# IGCSE Higher: Transformation Geometry

## Scheme of work: IGCSE Higher: Year 10: Term 3: Transformation Geometry

#### Prerequisite Knowledge

• Concept: Understanding how to plot and read coordinates in all four quadrants of the Cartesian plane.
• Example: $A(3, -2), \quad B(-4, 5)$
2. 2D and 3D Symmetry
• Concept: Recognizing lines of symmetry in 2D shapes and planes of symmetry in 3D shapes.
• Example: $\text{A square has 4 lines of symmetry}, \quad \text{A cube has 9 planes of symmetry}$
3. Rotational Symmetry
• Concept: Understanding the concept of rotational symmetry and identifying the order of rotational symmetry in shapes.
• Example: $\text{A regular pentagon has rotational symmetry of order 5}$

#### Success Criteria

1. Translating Shapes
• Performing Translation: Students should be able to translate a shape on the Cartesian plane by a given vector.
• Example: $\text{Translate the shape } A(1, 2), B(4, 2), C(4, 5) \text{ by the vector } \begin{pmatrix} 3 \\ -2 \end{pmatrix}$
• Describing Translation: Students should be able to describe a translation using a vector.
• Example: $\text{The shape } A'(4, 0), B'(7, 0), C'(7, 3) \text{ is a translation of } A(1, 2), B(4, 2), C(4, 5) \text{ by } \begin{pmatrix} 3 \\ -2 \end{pmatrix}$
2. Reflecting Shapes
• Performing Reflection: Students should be able to reflect a shape in a given line (e.g., x-axis, y-axis, y = x).
• Example: $\text{Reflect the shape } A(2, 3), B(5, 3), C(5, 6) \text{ in the y-axis}$
• Describing Reflection: Students should be able to describe a reflection in a given line.
• Example: $\text{The shape } A'(-2, 3), B'(-5, 3), C'(-5, 6) \text{ is a reflection of } A(2, 3), B(5, 3), C(5, 6) \text{ in the y-axis}$
3. Rotating Shapes
• Performing Rotation: Students should be able to rotate a shape around a given point by a specified angle and direction (clockwise or counterclockwise).
• Example: $\text{Rotate the shape } A(1, 2), B(4, 2), C(4, 5) \text{ 90 degrees clockwise around the origin}$
• Describing Rotation: Students should be able to describe a rotation using the center of rotation, angle, and direction.
• Example: $\text{The shape } A'(-2, 1), B'(-2, 4), C'(-5, 4) \text{ is a 90-degree clockwise rotation of } A(1, 2), B(4, 2), C(4, 5) \text{ around the origin}$
4. Enlarging Shapes
• Performing Enlargement: Students should be able to enlarge a shape from a given center by a specified scale factor.
• Example: $\text{Enlarge the shape } A(1, 2), B(4, 2), C(4, 5) \text{ from the center } (0,0) \text{ by a scale factor of 2}$
• Describing Enlargement: Students should be able to describe an enlargement using the center and scale factor.
• Example: $\text{The shape } A'(2, 4), B'(8, 4), C'(8, 10) \text{ is an enlargement of } A(1, 2), B(4, 2), C(4, 5) \text{ from the center } (0,0) \text{ by a scale factor of 2}$

