[list=1][*]Perform the four isometric transformations on the shape below: translation, rotation, reflection, and glide reflection.[/*][*]Record what characteristic changes and what stays the same (side lengths, angles, orientation, shape etc).[/*][/list]

Goal: In transformations, there are some movements that have points that do not move. Can you identify them?[br][br][br]Invariant points: a fixed point that do not change its position/ stay still when a transformation is performed.[br][list=1][*]Create a shape on the grid below. Perform a reflection.[/*][*]Does any point on the shape stay in the same place?[/*][*]Drag the shape so it touches/crosses the line of reflection. Does any point stay in the same place?[br][/*][*]Now, perform a rotation. Move the center point of rotation. What happens to the center point itself?[/*][*]Lastly, perform a translation. Do any point stay still?[/*][*]Can you conclude which isometric transformation has invariant point(s)?[/*][/list]

[b]The 'Home' Move (Inverse):[/b][br][list=1][*]Create a shape on the grid.[/*][*]Perform translation, [math]T=\left(\begin{matrix}5\\0\end{matrix}\right)[/math]. Now, perform another transformation so that it maps the image back onto the original object (perform the 'home' move). What is a single transformation that moves the image exactly back onto its object position?[/*][*]Repeat with a [math]90^{\circ}[/math] clockwise rotation. What is a single transformation that moves the image exactly back onto its object position?[/*][/list]

[b]The 'Double' Move (Product):[/b][br][list=1][*]Create a shape on the grid.[/*][*]Perform a reflection over a line [math]x=2[/math]. Reflect that new image over another vertical line [math]x=5[/math].[/*][*]Observe the original object and the final image. What single transformation could have skipped the middle step?[/*][*]Repeat with a [math]90^{\circ}[/math] rotation followed by another [math]90^\circ[/math] rotation. What single transformation could have skipped the middle step?[/*][/list][br][b]The 'Do Nothing' Move (Identity):[/b][br][list=1][*]Create a shape on the grid.[/*][*]Perform a [math]360^{\circ}[/math] clockwise rotation.[/*][*]Observe what changed.[/*][/list]

[b]Identity transformation[/b]: is a mapping where every point stays in its original position. It is the "zero" of transformations.[br]For any transformation [math]T[/math], the composition [math]T\cdot I=T[/math] and [math]I\cdot T=T[/math] where the identity results in no change.

The [b]Inverse [/b]is the transformation that maps the image back to the original object.[br]Every transformation has 'reverse' move, where the inverse of a transformation [math]T[/math] is [math]T^{-1}[/math], such that the composition [math]T\cdot T^{-1}=I[/math] or [math]T^{-1}\cdot T=I[/math][br][br]The [b]Product [/b]is a composition of two or more transformations in a sequence to create a new isometry.[br]For transformation [math]T_1[/math] followed by [math]T_2[/math], their product [math]T_3[/math] is also an isometry.

[br][size=85][i]Note:[br]- Rotations have one invariant point; reflections have a line of invariant points; translations have none.[br][/i][/size][size=85][i]- A [math]360^\circ[/math] rotation or a double reflection over the same line results in the identity.[/i][/size]

[b]Goal:[/b] Use vertex labeling to distinguish between "sliding/turning" and "flipping."[br][br][list=1][*]Create a shape. Label all vertices in clockwise order (A, B, C, ...).[/*][*]Perform each isometric transformation on the shape. Record the orientation of the image. Which transformation results in the order remaining clockwise, and which is counter-clockwise? Did it 'flip'?[/*][/list][br][b]Direct isometry[/b]: a transformation that preserves the orientation of the object.[br][b]Opposite isometry[/b]: a transformation that reverses the orientation of the object.[br][br][br][size=85][i]Note: [br]Direct isometry - translation, rotation[br]Opposite isometry - reflection, glide reflection[/i][/size]