Powers of i Game

The Subtraction within 20 game builds on basic ten-frame logic but introduces a second frame to handle “teen” numbers. This is a critical bridge in early mathematics because it helps students visualize numbers as Base-10 structures—essentially seeing 14 as “one ten and four ones.”

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Powers of i Game
The Powers of i Game is a mathematical training tool designed to help students master the cyclical nature of the imaginary unit i. The powers of i do not grow infinitely; instead, they rotate through a specific four-step cycle.
Scroll down the page for a more detailed explanation.

How the “Powers of i Game” Works

The Mathematical Cycle
The core of the game is based on the fact that i = √-1. As you multiply i by itself, it repeats every four steps:
i¹: Just i (The starting point).
i²: -1 (By definition).
i³: -i (Because i² × i = -1 × i).
i⁴: 1 (Because i² × i² = -1 × -1).
Handling Large and Negative Exponents
The game challenges the player’s ability to simplify any exponent (n) into one of these four results.
Positive Exponents: The game models the Modulo 4 rule. To find i²², you divide 22 by 4. The remainder is 2, so i²² is the same as i², which is -1.
Negative Exponents: For negative powers like i^-1, the game logic moves counter-clockwise around the cycle (or adds 4 until the number is positive). Thus, i^-1 is the same as i³, which is -i.
Gameplay Mechanics
Mode Selection: Users can focus on specific areas (Positive, Negative, or Mixed) to build confidence before tackling the full range of complex numbers.
Visual Feedback: When a user selects an answer, the game reveals the correct simplification in the “answer blank.” If the user is incorrect, the right answer is highlighted in the grid.

This video gives a clear, step-by-step approach to how to simplify imaginary numbers with large exponents.

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