Friday
Friday: Motions Everywhere
Comet holds a sheet of the set's wallpaper to the light. "The same leaf over and over. Translations."
"And the tiles by the shop sink," Wren says. "Each one is its neighbor flipped. What do you notice about the letters b and d?"
"A reflection!" Comet says. "And p is b turned a half turn. The alphabet is full of rigid motions."
Nova projects the whole set plan: flats sliding, flipping and turning into place. "Would you like a hint? Count the motions."
"Slide, flip, slide, turn," Comet says. "Every flat lands where the plan says."
"Review today," Wren says. "Our designer, every rule and every definition. Then next week we prove the braces match."
Rigid motions in everyday places
| Where | Motion | Why |
|---|---|---|
| a repeating wallpaper pattern | translation | the same shape slides along the wall |
| the letters b and d | reflection | one is the mirror image of the other |
| the letters b and q | rotation | a half turn takes one to the other |
| a row of floor tiles | translation and reflection | each tile slides or flips onto the next |
| a pinwheel at rest | rotation | turning it by one arm angle lands it on itself |
Every one of these keeps the shape the same size with the same angles. That is why patterns repeat and tiles fit.
The week in five lines
- A transformation is a function on points. Rigid motions keep every distance and angle; stretches do not.
- Translation: equal parallel segments. Reflection: equal distances on a perpendicular. Rotation: equal angles on circles.
- Rules: (x + a, y + b), (x, -y), (-x, y), (y, x), (-y, x), (-x, -y).
- Symmetry: the rotations and reflections that carry a figure onto itself. Test each one; do not guess.
- A sequence of motions can carry one figure onto another.
- Read the question.
- Tap your answer.
- Read the question.
- Tap your answer.
| Week review | True or false? |
|---|---|
| A translation keeps every length the same. | ? |
| A reflection over the x-axis changes the sign of y. | ? |
| A rotation changes the angles of a figure. | ? |
| The stretch (2x, y) is a rigid motion. | ? |
| A square has exactly 4 lines of reflection. | ? |
- ?We saw a transformation as a function and met the three rigid motions on the flat and its twin.
- ?We wrote a precise definition and a coordinate rule for each motion.
- ?We measured the flat before and after each motion in Shop Lab, and the stretch broke the lengths.
- ?We read the flat log, used sequences of motions, and counted the symmetries of four flats.
Write two capital letters that are reflections of each other, and two that are half turns of each other.
Rigid motions are everywhere. Tomorrow is Shop Day: the family moves flats on the kitchen table.