Recursive Designs and Fractional Thinking The Nature of Recursive Thinking One of the most exciting topics in mathematics is that of recursion. The concept can be used to describe plant growth along with having rich application in many areas or thinking. Recursion in a simple form can be thought of as establishing a rule and then using that rule repeatedly to continue creating complexity in a pattern. This project will use recursive divisions of a rectangle to create mosaics of patterns. We start with the rectangle and the rule. To Start: Draw a large rectangle! The Rule(s): 1. Divide the rectangle into thirds in each direction 2. Choose a pattern of rectangles to color (We have chosen the sides, top middle and bottom middle) 3. With the uncolored rectangles go to step 1 Rosie Ayers & Richard Seitz, Fine Art Handouts RSeitz, October 2016
K-12: In figure one, what can in the picture? Pick another notice about the fractions. you tell about the fractions seen picture and share what you 9-12: What sequence of shaded squares at each step? If you were to go on to infinity what fraction of the entire rectangle is shaded? Explorations: Other shapes, other divisions, unequal divisions, other fractals, 2
Musical Mathematics with a Monochord The Nature of Musical Harmonies Studying a bit of the history of music opens us to a world where mathematics and musicians collide in a rich world of sounds. Examining the work on composers of music is an enticing place to begin to think about the nature of harmony and the beauty of song. This activity will use a monochord to explore just two of the ideas from music, the octave and fifths to see how some mathematical ratios just naturally create the beauty of harmony. Some Definitions from Wikipedia, accessed 9/22/2016 In music, an octave (Latin: octavus: eighth) or perfect octave is the interval between one musical pitch and another with half or double its frequency. In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so. When the A440 pitch standard is used to tune a musical instrument, Middle C has a frequency around 261.6 Hz. Middle C is designated C 4 in scientific pitch notation because of the note's position as the fourth C key from left on a standard 88-key piano keyboard. In vocal music, the term Soprano C, sometimes called "High C" or "Top C," is the C two octaves above Middle C. It is so named because it is considered the defining note of the soprano voice type. It is C 6 in scientific pitch notation (1046.502 Hz) and c''' in Helmholtz notation. Explore the Monochord 1. Our monochord has a scale of 60. Explore the ratios of 1:2 and make some notes of your findings. 2. What do you think they mean by a ratio of 3:2? Explore your thinking with the monochord. K-12: In figure one, what can you tell about the fractions seen in the picture? Pick another picture and share what you notice about the fractions. 9-12: What sequence of octaves and fifths? Create chord combinations for Twinkle Twinkle Little Star using octaves and check the result on Wikipedia. Explore More: Another musical combination is fourths. Explore and explain what Fourths are all about. 3
From Circles to Fractions to Cones to Megaphones 4
Use compasses or this template to make two large circles. Draw a radius, cut them out, put them together and demonstrate fractions. Make your own smaller circles or use this template to have students make their own fraction circles. Can you make cones? How would you make megaphones? 5
Megaphones Figure 1 Bing Search Oct 2016 (c) Creative Common 6
Megaphone Construction is a wonderful way to discuss sound waves and practice using geometry to create a model. Note the center angles of 20 the inside circle, the 20 outside sided polygon where you will only need 8 or 9 sides for the pattern. Try making this out of cardboard! 7
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Super Stories and Fractional Thinking The Nature of Thinking about Stories Engaging students in their love of storytelling, symbolic representations, and visual drawings can help student to clarify their thinking, explore the language of mathematics and improve their self-confidence and ability to communicate. Directing: 1 st Tell a fraction story As the woolly mammoth continued to sink into the tar pit at a rate of 1/48 of its mass every hour, the wolves and saber tooth tiger continued to debate over who should lay claim to the meal. It looked as if the mammoth had already sunken 1/3 of the way and that one of the wolves was on its way to join the giant creature in death. Figure 2: Robert Bruce Horsfall's 1911 drawing What questions does this raise in your mind? Are their questions you could pose and solve using the mathematics of fractions. For example, what percent of the animals pictured are herbivores? 10