matrix animation

FST Matrix Animation Project (83/84 version) Goals of the project: 1) To become familiar with programming on a graphing calculator 2) To utilize transformation matrices 3) To have an opportunity to use creativity, technology, and mathematical problem-solving to create a product Many of you spend your free time either playing video games or watching movies that contain special effects. Much of what you see on the television or movie screen is done by computer animation. Using the tools of transformations and matrices you can produce a simple animation similar to that used in video games and movies. In this project, you will create an animation using matrices and then present it to the class using the overhead projector calculator. In this project, you will use your knowledge of matrices and transformations to write a program which: 1) draws a unique shape (HINT: a square is not unique!), 2) transforms the shape in at least 5 ways using at least 3 of the following types of transformations: size change; rotation; reflection; and translation, and write a description of the program that lists the matrices used and describes the transformations. Your project will be evaluated on whether or not your program successfully performs all the transformations, on the clarity of the written description, and on the quality and creativity of the overall effect. You will be assessed by the following rubric: SCORING RUBRIC 4 Transformations Quality & Creativity All 5 worked successfully Superbly effective Stays on screen Very creative Correct grammar and vocabulary, listed all matrices and their uses 3 Only 4 worked Highly effective Stays on screen Creative Grammar and vocab. mostly correct, listed most matrices and their uses 2 Only 3 worked Effective Not always on screen Somewhat creative Quite a few errors in grammar/vocab., some matrices not listed 1 2 or less worked Ineffective Most action off screen Not creative Significant errors in grammar/vocab., most matrices not discussed Written Description How to do it: 1) On graph paper, draw a shape of your own choosing and label all vertices with coordinates. 2) Represent your shape as a matrix. 3) Write the transformation matrices for the animation that you want to create. Remember we have learned about rotations, reflections, translations, and size changes. 4) Enter the matrices into the memory of your calculator. Be sure to enter your shape matrix in matrix A. Use matrices D through J for your selected transformation matrices. DO NOT use matrix B because it will be used for temporary storage in your program. 5) Enter the program TURN 45 into your calculator. To write a program, select PRGM and NEW. You will then be asked to type the name of the program. You will not need the ALPHA key to do this. This list will help you find some of the programming commands: COMMAND AxesOff/AxesON For Line [ and ] WHERE TO LOOK FORMAT PRGM CTL DRAW above X and -- COMMAND ClrDraw END WHERE TO LOOK DRAW PRGM CTL STO in lower left the key above the 7 → , Very Important Note: When entering a matrix in a program, you need to hit the MATRX key, select names, and then select the name of the matrix you want. You can’t enter a matrix in a program directly from the keyboard. TURN 45 AxesOff ClrDraw [A] → [B] For (N, 1, 8, 1) Clears the axes from the screen. Clears any previous drawing from the screen Places the shape matrix into matrix B. Beginning of a loop that will repeat 8 times. Line ( [B] (1,1), [B] (2,1), [B] (1,2), [B] (2,2) ) Draws a line segment from the point whose coordinates are in the 1st column of the shape matrix stored in B to the point whose coordinates are in the 2nd column. Line ( [B] (1,2), [B] (2,2), [B] (1,3), [B] (2,3) ) Draws a line segment from the point whose coordinates are in the 2nd column of the shape matrix stored in B to the point whose coordinates are in the 3rd column. Line ( [B] (1,3), [B] (2,3), [B] (1,4), [B] (2,4) ) Continues to draw line segments. [C] * [B] → [B] Stores the product of the transformation matrix C times matrix B in matrix B. THIS IS THE TRANSFORMATION LINE. Beginning of a loop that causes the program to wait a moment before changing the drawing. End of pause loop. Clears drawing to get ready for the next one. End of loop (N, 1, 8, 1) For (K, 1, 50, 1) End ClrDraw End 6) Before running the program, you have to store the rotation matrix for R45o in matrix C and the shape matrix in matrix A. The shape matrix for this example is ⎢ ⎡0 0 3 0 ⎤ ⎥ . Run the program and see if it works. You may need to ⎣0 8 6 4⎦ adjust the window to get it to work. Make sure this program is running correctly before you proceed. 7) Check that this program runs using your own shape matrix in Matrix A and the R45o already stored in Matrix C. You will have to edit the lines to make the proper connections. Refer to your drawing. 8) Set up a new program that has YOUR NAME for a title. 9) Recall TURN 45 into this new program. To recall a program into another program, hit RCL (it is above STO) then go to PRGM and EXEC and select the program you want to recall. When the program name appears on the screen after RCL, then hit enter. 