Unit Activity Unit: Mathematical Models in Art and Music This activity will help you meet these educational goals: Mathematical PracticesYou will make sense of problems and solve them, reason abstractly and quantitatively, model with mathematics, look for and make use of structure, and look for and express regularity in repeated reasoning. Introduction In this activity, you will see how mathematics can be used to model and understand realworld scenarios and concepts. In the first task, you will explore how different sound frequencies can be mapped using a periodic function. In the second task, you will see how similarity and symmetry promote visual harmony in artwork. __________________________________________________________________________ Directions and Analysis Task 1: Sound Waves Have you ever looked closely at sound waves? They may look like random lines at first glance. But if you zoom in, you will see that they are very similar to graphs of trigonometric functions. Like trigonometric functions, they appear as oscillating waves with a measurable frequency and amplitude. In music, each note has a specific frequency, measured in hertz. The units for hertz are cycles per second (1 sec), or sec-1. The most common note used for tuning an instrument is 1 2014 EDMENTUM, INC. the A next to middle C. Pianists often call this note A 4. This note has a frequency of 440 Hz. This means that the note A4 has 440 cycles in one second. Any musical note can be graphed using the function f(x) = sin(y 2x), where y is the frequency of the note and x is the time in seconds. a. Using this graphing tool, graph the function for note A4. Paste a copy of the graph below, keeping the default scale. What does it look like? Why does it appear like that? Type your response here: b. Now change the scale of the graph. Go back to the graphing tool and scroll down to adjust the minimum and maximum values of x and y. Change the minimum value of x to -0.01 and the maximum value to 0.01. Change the minimum value for y to -1.5 and the maximum value to 1.5. Graph the function again, and paste a screenshot of it below. What does the graph look like now? What is the amplitude of the wave? What is the period of the wave? The period of a wave is the time it takes for one wavelength, or cycle, to pass a given point. It is the inverse of the wave's frequency. Type your response here: c. A musician usually plays more than one note at a time; these are called harmonies. If you play three or more notes at once, you get a chord. An example of a chord is A-C# -E. Each of these notes has a different frequency. The frequency of A is 440 Hz, the frequency of C# is 554.37 Hz, and the frequency of E is 659.25 Hz. Create a function for each note, and then add them to create a combination function for the chord A-C#-E. Write all four of these functions in your response. Graph the combination function, keeping the scale of the graph similar to the scale you used in part b. Paste a copy of the graph below. Type your response here: d. Compare the graphs from parts b and c. What are some similarities and differences between the two graphs? Type your response here: e. Noise-cancelling headphones target the undesirable sound waves that you would hear without the headphones. Then they emit a sound wave that is a reflection across the xaxis of the undesirable sound wave. To see this on a graph, consider the note A 4 again. The sound wave needed to cancel this note out is represented by the function 1 f ( x ) sin 880 x . Add this function to the function for A4 and graph the 880 combination of these functions using the default dimensions. Paste the graph below. What does the graph look like? What does it indicate about the sound produced? 2 Type your response here: Task 2: Art and Photography The concepts of symmetry and similarity play a role in art and photography. To explore these concepts, use this picture of the Eiffel Tower in Paris, France. a. Measure the width at the base of the tower and height of the tower in the image to the nearest millimeter. For the height, measure all the way to the top of the spire. You can use an actual ruler and place it on the image to make measurements. Note your observations below. Type your response here: b. What is the ratio of the base width to the height of the tower in this image? Type your response here: c. In reality, the Eiffel Tower is 325 meters high and 125 meters wide at its base. What is the ratio of the base width to the height for the actual dimensions? Type your response here: d. Is the ratio from part b about equal to the ratio from part c? Why or why not? Type your response here: 3 e. Research more about the Eiffel Towers and its dimensions. Does the Eiffel Tower have rotational symmetry about a line? If so, which line? Type your response here: f. Is the Eiffel Tower symmetrical across any planes? If so, how many? Type your response here: g. Why would artists use symmetry in their work? What are some of the benefits of using symmetry? Express your opinion or do additional research. Type your response here: __________________________________________________________________________ Resources Document any references you used for this project below. At minimum, include a title and URL for any Internet resource: __________________________________________________________________________ 4 Evaluation Your teacher will use this rubric to evaluate the completeness of your work as well as the clarity of thinking you exhibit. Total Points: 10 Task 1: Sound Waves Task points: 5 a. Graph note A4, and describe its shape. 1 b. Find the period and amplitude for note A4. 1 c. Graph chord A-C#-E. 1 d. Compare the graphs of A4 and A-C#-E. 1 e. Explain the graph from noise-canceling headphones. 1 Task 2: Art and Photography Task points: 5 a. Measure the Eiffel Tower in the image. 0.5 b. Find the ratio of measurements of the tower in the image. 0.5 c. Find the ratio of the actual dimensions of the tower. 0.5 d. Compare the ratios from parts b and c. 1 e. Describe the rotational symmetry in the Eiffel tower. 1 f. 1 Describe the planes of symmetry in the Eiffel tower. g. Explain why an artist might use symmetry. 0.5 5