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Geometry and Measurement 6-8 / Week 09 / Tuesday
2/6
Week 09 · Slides, Flips and Turns

Tuesday

The rule for every corner
// Moving a shape without changing it
⏱ about 20 min

Tuesday: The Rule for Every Corner

Raven pins the triangle to the origin of the paper grid with a paper fastener. "Quarter turn. Counterclockwise."

Rocket turns it. "B was at (4, 1). Now it is at (-1, 4). The numbers swapped and one went negative."

"Try another corner before you decide the rule," Raven says. "What do you notice?"

"C was (1, 3). Now (-3, 1). Yes, swap them, then make the first one negative." He writes (-y, x).

Nova hovers over the pinned corner, her light a small circle. "Now the flip," she says. "Across the y-axis."

Rocket flips the triangle over the axis. "(4, 1) goes to (-4, 1). Only x changes sign." Nova hums. "Rules, not guesses."

Coordinate rules

MoveRuleExample
slide a right, b up(x + a, y + b)slide 3 right, 2 up: (4, 1) to (7, 3)
flip across the x-axis(x, -y)(4, 1) to (4, -1)
flip across the y-axis(-x, y)(4, 1) to (-4, 1)
quarter turn counterclockwise about the origin(-y, x)(4, 1) to (-1, 4)
half turn about the origin(-x, -y)(4, 1) to (-4, -1)

A slide adds the same numbers to every corner. A flip changes the sign of one coordinate. A turn swaps and changes signs.

Left and down are negative: a slide 2 left and 5 down is (x - 2, y - 5).

A triangle at (1, 1), (4, 1), (1, 3) and its image after a quarter turn about the origin
  1. Name the move and write its rule.
  2. Take one corner and put its x and y into the rule.
  3. Repeat for every corner. Plot the new points and join them.
  4. Check: lay the original on the image. Same lengths, same angles.
PICK THE RULE
  • Read the question.
  • Tap your answer.
Which rule gives the new coordinates after a flip across the y-axis?
Which rule gives the new coordinates after a turn of 90° counterclockwise about the origin?
Which rule gives the new coordinates after a slide 3 right and 2 down?
Which rule gives the new coordinates after a half turn about the origin?
USE THE RULE
  • Read the question.
  • Tap your answer.
Corner B at (4, 1) makes a quarter turn counterclockwise about the origin. Where does B′ land?
Corner B at (4, 1) flips across the y-axis. Where does B′ land?
Corner C at (1, 3) flips across the x-axis. Where does C′ land?
Point (2, 5) slides 2 left and 3 down. Where does it land?
TURNS AND HALF TURNS
  • Read the question.
  • Tap your answer.
Point (3, 2) makes a half turn about the origin. Where does it land?
Point (-2, 4) makes a quarter turn counterclockwise about the origin. Where does it land?
Point (0, 3) sits on the y-axis. Where does it land after a flip across the y-axis?
StatementTrue or false?
A flip across the y-axis changes the sign of x and keeps y.?
A slide changes the sign of both coordinates.?
A quarter turn counterclockwise sends (x, y) to (-y, x).?
A point on the y-axis stays put when flipped across the y-axis.?
WHY THIS EXERCISEA rule lets you move a tile with numbers only, which is how the floor plan gets drawn before any cutting.
Try it
On grid paper, draw a small triangle and mark the origin. A grown-up pushes a paper fastener through the origin.
Turn the triangle a quarter turn. Read each new corner and check it against (-y, x).
Safety first
A grown-up pushes the paper fastener through and folds its legs flat. Keep fingers clear of the point.
Paper fasteners stay away from children under three and pets.
Draw the triangle A (1, 1), B (4, 1), C (1, 3) and its image after a flip across the y-axis. Label A′, B′, C′.

Rules, not guesses. Tomorrow is Build Lab: the tiles move on the floor grid, and you measure what stays the same.

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