In this task, we aim to apply error margins to a [b]rotation scenario[/b]. Specifically, we will rotate a point [b]A[/b] around a fixed point [b]O[/b] by a given angle, introducing margins of error to increase or decrease the precision of the solution.[br][br][b]Key Concepts:[/b][br][br] • [b]A’[/b]: The point that represents the rotation of [b]A[/b] around the point [b]O[/b] by an angle [b]α[/b].[br] • [b]α[/b]: The random angle of rotation, measured in degrees, which determines how far [b]A[/b] rotates around [b]O[/b].[br][br]The goal is to find [b]A’[/b] (the rotated point) after rotating [b]A[/b] by [b]α[/b] degrees around [b]O[/b], while applying the appropriate margin of error.[br][br][b]Steps for Implementation:[/b][br][br] 1. [b]Determine the Random Angle (α)[/b]:[br] • [b]α[/b] is the angle through which point [b]A[/b] is rotated around point [b]O[/b] to reach point [b]A’[/b]. This angle is randomly assigned for each question. [br][br] 2. [b]Apply the Margin of Error (n and m)[/b]:[br] • In a typical rotation problem, [b]A’[/b] would be precisely located based on the exact value of [b]α[/b].[br] • However, in this task, we introduce [b]two margins of error[/b]—[b]n[/b] (green) for more precise answers and [b]m[/b] (orange) for less precise answers.[br][br] 3. [b]Adjusting the Rotation with Margins[/b]:[br] • The [b]n margin[/b] represents a smaller range of acceptable angles around the exact value of [b]α[/b], making the task more challenging by requiring greater precision.[br] • The [b]m margin[/b] allows a broader range of acceptable angles around [b]α[/b], making the task easier as answers within this larger range are still considered correct.[br][br] 4. [b]Implementation with Sliders[/b]:[br] • The sliders for [b]n[/b] and [b]m[/b] allow the dynamic adjustment of these margins. [b]m[/b] should always be larger than [b]n[/b], and the rotation error allowed for [b]A’[/b] must fall within these margins:[br] • [b]n[/b]: Controls the smaller, more precise range.[br] • [b]m[/b]: Controls the larger, less precise range.[br] • When applying the margin, the randomly generated angle [b]α’[/b] will fall within [b]α ± n[/b] for precise responses or [b]α ± m[/b] for broader answers.[br][br] 5. [b]Ensuring Consistency[/b]:[br] • To maintain logical consistency, the error bounds must ensure that [b]m [/b][math]\ge[/math][b] n[/b], as the broader margin should always include the more precise margin.[br] • The calculation will involve checking the random angle [b]α’[/b] and ensuring it satisfies the margin conditions before determining if the student’s answer is acceptable.[br][br]By rotating [b]A[/b] around [b]O[/b] and incorporating dynamic error margins, we allow students to engage with both precise and flexible answers, enhancing the learning experience through adjustable difficulty levels.