Skip-counting puzzles

Learning Objectives Define an elliptical skip-counting sequence using parametric equations. Calculate the coordinates of points in a sequence on an ellipse given a starting parameter and an angular step. Determine the number of unique points in a skip-counting sequence on an ellipse before it repeats. Analyze the geometric patterns formed by connecting sequential points on an ellipse. Solve for an unknown angular step given consecutive points in an elliptical sequence. Apply the distance formula to calculate the length of chords connecting points in an elliptical sequence. Ever wondered how planets trace their orbital paths or how a Spirograph toy creates such intricate patterns? 🌌 We can model these as a kind of 'skip-counting' journey around an ellipse! In this...

TermDefinitionExample Parametric Equations of an EllipseA method of representing an ellipse where the x and y coordinates of each point are defined as functions of a third variable, called a parameter, usually denoted by 't'. For an ellipse centered at the origin, the equations are x = a cos(t) and y = b sin(t).The ellipse given by x²/25 + y²/9 = 1 can be represented parametrically as x = 5 cos(t) and y = 3 sin(t). Parameter (t)The independent variable in the parametric equations of an ellipse. It is an angle, but it does not directly correspond to the geometric angle of the point from the center, unless the ellipse is a circle (a=b).For the point P on the ellipse x = 5 cos(t), y = 3 sin(t), if t = π/2, the coordinates are (5 cos(π/2), 3 sin(π/2)) = (0, 3). Elliptical Skip-Count...

Parametric Point Generation Formula P_n = (x_n, y_n) = (a \cos(t_0 + n\Delta t), b \sin(t_0 + n\Delta t)) Use this formula to find the coordinates of the nth point (where n starts from 0) in a skip-counting sequence. Here, (a, b) are the semi-axes, t₀ is the starting parameter, n is the step number, and Δt is the angular step. Periodicity Condition A sequence repeats when k\Delta t = 2\pi m, for integers k and m. To find the number of unique points, k, find the smallest positive integer k such that k times the angular step (Δt) is an integer multiple of 2π. If Δt = (p/q) * 2π, where p/q is a reduced fraction, the number of unique points is q. Chord Length Formula d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} Use the standard distance formula to find the length of the stra...