Click on the arrows at the bottom of the applet to animate the rectangle construction!
Consider the rectangles from the last activity.
What were the dimensions of the rectangles?
1x2, 2x3, 3x5, 5x8, 8x13, 13x21
Use the connections you made between the Fibonacci Sequence and the rectangles from the last activity.
What will the dimensions of the next three rectangles be?
21x34, 34x55, 55x89
Rewrite the dimensions of each rectangle as a ratio (using a vinculum) of the larger dimension to the smaller dimension. Start with the 1x2 rectangle and end with the 55x89 rectangle. Convert each ratio to a decimal.
What do you notice about these ratios and decimals?
1x2 is 2/1 = 2[br]2x3 is 3/2 = 1.5[br]3x5 is 5/3 = 1.6666666...[br]5x8 is 8/5 = 1.6[br]8x13 is 13/8 = 1.625[br]13x21 is 21/13 = 1.615384...[br]21x34 is 34/21 = 1.619047...[br]34x55 is 55/34 = 1.6176470...[br]55x89 is 89/55 = 1.6181818...[br]We will discuss this as a whole class, so be ready to share your noticings!
Rewrite the dimensions of each rectangle as an ordered pair with the smaller dimension as the x-coordinate and the larger dimension as the y-coordinate. Start with the 1x2 rectangle and end with the 55x89 rectangle. Graph the points and create a line of b
Geogebra writes the equation for the line of best fit in standard form (Ax + By = C). Rewrite the equation in slope-intercept form (y = mx + b).
What do you notice about the slope of the line? Where have you seen this number before? Why do you think this number is important?
We will discuss this as a whole class, so be ready to share your thinking!