An Introduction to Imaginary Numbers

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In this engaging activity, students first delve into the concept of imaginary numbers i. They are presented with a fascinating spinner featuring i raised to the 0 through 3 powers. Students then embark on discovering the pattern for powers of i and learning how to calculate i raised to any non-negative power. Print Settings Printer Brand: MakerBot Printer: MakerBot Replicator Rafts: Yes Supports: No Resolution: standard Infill: 20% How I Designed This This spinner was meticulously created in TinkerCad. Initially, a circle with a diameter of 80 mm and height of 10 mm was designed. The text "i =" was added to this circle. Two square holes were carefully placed in the circle, one above the i to act as the exponent and another after the equals sign. Next, an additional circle was created with a diameter of 75 mm and height of 3 mm. This circle was then positioned on top of the first one, and a small circular hole was made in the center of both circles. Subsequently, where the square holes were located in the top circle, a 0 for the exponent and a 1 after the equals sign were placed in the bottom circle. These holes were designed to be partially transparent. The top circle was then rotated, and this process was repeated for the other exponents and solutions. Standards CCSS.Math.Content.HSN.CN.A.1 Overview and Background This activity aligns with the high school Number and Quantity standard below: Know there is a complex number i such that i2 = -1, and every complex number has the form a + bi with a and b real. Lesson Plan and Activity This activity begins with a "Do Now" section where students evaluate √25, √9, and √36. They are then asked to estimate √50 and √80, followed by pondering what the square root of -25 is. After the "Do Now," students can discuss their answers. First, ensure they grasp the concept of the square root function. Then, have students share their thoughts on what the square root of -25 is in a class discussion. Once they realize that there's no straightforward answer to this question, introduce i as the square root of negative 1. Next, students can work in groups with a 3D printed spinner. First, they are asked to find i raised to the 0, 1, 2, and 3 powers. These values can be directly obtained from the spinner. Then, they are tasked with finding i raised to the 4, 5, 6, and 7 powers. As they realize that the pattern repeats, hopefully using the spinner will help them see that the values keep repeating (1, i, -1, -i). Students are then asked to come up with a formula for finding any non-negative power of i and use this formula to determine higher powers of i. Materials Needed Each group requires a 3D printed spinner. Additionally, each spinner needs a metal paper fastener to keep it together. Skills Learned imaginary numbers Duration of Lesson One class or less