#### Key Concepts

1. Translating Shapes
• Performing Translation: Understanding that translating a shape involves moving every point of the shape by the same distance in the same direction.
• Example: $\text{Translate the shape } A(1, 2), B(4, 2), C(4, 5) \text{ by the vector } \begin{pmatrix} 3 \\ -2 \end{pmatrix}$
• Describing Translation: Describing the movement using vectors.
• Example: $\text{The shape } A'(4, 0), B'(7, 0), C'(7, 3) \text{ is a translation of } A(1, 2), B(4, 2), C(4, 5) \text{ by } \begin{pmatrix} 3 \\ -2 \end{pmatrix}$
2. Reflecting Shapes
• Performing Reflection: Understanding that reflecting a shape involves flipping the shape over a specified line so that each point on the shape and its image are equidistant from the line.
• Example: $\text{Reflect the shape } A(2, 3), B(5, 3), C(5, 6) \text{ in the y-axis}$
• Describing Reflection: Describing the reflection using the line of reflection.
• Example: $\text{The shape } A'(-2, 3), B'(-5, 3), C'(-5, 6) \text{ is a reflection of } A(2, 3), B(5, 3), C(5, 6) \text{ in the y-axis}$
3. Rotating Shapes
• Performing Rotation: Understanding that rotating a shape involves turning the shape around a specified point by a specified angle and direction.
• Example: $\text{Rotate the shape } A(1, 2), B(4, 2), C(4, 5) \text{ 90 degrees clockwise around the origin}$
• Describing Rotation: Describing the rotation using the center of rotation, angle, and direction.
• Example: $\text{The shape } A'(-2, 1), B'(-2, 4), C'(-5, 4) \text{ is a 90-degree clockwise rotation of } A(1, 2), B(4, 2), C(4, 5) \text{ around the origin}$
4. Enlarging Shapes
• Performing Enlargement: Understanding that enlarging a shape involves scaling the shape from a given center by a specified scale factor.
• Example: $\text{Enlarge the shape } A(1, 2), B(4, 2), C(4, 5) \text{ from the center } (0,0) \text{ by a scale factor of 2}$
• Describing Enlargement: Describing the enlargement using the center and scale factor.
• Example: $\text{The shape } A'(2, 4), B'(8, 4), C'(8, 10) \text{ is an enlargement of } A(1, 2), B(4, 2), C(4, 5) \text{ from the center } (0,0) \text{ by a scale factor of 2}$

#### Common Misconceptions

1. Translating Shapes
• Common Mistake: Students might incorrectly apply the translation vector, leading to incorrect positions of the translated shape.
• Example: Incorrectly translating the shape $$A(1, 2), B(4, 2), C(4, 5)$$ by the vector $$\begin{pmatrix} 3 \\ -2 \end{pmatrix}$$ as $$A'(4, 0), B'(7, 0), C'(7, 3)$$ instead of $$A'(4, 0), B'(7, 0), C'(7, 3)$$.
• Common Mistake: Students might describe the translation vector incorrectly.
• Example: Describing the translation vector as $$\begin{pmatrix} 3 \\ 2 \end{pmatrix}$$ instead of $$\begin{pmatrix} 3 \\ -2 \end{pmatrix}$$.
2. Reflecting Shapes
• Common Mistake: Students might incorrectly reflect the shape over the specified line, leading to incorrect positions of the reflected shape.
• Example: Incorrectly reflecting the shape $$A(2, 3), B(5, 3), C(5, 6)$$ in the y-axis as $$A'(-2, -3), B'(-5, -3), C'(-5, -6)$$ instead of $$A'(-2, 3), B'(-5, 3), C'(-5, 6)$$.
• Common Mistake: Students might describe the line of reflection incorrectly.
• Example: Describing the line of reflection as x-axis instead of y-axis.
3. Rotating Shapes
• Common Mistake: Students might incorrectly apply the rotation, leading to incorrect positions of the rotated shape.
• Example: Incorrectly rotating the shape $$A(1, 2), B(4, 2), C(4, 5)$$ 90 degrees clockwise around the origin as $$A'(2, -1), B'(2, -4), C'(5, -4)$$ instead of $$A'(-2, 1), B'(-2, 4), C'(-5, 4)$$.
• Common Mistake: Students might describe the rotation angle or direction incorrectly.
• Example: Describing the rotation as 90 degrees counterclockwise instead of 90 degrees clockwise.
4. Enlarging Shapes
• Common Mistake: Students might incorrectly apply the scale factor, leading to incorrect positions of the enlarged shape.
• Example: Incorrectly enlarging the shape $$A(1, 2), B(4, 2), C(4, 5)$$ from the center (0,0) by a scale factor of 2 as $$A'(1, 4), B'(2, 4), C'(2, 10)$$ instead of $$A'(2, 4), B'(8, 4), C'(8, 10)$$.
• Common Mistake: Students might describe the scale factor or center of enlargement incorrectly.
• Example: Describing the scale factor as 3 instead of 2, or the center of enlargement as (1,1) instead of (0,0).

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