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Pistons and Trig. Functions 30.00 25.00 h 20.00 e i 15.00 g h 10.00 t 5.00 0.00 angle Tom Berg Mitchell High School 920 N. Capital, Mitchell, SD 57301 email: [email protected] phone: (605) 995-3034 Integrated Math II, Grades 10-12 1 Class setup: Students will be divided into heterogeneous groups of 3 to 4 students. Teacher’s role: The teacher’s role is to present the lab to the students. While the students are working on the lab, the teacher will be walking around the room monitoring the students. Textbook and other resources: CORD Applied Mathematics (Center for Occupational Research and Development), Waco, Texas (1994), Unit 22-Using Trigonometric Functions. Content/topic/theme: Trigonometric Functions Real-world experience: I don’t believe that this will give them a real-world experience but rather a real hands-on experience. As students rotate the piston, they will see how the wave is produced and how to find the amplitude, frequency, and period. Hopefully, they will relate the modeling of this wave to other waveforms. Why do students need to know this? Students will learn about sinusoidal waves and their connection to sound waves, vibration waves, electrical waves (AC current), ultrasonic waves, and microwaves. What is the connection to prior learning experience ? Students will use skills previously taught in Unit 1: Learning Problem-Solving Techniques, Unit 2: Estimating Answers, Unit 6: Working with Lines and Angles, Unit 9: Using Ratios and Proportions, Unit 11: Using Signed Numbers and Vectors, Unit 15: Using Formulas to Solve Problems, Unit 17: Graphing Data and Unit 21: Using Right-Triangle Relations. Discipline-specific standard(s): National Math Standards: Algebra: Understand Patterns and Use Mathematical Models Problem Solving Connections Representation South Dakota State Math Standards: Goal 1: Algebra; Indicator 1: Analyze procedures to transform algebraic expressions; Standard 7: extend the concepts of algebra to other types of functions, e.g., trigonometric, exponential, and logarithmic Goal 5: Patterns, Relations, and Functions; Indicator 1: Analyze and describe the properties and behaviors of relations, functions, and related inverses; Standard 1: use various representations of functions, e.g., graphs, tables, symbolic forms. Goal 5: Patterns, Relations, and Functions; Indicator 1: Analyze and describe the properties and behaviors of 2 relations, functions, and related inverses; Standard 8: create tables or graphs to interpret relations and/or functions. Goal 5: Patterns, Relations, and Functions; Indicator 2: Apply relations and functions to complex problem-solving situations; Standard 5: identify natural phenomena that are cyclic. Activities (What will the students do?) The students will do a hands-on math lab involving a piston and relating this to a cosine wave. Strategy (How will they do it?) The students will have a model of a piston. The piston will be in its tallest position (top-dead-center). The students will then measure how far the top of the piston is from the base (height). The students will turn the piston in 30° increments and take the height measurements. They will continue turning the piston until two revolutions (720°) have been achieved. The students will make graphs from their tables of values and answer the questions. The students will also use the graphing calculator to plot the points. They will also type in their equations and see if their graphed equations match their plotted points. Assessment strategies: The students will be assessed on how they answer the questions in the lab. The students need to fill in the table of values; make graphs (including labeling of the x and y axes); answering questions about the kind of curve it is, the amplitude, the period, the frequency, and the vertical shift. They are also graded on the correct placement of these items into the equations. Warm-up or closure strategy: At the beginning of the class, the students will be given a graph of a wave. The students must find the amplitude, period, and frequency of this wave. They must also identify whether it is a sine wave or a cosine wave. 3 From Unit 22 - Using Trigonometric Function Exercise 20: During the stroke of a piston, the connecting rod pushes the crankshaft around 360°, as illustrated here. As the crankshaft turns, the radius of the "throw" makes an angle T with a line between the shaft and the end of the rod (as shown in the drawing). The changing distance between the end of the rod and the crankshaft center can be computed using the formula below. (This changing distance can be used to determine the volume of gas/air mixture ahead of the piston as the crankshaft rotates.) where h is the distance between the end of the rod and the crankshaft center, r is the radius from the crankshaft center to the throw, d is the length of the rod, and T is the angle made by the radius r and a line between the crankshaft center and the end of the rod. a. Using a value of r = 2.5 inches and d = 6.0 inches, make a table of values for h for angles of T equal to 0°, 30°, 60°, and so on, up to 360°. b. Draw a graph of these values for d, and connect the plotted points with a smooth curve. c. Referring to the drawing, you should be able to predict the value for d when T = 0° and T = 180°. Check the results from the formula with your expectations. d. Examine the shape of the curve in your graph. Which curve does it most resemble—a sine curve or a cosine curve? 4 PISTON LAB Start with the piston in the top-dead-center (TDC) position. This is the 0° position. Measure the height of the piston head from the base. Turn the gear 30° clockwise, measure the height. Continue turning the gear 30° for 2 revolutions and take the height readings (in cm). a. Fill in the chart and graph the results. Gear angle height 0 30 60 90 120 150 180 210 240 270 300 330 360 390 420 450 480 510 540 570 600 630 660 690 720 b. Does this look like a sin or cos curve? c. What is the amplitude? d. What is the period? e. What is the frequency? f. What is the vertical shift? g. Write the equation: ht = 5 6