Imaginary Island

Hook 5 min

1. "What is the square root of −1?" Let students argue that it doesn't exist.
2. "Mathematicians named it i and built a whole island with it. Today you explore the island."

Explore 17 min

1. On the projector, read the panel: "Traveler z = 3 + 2i, distance 3.61, heading 33.7 degrees."
2. Press × i once and read the worked line: (3 + 2i) × i = 3i + 2i² = −2 + 3i. "Where did the lantern go? Why did i² turn into −2?"
3. Pairs turn on Distance rings and press × i three more times. The lantern visits the Fox Den, the Salt Flats and the Waterfall, then lands back on the Orchard. "What stayed the same? What changed?"
4. Pairs try Reflect, Square it, + w and × w from different landmarks and record what each does to distance and heading.

Make sense 10 min

1. Show the Powers of i dial: i¹ = i, i² = −1, i³ = −i, i⁴ = 1. "Multiplying by i is a quarter turn."
2. "Adding w moves every point the same way. Multiplying by w does not. What does × w do to a far point compared with a near one?" (It turns and stretches.)
3. "Multiply a landmark by its own conjugate. Why is the answer always real?"

Apply 13 min

1. Pairs do four clues in Clue Trail: predict on paper, drop a guess pin, then press Walk it. A guess within 0.25 counts as landed.
2. Pairs switch to Springs. With b = 0, they raise c from −4 and watch the springs slide together on the Real Road, meet at 0, then split north and south.
3. "Why do the two complex solutions always come as a mirror pair across the Real Road?"

Exit check 5 min

On a card: "Compute (2 − i)(3 + 4i) and plot it. Then solve x² + 9 = 0 and say where its solutions sit on the map." Cards that leave i² as i² show who needs the worked lines in Explore again.