Geometry

Guiding Question: How can congruence support interpretation of symmetry?

symmetry: rotation & reflection

Reflection Symmetry (Line Symmetry):

Think of this as the "Mirror Rule." If you can fold a shape over a line and the two halves match perfectly, it has reflection symmetry.
The line you fold on is called the Line of Symmetry.

Rotation Symmetry (Turn Symmetry):

Think of this as the "Pinwheel Rule." If you can rotate a shape around a center point (less than a full 360° turn) and it looks exactly like it did when you started, it has rotation symmetry.
Order of Rotation: This is how many times the shape looks the same during one full 360° turn. (e.g., A square looks the same 4 times, so it has an order of 4).
Mapping Transformations:
- - Symmetrical shapes can be "mapped" (moved) onto each other using a combination of reflections (flips) and rotations (turns).
  - Example: You might have to flip a triangle AND then turn it 90 degrees to make it land perfectly on its symmetrical partner.
  - More information about transformations in the Coordinate Geometry section!

The Definition: Two shapes are congruent if they are the exact same size and the exact same shape.
Orientation doesn't matter: A triangle is still congruent to its twin even if one is standing on its head or hidden in the corner of the page.
The Relationship:
- - If two shapes are symmetrical, they are always congruent.
  - However, two shapes can be congruent without being symmetrical (they might just be two identical squares sitting far apart on a page).

What is a Tessellation?
- - It is the tiling of a flat surface using one or more geometric shapes (tiles) with no overlaps and no gaps.
  - Many tessellations use symmetrical shapes to create a repeating, infinite pattern.
First Nations and Métis Star Blankets:
- - The Design: A traditional star blanket often features an 8-pointed star made of many small rhombi (plural of rhombus).
  - Geometry in Action: These blankets are perfect examples of tessellations and symmetry. An 8-pointed star has 8 lines of reflection symmetry and rotation symmetry of order 8.
  - The Purpose: Beyond math, these blankets are sacred gifts given to honor people during life-changing events (births, graduations, marriages). The 8 points often represent the four directions and the stages of life.

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