Catapult Trajectories: Don t Let Parabolas Throw You


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1 Catapult Trajectories: Don t Let Parabolas Throw You TEKS Aa (2) Symbolic reasoning plays a critical role in algebra; symbols provide powerful ways to represent mathematical situations and to express generalizations. Students use symbols in a variety of ways to study relationships among quantities. Aa (5) Students use a variety of representations (concrete, pictorial, numerical, symbolic, graphical, and verbal), tools, and technology (including, but not limited to, calculators with graphing capabilities, data collection devices, and computers) to model mathematical situations to solve meaningful problems. Aa (6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problemsolving, language and communication, and reasoning (justification and proof) to make connections within and outside mathematics. Students also use multiple representations, technology, applications and modeling, Ab4 (B) Write the equation of a parabola using given attributes, including vertex, focus, directrix, axis of symmetry, and direction of opening; Ab8 (C) Predict and make decisions and critical judgments from a given set of data using linear, quadratic, and exponential models. Duration: Approximately 3 days Materials and Resources ipads/computers one per group of students 3D printed catapult Two types of projectiles (1 set per group) Graph paper (1 inch grid) Colored markers Tape Worksheets (See attached) 1 yard or meter of craft paper
2 Engage The Engage portion of the lesson is designed to create student interest in the concepts addressed and make connections to past and present learning. See the first 2 minutes of the YouTube video Dan Meyer on Real World Math at Show about 2 minutes of the YouTube video to the class: Trebuchet Siege Artillery Battle Castle with Dan Snow or ask questions like these: 1) What is a trebuchet? What is a catapult? Describe them. 2) Do you know any real life examples of catapults? What can they be used for? 1. (Catapults are any device that throws an object, although it commonly refers to the medieval siege weapon.  Trebuchets are a TYPE of catapult, using gravity (with a counterweight) or traction (men pulling down), to propel the arm and often employing a sling at the end of the arm for greater distance) 2. Catapults are used for dropping bombs from boats on submarines, to propel aircrafts into the air from aircraft carriers, and for fun. Kids like to hurl water balloons at each other using catapults Explore The Explore portion of the lesson provides the student with an opportunity to be actively involved in the exploration of the mathematical concepts addressed. This part of the lesson is designed for groups of three or four students. For the data collection portion, if there are 4 students in the group, they will identify 3 observers and one recorder. If there are 3 students, one will both observe and record. Provide instruction, as students need guidance and assistance. Have them create a 1 yard or 1 meter number line on the floor. Have them use a sheet of large graph paper (1 grid). Tape it to the edge of a table or two desks with the bottom edge as close to the floor as possible. Once in this position move the table or desks to align the number line on the floor with the grid lines on the paper taped to the table or desks. Align the catapult at the zero mark on the number line. Practice launching the first projectile and having one spotter finding the point where it lands and another spotter looking for the apex of flight on the craft paper. Once the teams become successful, have students start collecting data using Worksheet #1. Repeat the process for the second projectile. Be sure to mark the points on the number line and identify the projectile and launch order. They should have 3 trials for each projectile so they need to be sure to label their data accurately.
3 Worksheet #1 Catapult Lab Investigation Name Date Purpose 1) To investigate characteristics of the function resulting from height vs horizontal distance of a projectile, and 2) To find the equation of the curve of best fit formed by a projectile from an algebraic approach. Materials (for each group) 1 3D printed catapult 1 small projectile 1 large projectile tape colored markers 1 sheet of large graph paper (1 inch grid) Pre Lab Questions Draw a sketch of what you think will be the path of the projectile from the catapult to the ground. Setup NOTE: Each group should set up the lab station in the same manner. Any deviation should be discouraged if class averages will be calculated. All catapult should be taped the same distance from the edge of the number line. The vertical scale is height in inches.
