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Geometry 9-12 / Week 02 / Shop Day
6/6
Week 02 · Rigid Motions and Symmetry

Shop Day

Shop Day: the family flat pattern
// The flats slide, flip and turn
⏱ about 20 min

Shop Day: The Family Flat Pattern

Saturday the kitchen table becomes the stage. A paper grid is taped flat and a stack of paper flats waits.

"Rule first," Wren says. "Pick the motion, write the corners, then place the flat."

"Flat one at A (1, 1)," Comet says. "Flat two is flat one slid 4 right. Flat three is flat two flipped."

"What do you notice when they meet?" Wren asks. Comet's cousin lays them down. "No gaps. Every flat is the same flat."

Nova projects the pattern across the table. "Would you like a hint? Try a half turn for the next one."

"Slide, flip, turn, slide," the cousin says. "Our designer, your family can build this too."

  1. A transformation is a function on points. Translations, reflections and rotations are rigid: they keep every distance and angle.
  2. Translation: equal parallel segments. Reflection: equal distances on a perpendicular. Rotation: equal angles on circles.
  3. Rules on the grid: (x + a, y + b), (x, -y), (-x, y), (y, x), (-y, x), (-x, -y).
  4. The symmetries of a figure are the rotations and reflections that carry it onto itself. Test each one.
  5. A sequence of motions carries one figure onto another, and that is how congruence will be defined.
◇ FAMILY FLAT PATTERN
Together, cut six paper flats the same shape and size. A grown-up cuts any cardboard ones.
Tape a paper grid to the table. Place flat one and write its corners.
Each person chooses the next motion: a slide, a flip or a turn. Write the rule, predict the corners, then place the flat.
Check that every flat matches flat one by laying flat one on each of the others. Then tape the pattern down.
For a grown-up
This week your student learned translations, reflections and rotations, their definitions, their coordinate rules and symmetry.
Ask them what a reflection changes and what it keeps. The answer should include lengths and angles.
You cut cardboard with scissors and push any paper fasteners. Tape every grid flat.
Safety first
A grown-up cuts cardboard with scissors. Students cut paper, sitting down.
Tape the grid and the flats down flat. A grown-up holds any ladder, and nothing heavy is lifted alone.
SHOP DAY CHECK
  • Read the question.
  • Tap your answer.
Flat one has corner A at (1, 1). Flat two is flat one slid 4 right. Where is A′?
A square paper flat is centered on the origin. How many rotations of less than 360° carry it onto itself?
A triangle flat sits at (1, 1), (4, 1), (1, 3). Its image is at (3, -1), (6, -1), (3, -3). Which sequence carries it there?
How many lines of reflection carry a square onto itself? Type the number.
WHY THIS EXERCISECounting symmetries by testing each flip is how the family knows a square flat can be placed four ways.
Draw your family's flat pattern on a grid. Write the motion between each pair of flats.
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