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Algebra 2 9-12 / Week 10 / Garden Day
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Week 10 · Periodic Models

Garden Day

Garden Day: the clock hand wave
// The vent timer and the temperature wave
⏱ about 20 min

Garden Day: The Clock Hand Wave

Saturday morning the kitchen clock comes off the wall and onto the table.

"The minute hand is 8 centimeters long," Comet says. "Watch the tip. How high above the center is it?"

"At twelve, 8 up. At three, level with the center. At six, 8 down," someone answers.

"That is a wave," Wren says. "High eight, low negative eight, and it repeats every hour. What do you notice?"

"Amplitude 8, midline 0, period 60 minutes," Comet says, writing it on a card.

Nova hovers over the clock face. "Would you like a hint? At twelve the tip is at its peak, not the midline."

"So it is a shifted sine," Wren says. "Or a cosine. Either way, three numbers and a shift. Family, your turn."

  1. A periodic pattern repeats at a steady spacing. Its three numbers are amplitude, midline and period.
  2. Amplitude = (high - low) ÷ 2. Midline = (high + low) ÷ 2. Period = the distance before the pattern repeats.
  3. y = a sin(b(x - c)) + d, with b = 2π ÷ period and c the phase shift.
  4. Output changes (+ k, × k) move the wave up or stretch it. Input changes (x - k, kx) slide or squeeze it sideways.
  5. The model fills in values between the readings and shows where the wave rises, falls, peaks and dips.
◇ FAMILY WAVE HUNT
Find something at home that turns or swings: a clock hand, a sticker on a slow turning fan, a swinging string.
Pick a point on it and measure or estimate its height every quarter of a cycle. Record four or five readings.
Together, find the high, the low and the repeat. Write the amplitude, midline and period on a card.
Sketch the wave on graph paper. Decide where it starts: at the midline, the top or the bottom.
For a grown-up
Anything on a wall or a shelf is taken down by a grown-up. Nothing is measured on a ladder.
A wrong reading is useful: plot it and see where it falls off the wave.
The clock's three numbers are the family's own readings, like the crew's log. There is no fact to look up.
GARDEN DAY CHECK
  • Read the question.
  • Tap your answer.
The tip of the minute hand swings between -8 and 8 cm from the center. What is the amplitude?
The tip swings between -8 and 8 cm from the center. What is the midline value?
A wave starts at its midline, rises to 8, falls to -8 and repeats every 60 minutes. Which equation models it?
The minute hand's tip wave repeats every 60 minutes. What is its period, in minutes? Type the number.
WHY THIS EXERCISEThe period is the first number to find for any wave. It gives b through 2π ÷ period.
Draw the clock hand wave: minutes across, height of the tip up. Mark where it starts and one period.
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