4 Figure 1 Catapult Gradations Try marking every two inches In Figure 1 the dots represent the students observations as the projectile crosses the vertical lines. There are three students who each observe the location of the projectile as it passes each vertical lines on the graph paper. The dotted line represents the actual path of the projectile (only three points of which are recorded). NOTE: Each group should set up the lab station in the same manner so class averages can be calculated. All catapults should be taped the same distance from the edge of the number line. The vertical scale is height in inches. Procedure 1. Orient the paper vertically. 2. Make the origin the lower left corner. 3. Draw three vertical lines on the grid at 3, 6, and 9 inches, each in a different color. 4. Hang and tape graph paper from the desk edge. 5. Tape the catapult so that the tip is 4 inches from the edge of the number line. 6. Put catapult in front of graph as shown. 7. Place the large projectile in the catapult. 8. Release the projectile from catapult and observe the projectile path. Each of the other three students must mark the spot where the projectile crosses a given colored line. 9. Record the heights in data table. 10. Repeat steps 8 and 9 twice more for a total of three trials. 11. Repeat steps 7 to 10 using small projectile. Projectile 1 Launch at x= 0 at x = d 1 at x = d 2 at x = d 3 Maximum Trial 1 0 d 1= Trial 2 0 d 1= Trial 3 0 d 1= Group Average 0
5 Projectile 2 Launch at x= 0 at x = d 1 at x = d 2 at x = d 3 Trial 1 0 Trial 2 0 Trial 3 0 Group Average 0 Analysis: In this situation identify the independent variable. What is the dependent variable? 1) Enter the four data points for the forward distance in one spreadsheet column (or list) in the calculator and the corresponding four data points for the group average data for vertical distance in a second column or list, using the small projectile data. 2) Enter the corresponding four data points for the class average data for vertical distance in a third column or list. 3) Graph the vertical distance versus the forward distance as a scatter plot. Determine the scale, label the axes, and sketch the graph. Draw a smooth curve that goes through the points as much as possible. 4) List the types of functions you have studied previously (might include linear, exponential, or quadratic). Write the characteristics of your graph that are the same as each function you listed. Give the characteristics that are different.
6 5) Which functions that you have studied previously have a highest point to the graph and/or a lowest point to the graph? 6) Does your graph have a highest or lowest point? If so, which does it have? 7) Trace the points on your scatter plot and estimate the coordinates of any high or low points. 8) Use a dotted line to draw any lines of symmetry you can find on your graph. Give the equation of the line(s) of symmetry. 9) From the smooth curve you drew, estimate the xintercepts of the graph. 10) Is there a possible relationship between the xvalues of the intercepts and the high/low point on your graph? Explain. 11) Find the quadratic equation of best fit using quadratic regression (QUADREG) on the graphing calculator. Confirm your curve with the data points. Record your equation in the appropriate cell below (round to the nearest tenth). 4) Repeat this procedure with L1 and L3. 5) Repeat this entire process for the large projectile and record the curve equation. Projectile Size Averaged Data Fitted Curve Equation Small Group Large Group
7 Explain The Explain portion of the lesson is directed by the teacher to allow the students to formalize their understanding of the TEKS addressed in the lesson. Use the Facilitation Questions to prompt student groups to share their responses. The teacher will introduce the term parabola (a shape defined by a quadratic equation), describe the characteristics of a parabola, and the general equation yielding a parabolic graph (y = ax 2 + bx c). The teacher will define terms, such as maximum and minimum, which students will use later. The teacher will also distribute the Introduction to Parabolas Worksheet (WS #2). This worksheet will apply prior knowledge of graphing on the xyplane, using ordered pair notation, to parabolas. The teacher will introduce the vocabulary keywords and concepts by discussing characteristics of quadratics key terms: Parabola: shape defined by a quadratic equation Quadratic equation: equation that can be written in the standard form ax 2 + bx + c = 0, where a, b, and c are real numbers and a does not equal zero. Leading coefficient: In a polynomial, the coefficient of the term with the highest degree is called the leading coefficient. Coordinates: pairs of numbers, which specify the position or location of a point or of an object. Vertex: the peak in the curve. The peak will be pointing either downwards or upwards depending on the sign of the x 2 term. Maximum: The maximum value of a quadratic function f(x) = ax 2 + bx + c where a < 0, is the y coordinate of the vertex. Minimum  The minimum value of a quadratic function f(x) = ax 2 + bx + c where a > 0, is the y coordinate of the vertex. Roots: solutions of an equation (also xvalues of the xintercepts) xintercept: an xintercept of a curve is a point at which the line crosses the xaxis. The teacher will explain polynomial equation progression from linear to quadratic, comparing and contrasting shape of the graph, degree of x, number of terms, slopes, etc. In addition, the students will see that while two points can be used to find the equation of a line, three points are needed for a parabola. The standard form will be used for the basic lessons. The vertex form could be as an extension. The teacher will ask the students to give an example of an object that is roughly parabolic and opens up. (Does it have a minimum or maximum value? Similarly, does a parabolicshaped object opening down have a minimum or maximum?) Facilitation Questions Explain Phase The teacher will link the behavior of parabolas to a trajectory caused by catapulting an object. The teacher should ask questions to tie concepts together: Why does a projected object take the path of a parabola? Which upward or downward path might it have under what conditions?
