Notice that the pattern, at large scales, appears to be linear. Then rescale the coordinate system (by dragging one of the numbers on the x-axis) until you can see all the points. When the points appear, verify that the first few points are at (2, 3), (3, 5), and so forth. (This command is available in Sketchpad release 4.02, available here.) In the dialog, set x to seed1 + seed2, and set yto seed2 + (seed1 + seed2) and click Plot. Increase the depth of the table to 16 or 18 rows, and select the table and choose Graph | Plot Table Data. (In other words, we’ll plot one element of the sequence against the next, rather than plotting each element against the corresponding depth of the iteration.) Iterate seed1 to seed2 and seed2 to seed1 + seed2. This time we’ll plot the function representing the iteration rule, rather than the function representing the sequence. Graphing the Fibonacci Sequence: Here we'll plot data directly from the table itself. Try changing the seed and the difference, and see if you can find a connection with the slope and intercept of the graph. When you iterate n to n + 1 and seed to seed + difference, the result of the iteration is both a table of the values and an iterated image of the plotted point, showing the graph of the value of the sequence as a function of n. Graphing a Sequence: This sketch contains a plotted point at (n, seed). What happens if the initial seed is zero? Be sure to try zero and negative numbers as radicands, and observe the convergence of the result. For instance, with the original radicand of 5, try a really bad first guess, in the hundreds or even thousands, and observe the behavior of the iteration. Investigate how quickly the calculation converges, trying different values for the initial seed and for the radicand.
Iterate the seed ( x 0) to the result of the calculation, and increase the number of iterations shown in the table. Given a radicand s and a seed x 0, we divide the radicand by the seed, and then find the average of the seed and the result of the division. The Babylonian iterative square root algorithm (which may be almost 4000 years old) makes more sense and is easier to remember. Roots: Few of us now teach the square root algorithm from years ago that some students learned, many students hated, and almost none understood. The animation on this page is the graph of an arithmetic sequence, with a depth that changes. In this portion of the presentation, we look at numeric iterations as functions, we use them to plot some graphs, and we discover some interesting results. Select the table and change the depth of iteration by pressing the + or – keys on the keyboard. Make sure the resulting table has constant second differences. Map seed to seed + diff, and map diff to diff + diff2. Select both the seed and the diff parameters and choose Iterate. Quadratic Sequence: Calculate seed + diff, and also calculate diff + diffs. Select the resulting table and change the depth of iteration by pressing the + or – keys on the keyboard. Map seed1 to seed2, and map seed2 to seed1 + seed2. Select the resulting table and change the depth of iteration by pressing the + or – keys on the keyboard.įibonacci Sequence: Select both seed1 and seed2 parameters and choose Iterate. Then select the seed parameter and iterate it to the seed*factor calculation. Geometric Sequence: Use the calculator to calculate seed*factor.
The animation on this page is an arithmetic-sequence iteration using Iterate To Depth, so that the depth of the iteration is controlled by an animated parameter.Īrithmetic Sequence: Select the seed parameter and iterate it to the seed + difference calculation. Numeric iteration is the first topic that we’ll use Sketchpad to explore. The animation on this page is a simulation of projectile motion. Topics: The presentation provides examples and applications of iteration in each of these topic areas. The animation on this page is a spiral formed by rotating and dilating a triangle. Ideas: Iteration is one of the big mathematical ideas that crosses the lines among all the different areas of mathematics. (Be sure to use the Next button when going from page to page, in order to activate the animation on the page you’re going to.) Click the Next button in the lower right corner of the sketch to go on to the next page. Title: The animation on this page is a Pythagorean fractal tree. Iteration Through the Math Curriculum: Sketchpad Does It Again and Again Scott Steketee
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