 16 units².
- Scale factor: 2.
- New area: 16 × 2² = 64 units².
Scale Drawing Example:
- Empire State Building: 1 cm : 50 m scale.
- Drawing height: 10 cm (actual height: 500 m).
Map Example:
- 1 inch : 500 feet map is larger than 1 inch : 1000 feet map.
Step-by-Step: Area with Scale Factor
1. Find the original area.
2. Identify the scale factor.
3. Square the scale factor.
4. Multiply the original area by the squared scale factor.

Action Items / Next Steps

Practice identifying scaled copies in diagrams and calculating scale factors.
Draw scaled copies using different scale factors and check corresponding parts.
Complete homework on area and perimeter calculations with scale factors.
Solve problems involving scale drawings and maps, including changing scales.
Review and memorize key terms and definitions for upcoming quizzes.
Use tables and visuals to organize proportional relationships and ratios.
Work through example problems from the unit for additional practice.

Visuals and Step-by-Step Problem Solving in Every Chapter

Chapter 1: Scaled Copies
- Visual: Draw two "A" shapes, one original and one scaled.
- Step-by-step: Measure original, multiply by scale factor, redraw.
Chapter 2: Corresponding Parts
- Visual: Use colored lines to connect matching sides.
- Step-by-step: Label points, compare side lengths, check angles.
Chapter 3: Making Scaled Copies
- Visual: Show original and scaled polygons on grid paper.
- Step-by-step: List side lengths, multiply by scale factor, draw new shape.
Chapter 4: Scaled Relationships
- Visual: Table comparing side lengths and angles.
- Step-by-step: Identify scale factor, check all sides and angles.
Chapter 5: Size of Scale Factor
- Visual: Draw rectangles of different sizes.
- Step-by-step: Compare original and scaled, note if scale factor is >1, <1, or =1.
Chapter 6: Scaling and Area
- Visual: Show original and scaled polygons, shade area.
- Step-by-step: Find original area, square scale factor, multiply.
Chapter 7: Scale Drawings
- Visual: Use a ruler to measure drawing, show scale bar.
- Step-by-step: Measure, multiply by scale, label real-world size.
Chapter 8: Scale Drawings and Maps
- Visual: Map with scale bar, mark distances.
- Step-by-step: Count scale units, multiply, adjust for curves.
Chapter 9: Creating Scale Drawings
- Visual: Draw flag or book to scale.
- Step-by-step: Find scale, calculate drawing size, sketch.
Chapter 10: Changing Scales
- Visual: Show two drawings of same object at different scales.
- Step-by-step: Convert between scales, redraw.
Chapter 11: Scales Without Units
- Visual: Compare scale bars with different units.
- Step-by-step: Convert units, compare sizes.
Chapter 12: Units in Scale Drawings
- Visual: Table for area, cost, or recipe scaling.
- Step-by-step: Set up proportion, solve, check with table.

Continue to use visuals, tables, and step-by-step methods for all practice and homework problems to reinforce understanding.