8 Student pairs will discuss how key concepts of vertex and x intercepts might have practical meaning in a trajectory situation. Extension In this portion of the lesson, students will learn more about quadratic functions and practice skills related to the concepts they learned in the Explore and Explain sections. Place students into groups of four. Pass out ipads, or computers to groups. Allow students to use smart phones, or their own laptops to explore internet sites about real world parabolas. Some suggested sites could be Have each student write down 2 things they learned about parabolas and have each student draw/copy/print at least 1 image of what that they have seen around them (no duplications from the shared video) of parabolas. Have students share in groups and then with the whole class what they have found out about parabolas. Facilitation Questions Do all parabolas look like McDonald arches? Why or why not? Draw in the air with your finger two types of parabolas (think about a rollercoaster) Depending on what students say be aware that a flattened catenary is common in construction. The reason is that for engineering purposes and load distribution, a flattened catenary provides greater load distribution with vectors going more directly into the substrate avoiding deviation from 0 degrees. From the examples found in your group, show 2 types of examples of parabolas. Is the vertex of a parabola always at the highest point? Why or why not? Can the vertex be at the lowest point? Show an example and explain. Will this parabola open upward or downward: y = 3x x 2? How can you tell? Will this parabola open upward or downward: y = 2x 2 + 4x 4? How are you sure? To connect the learning to real life, the teacher will play the video for the class and or post it to the class website: YouTube video of Algebra 2: Parabolas in the Real fm4 and/or Parabolas in the Real and/or Parabola s in the real
9 Students will recognize catapult paths as tracing downwardopening quadratic equation graphs. They will be able to identify parameters of quadratic equations (i.e., a, b, and c) and special parts of the graphs (e.g., x and yintercepts, maximum). How could catapults apply in real life? Why would accuracy of the projected path be important? The teacher will link the path of a catapult to parameter changes in quadratic functions. Extension/ Homework. As a homework assignment, each student can describe a situation demonstrating a parabolic motion, give specific dimensions for the situation, graph a parabola that fits the situation, and determine the quadratic function for the situation. Additional Extension. Use Worksheet #2 and have students use the calculator to explore changes in parameters. Give students a set of quadratic functions that have different values for the parameters as ones they could use. They can also use textbook problems in this portion. The most important part to do is the exploration and connections.
10 Worksheet #2 Name Date Identify the marked coordinates of the following parabola in ordered pair notation. 1) point x y A B C D E F G 2. point x y A B C D 3). If a parabola opens down (like a lampshade), does it have a maximum or minimum? How do you know this? 4). If a parabola opens up (like a tulip), does it have a maximum or minimum? Write how you can you prove this to another student.
11 Answers for Worksheet #2 1. point x y A 3 9 B 2 4 C 1 1 D 0 0 E 1 1 F 2 4 G point x y A 35 B 1 3 C 0 4 D 2 0 3). If a parabola opens down (like a lampshade), does it have a maximum or minimum? 4). If a parabola opens up (like a tulip), does it have a maximum or minimum?
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