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Geometry and Measurement 6-8 / Week 09 / Thursday
4/6
Week 09 · Slides, Flips and Turns

Thursday

Raven's tile log
// Moving a shape without changing it
⏱ about 20 min

Thursday: Raven's Tile Log

Raven opens the Blueprint Book to a full page of coordinates. "Every corner of the L, before and after each move."

"Six corners, three moves, eighteen new points," Rocket says. "You wrote them all?"

"The rules wrote them," Raven says. "I checked a few on the floor. What do you notice in the flip column?"

"Every x flipped its sign. Every y stayed," Rocket says. "And in the turn column they swap."

Nova hovers over the log, her light moving down the page. "Now a harder one," she says. "Two moves in a row."

"Flip, then slide," Rocket says. "Or slide, then flip. Do they land in the same place?" Nova hums. "Find out."

Reading the log

The crew's tile log, made at the Blueprint Table. Each row is one corner of the L and where it lands after each move.

CornerBeforeSlide (x + 3, y - 5)Flip (-x, y)Turn (-y, x)
A(1, 1)(4, -4)(-1, 1)(-1, 1)
B(3, 1)(6, -4)(-3, 1)(-1, 3)
C(3, 2)(6, -3)(-3, 2)(-2, 3)
D(2, 2)(5, -3)(-2, 2)(-2, 2)
E(2, 4)(5, -1)(-2, 4)(-4, 2)
F(1, 4)(4, -1)(-1, 4)(-4, 1)

Read a row: corner B starts at (3, 1). The slide sends it to (6, -4), the flip to (-3, 1), the quarter turn to (-1, 3).

Every row follows the same three rules. That is what makes a log like this quick to write and easy to check.

Two moves in a row, and congruence

The L-tile in gold and, in blue, its image after a flip across the y-axis then a slide 5 down

Flip the tile across the y-axis, then slide it 5 down. Corner A at (1, 1) goes to (-1, 1), then to (-1, -4).

Two figures are congruent when some sequence of slides, flips and turns lands one exactly on the other. The blue tile is congruent to the gold one.

To show two tiles are congruent, name the moves. To show two tiles are not, find a side or an angle that differs.

READ THE TILE LOG
  • Read the question.
  • Tap your answer.
Corner C at (3, 2) takes the slide (x + 3, y - 5). Where does C′ land?
Corner F at (1, 4) takes the quarter turn. Where does F′ land?
Corner D at (2, 2) takes the flip across the y-axis. Where does D′ land?
TWO MOVES IN A ROW
  • Read the question.
  • Tap your answer.
The tile sits at (1, 1), (4, 1), (1, 3). Its image is at (-1, -4), (-4, -4), (-1, -2). Which sequence of moves takes it there?
The tile sits at (1, 1), (4, 1), (1, 3). Its image is at (-1, -1), (-4, -1), (-1, -3). Which sequence of moves takes it there?
The tile sits at (1, 1), (3, 1), (3, 2), (2, 2), (2, 4), (1, 4). Its image is at (3, -1), (5, -1), (5, -2), (4, -2), (4, -4), (3, -4). Which sequence of moves takes it there?
Corner E at (2, 4) takes the quarter turn (-y, x). What is the y-coordinate of E′? Type the number.
Corner B at (3, 1) takes the slide (x + 3, y - 5). What is the y-coordinate of B′? Type the number.
StatementTrue or false?
Two tiles are congruent if a sequence of slides, flips and turns lands one on the other.?
A flip then a slide always gives the same image as a slide then the same flip.?
The image of the L after a quarter turn is congruent to the L.?
Stretching a tile to twice its size is one of this week's three moves.?
WHY THIS EXERCISECongruence is about the three moves only. Next week a fourth move, the dilation, changes size on purpose.
Try it
Put a paper L at the crew's start position. Flip it across the y-axis, then slide it 5 down. Mark the corners.
Start over. Slide 5 down first, then flip. Compare the two images. Are they in the same place?
Safety first
Tape stays on the table or a taped-down grid. Pick up paper tiles when you finish so no one slips.
Draw the L-tile, then its image after the flip across the y-axis, then that image slid 5 down. Label A, A′ and A″.

Eighteen new points, three rules. Tomorrow you spot slides, flips and turns around the house and review.

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