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Geometry 9-12 / Week 10 / Shop Day
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Week 10 · Equations of Circles and Parabolas

Shop Day

Shop Day: the backyard turntable
// The turntable and the lamp dish on the stage grid
⏱ about 20 min

Shop Day: The Backyard Turntable

Comet carries chalk, string and the tape measure out to the driveway. "Shop Day. A turntable with no stage."

"Center first," Wren says, drawing a cross. "Then the circle, then the check. What do you notice about the cracks?"

"They make a grid, almost," Comet says. "Good enough to count squares from the center."

Nova projects a faint grid over the driveway. "Would you like a hint? Pick points before you measure, not after."

Comet picks five. Wren measures. Three pass, two fail. "The string stretched," he says.

"The equation did not," Comet says. "Our designer, run this with your family today."

  1. A circle with center (h, k) and radius r is (x - h)² + (y - k)² = r².
  2. A point is on the circle when the sum of its squared differences from the center equals r².
  3. Complete the square to find a hidden center and radius.
  4. A parabola is every point equally far from the focus and the directrix: y = x²/(4p).
◇ CHALK CIRCLE, FAMILY CHECK
Together, tape or chalk a grid of squares outside, or use a big sheet of paper on a table.
Mark a center and draw a circle of radius 4 squares with string and chalk.
Each person picks two points that look like they are on the circle. Measure the changes in x and y.
Square, add and compare with 16. Which points passed? Then write the circle's equation together.
For a grown-up
Pick an outdoor spot away from any road, or keep the work on a table indoors.
The student does the arithmetic. You pick the points. Ask them to explain why 16 is the magic number.
Chalk washes off. A paper grid taped to a table works just as well on a rainy day.
SHOP DAY CHECK
  • Read the question.
  • Tap your answer.
A circle on a grid with center (0, 0) and radius 4Your backyard circle has center (0, 0) and radius 4. Which equation describes it?
A grid with a segment at (0, 0), (4, 3)A family member picks (4, 3). How far is it from the center (0, 0)?
A circle on a grid with center (2, 2) and radius 4You move the center to (2, 2) and keep radius 4. Which equation describes the new circle?
Your backyard circle has radius 4. What is the sum of the squares for every point on it? Type the number.
WHY THIS EXERCISEEvery circle has one magic number, r², that every point on it must match.
Draw your backyard grid and circle. Mark each family point and write its sum of squares beside it.

Week ten complete, designer. Next week the paint, the foam and the water jug show where every volume formula comes from.

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