Transformations (Term 4) Chapter Notes | Mathematics for Grade 6 PDF Download

Making Larger Copies of Figures

This section teaches students how to enlarge shapes using scale factors while preserving their shape.

Understanding Enlargements

Enlargement: Creating a larger copy of a shape where all side lengths are multiplied by a scale factor, keeping the shape identical.
Example: Figures A, B, and C (each a combination of two quadrilaterals):

• Figure B is 2 times as large as Figure A (scale factor = 2).
• Figure C is 3 times as large as Figure A (scale factor = 3).
• Figure C is 1.5 times as large as Figure B (scale factor = 1.5, since 3 ÷ 2 = 1.5).

Key similarity: All figures have the same shape (proportions), but differ in size.
Key difference: Side lengths increase with the scale factor.

Enlarging on Grids
Use grid paper to enlarge shapes accurately:

Example: A kite on a 0.5 cm grid:

• Enlarge by scale factor 3: Draw on a 1.5 cm grid (0.5 × 3 = 1.5 cm).
• Enlarge by scale factor 2: Draw on a 1 cm grid (0.5 × 2 = 1 cm).

Method: Multiply each side length by the scale factor and draw on the appropriately scaled grid.

Scale Factor Application
To enlarge a shape:

• Scale factor 2: Double all side lengths (e.g., 1 cm becomes 2 cm).
• Scale factor 4: Quadruple all side lengths (e.g., 1 cm becomes 4 cm).

Shapes not proportionally scaled (e.g., different shapes) are not enlargements.

Enlargements and Reductions

Enlargements vs. Non-Enlargements

Enlargement: All side lengths multiplied by the same scale factor, preserving shape.
Example: Rectangle C (12 cm × 9 cm) is an enlargement of Rectangle A (8 cm × 6 cm) by scale factor 1.5 (12 ÷ 8 = 9 ÷ 6 = 1.5).

Non-enlargement: Adding a fixed length to sides changes the shape.

• Example: Adding 16 units to all sides alters proportions, not an enlargement.
• Rebecca’s view: Correct—multiply side lengths by a scale factor (e.g., 3) to keep shape.
• Nkhangweleni’s view: Incorrect—adding fixed lengths changes shape.

Reduction: A smaller copy of a shape (scale factor < 1).
Example: Figure A is a reduction of Figure C by factor 1/3 (since C is 3 times A).

Reductions on Grids
Example: Yellow rhombus on a 1.5 cm grid:

• Reduction to 1.25 cm grid (scale factor = 1.25 ÷ 1.5 = 5/6).
• Reduction to 0.5 cm grid (scale factor = 0.5 ÷ 1.5 = 1/3).

Grid comparisons for quadrilaterals:

• Reduction by factor 5 from 1.25 cm grid: 1.25 ÷ 5 = 0.25 cm grid.
• 3/4 size of 1 cm grid: 1 × 3/4 = 0.75 cm grid.
• 3/5 size of 1.25 cm grid: 1.25 × 3/5 = 0.75 cm grid.
• 5/4 size of 1 cm grid: 1 × 5/4 = 1.25 cm grid.
• 4/3 size of 0.75 cm grid: 0.75 × 4/3 = 1 cm grid.

Scale Factor Relationships
Example: Figures A, B, C (page 350):

• B is 2 times A (scale factor 2).
• B is not 3 times A or 2/3 of C (check proportions).

Enlargements/reductions of a quadrilateral:

• Scale factor 1.5: All sides × 1.5.
• Scale factor 2/3: All sides × 2/3.
• Scale factor 1 2/3: All sides × 5/3.

Side lengths and diagonals scale by the same factor as the figure.

Rectangles and Diagonals
Rectangle A: 8 cm × 6 cm.
Enlargements:

• B: 12 cm × 9 cm (scale factor 1.5).
• C: 16 cm × 12 cm (scale factor 2).
• D: 20 cm × 15 cm (scale factor 2.5).
• E: 24 cm × 18 cm (scale factor 3).

Diagonal: Line from a vertex to the opposite vertex, dividing a rectangle into two triangles.
Diagonals scale with the same factor (e.g., diagonal of B is 1.5 times A’s diagonal).

Check enlargements:

• 9 cm × 12 cm: Scale factor 1.5 (12 ÷ 8 = 1.5, 9 ÷ 6 = 1.5) → Enlargement.
• 12 cm × 16 cm: Scale factor 2 (16 ÷ 8 = 2, 12 ÷ 6 = 2) → Enlargement.
• 9 cm × 11 cm, 14 cm × 16 cm: Not enlargements (inconsistent scale factors).

Relationships:

• D is 2.5 times A.
• B is 1/3 of E (9 ÷ 18 = 12 ÷ 24 = 1/3).
• C is 0.5 times E (12 ÷ 24 = 16 ÷ 24 = 0.5).

Increasing the Lengths of Two Sides Only

This section explores scaling only two opposite sides of a rectangle, altering its shape.

Partial Scaling

Standard enlargement/reduction: Multiply all four sides by the same scale factor to preserve shape.
Partial scaling: Multiply only two opposite sides by a scale factor, changing the shape.
Example: Rectangle with sides 8 cm × 6 cm:

• Scale only lengths (8 cm) by 2: New rectangle 16 cm × 6 cm.
• Scale only widths (6 cm) by 2: New rectangle 8 cm × 12 cm.

Result: The shape is no longer proportional to the original (not an enlargement).

Effect on Shape

• Scaling two sides creates a new rectangle with different proportions.
• Example: Original 8 cm × 6 cm (ratio 4:3), scaling length to 16 cm gives 16 cm × 6 cm (ratio 8:3).
• This transformation is distinct from true enlargements/reductions.

Points to Remember

• Enlargement: Larger copy of a shape with all sides multiplied by a scale factor (e.g., 2, 3).
• Reduction: Smaller copy of a shape (scale factor < 1, e.g., 1/3).
• Scale factor: Number by which side lengths are multiplied (e.g., 1.5 for 12 cm ÷ 8 cm).
• Grid scaling: Use grids to enlarge/reduce (e.g., 0.5 cm to 1.5 cm for scale factor 3).
• Diagonal: Line dividing a rectangle into two triangles, scales with the same factor.
• Proportional scaling: Multiply all sides to keep shape (not adding fixed lengths).
• Partial scaling: Scaling only two sides changes the shape.

Difficult Words

• Transformation: Changing a shape’s size or position (e.g., enlargement).
• Enlargement: Making a shape larger by a scale factor.
• Reduction: Making a shape smaller by a scale factor.
• Scale factor: Number used to multiply side lengths (e.g., 2, 1/3).
• Quadrilateral: 2-D shape with four sides (e.g., rectangle, rhombus).
• Rhombus: Quadrilateral with all sides equal.
• Diagonal: Line from a vertex to the opposite vertex in a polygon.

Summary