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Algebra 2 9-12 / Week 01 / Garden Day
6/6
Week 01 · Complex Numbers and the Arc That Never Lands

Garden Day

Garden Day: your own arc
// When the square root of a negative shows up
⏱ about 20 min

Garden Day: Your Own Arc

Saturday morning, and the Glass House is quiet. Comet carries a spray bottle home in her backpack.

"Garden Day," she tells Wren on the phone. "I am going to show my family the arc that never lands."

In her backyard she sets a cup on a low wall and sprays an arc toward it. Short. She measures the reach with a tape.

"Reach 6, top 3," she says to Nova, who hovers at the window. "What model fits?"

"Would you like a hint?" Nova says. "Start from h = a x (6 - x) and use the top."

Comet works it out at the picnic table, then asks her brother to guess a height the water never reaches.

"Ten," he says. Comet smiles and writes the discriminant.

  1. i is the imaginary unit: i² = -1. Every complex number is a + bi.
  2. Add and subtract by combining real parts and i parts. Multiply like binomials, then replace i² with -1.
  3. The discriminant b² - 4ac tells how many real solutions a quadratic has: two, one or none.
  4. A negative discriminant gives two complex solutions, found by the formula or by completing the square.
  5. In a build, a real solution is a place or a time. A complex solution means it never happens.
◇ SHOW YOUR FAMILY THE ARC
Fill a spray bottle with cold water. Outside, spray an arc along a wall or path and mark where it lands.
Measure the reach R and estimate the top T. Together, write the model h = a x (R - x) and find a.
Ask a family member to name a height. Set the model equal to it and work out the discriminant together.
If the discriminant is negative, explain what i is and why the water never gets there.
Take turns naming heights until you find one with exactly one solution: the top.

Comet's backyard numbers

Comet's readings are made up for this lesson. Reach 6 and top 3 give a × 3 × 3 = 3. So a = 1/3, and h = (1/3)x(6 - x).

COMET'S BACKYARD ARC
  • Read the question.
  • Tap your answer.
Comet's arc has reach 6. At what distance is the top of the arc?
Comet's brother picks height 10. The top is 3. What kind of solutions will the equation have?
For Comet's arc, which height gives exactly one real solution? Type the number.
WHY THIS EXERCISEThe top height is where the discriminant is exactly zero. Above it, the solutions turn complex.
Draw your backyard arc on graph paper. Mark the reach, the top and the height your family picked.

Well done this week. Next week Comet builds a planter box, and its volume turns out to be a cubic.

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