10) To create your final program, recall the program in TURN 45 as many times as you need to complete all of the transformations that you wish to show. You will need at least 5 transformations. Edit the transformation line of each section of the program (It currently reads [C] * [B] → [B].) to include all of the transformations that you wish to show. You should have already stored your transformation matrices in the remaining matrices beginning with D. Be sure that you use these matrices when editing the transformation line. But, don’t change [B], just be sure to include the transformation matrix in place of [C] and the proper operation. 11) You should add a final line to your program to reset the calculator back to normal. That line is AxesOn. 12) Write a one page paper describing what your program does. Use correct mathematical vocabulary and list all of the matrices used in your program. Be sure to tell what transformation each matrix is for. 13) Present your project to the class using the overhead calculator. Turn in the paper to your teacher. CONSTRUCTION PROJECT As part owners of a new small construction business it is your job to establish your business in the community. This is possible by fairly examining all of the parameters of a potential job, estimating the cost, computing the amount of a bid and then, if given the job, demonstrating quality workmanship. Typically, especially for government-funded work, the jobs are “bid” and the qualified contractor with the lowest bid gets the work. The government agency wants the job at the lowest price. The contractor wants the job for an acceptable profit. This job is for Mr. and Mrs. Bowles who want their crushed stone driveway and crumbling sidewalks excavated and their driveway and sidewalks replaced with new concrete. The dimensions for the proposed job are shown on the diagram in this document. Township code requires driveways to be 6 inches thick and sidewalks to be 4 inches thick. It will be your job to compute the company cost for the job, add a profit margin, and make a bid for the job. If you underbid and the company loses money, your business goes bankrupt, and you incur a penalty (minus 2 points) on your writing grade. If your bid is large enough so that your company doesn’t lose money but also does not achieve the desired profit margin, you may survive but not for long and your annual bonus is lost (for the writing grade too!). If you are at or above the desired profit margin and are among the three lowest bidders you make a BONUS. If you do make the desired profit margin but are not among the three lowest bidders you lose the job to your competitors, no BONUS for you! Direct Costs to the Contractor Delivered, concrete costs the contractor $77 per cubic yard Excavation and removal of debris costs $55 per cubic yard Forms and their installation cost $0.50 per linear foot and must frame the entire perimeter of the project. (Forms must separate fresh concrete from the surrounding ground and house. Forms should NOT separate fresh concrete from fresh concrete!) “Puddling,” the labor of spreading the fresh concrete, costs $25 per hour. “Puddling” requires about 1 minute per square yard of surface. “Finishing,” the art of putting the final smooth surface on the concrete, costs the contractor $0.13 per square foot of surface. Once you compute the cost of the job, you need to add your profit margin to the cost. Any profit margin less than 15% is generally unacceptable - 15% is considered the minimum profit margin to stay in business. Your project is to establish a name for your construction company. Then write out in a clear and neat fashion for the homeowner to understand, how you arrived at your BID for this job. Grading: The project is worth 25 points. 2 points for correct grammar, spelling, and structure. 2 points for writing style. 19 points for a clear, organized and accurate, step-by-step description of your mathematics. 2 points - the BONUS (0 points - No Bonus; minus 2 points - your business goes under due to underbidding) Extra Credit - 1 point for typed bid The Property Survey Illustrating the Driveway and Sidewalk and the required dimensions House 20 ft 18 ft Driveway Sidewalk is 36 inches wide Driveway is 15 feet wide and 60 feet long CUBE COLORING PROBLEM Number of cubes Number of cubes Number of cubes Number of cubes Number of cubes with 0 faces with exactly 1 with exactly 2 with exactly 3 needed to build painted face painted faces painted faces painted (volume) Size Tota fac (su 2x2x2 3x3x3 4x4x4 5x5x5 10 x 10 x 10 nxnxn Overview: Investigate what happens when different sized cubes are constructed from unit cubes, the surface areas are painted, and the large cubes are taken apart. How many of the 1x1x1 unit cubes are painted on three faces, two faces, one face, no faces? CUBE COLORING PROBLEM - ANSWERS Number of cubes needed to build (volume) Number of cubes with exactly 3 faces painted Number of cubes with exactly 2 faces painted Number of cubes with exactly 1 face painted Number of cubes with 0 faces painted Total number of faces painted (surface area) Size 2x2x2 3x3x3 4x4x4 8 27 64 8 8 8 0 12 24 0 6 24 0 1 8 24 54 96 5x5x5 10 x 10 x 10 nxnxn NOTE 125 8 36 54 27 150 1000 8 96 384 512 600 n3 8 12( n − 2) 6(n − 2) 2 ( n − 2)3 6n 2 a) ( n − 2)3 + 6(n − 2) 2 + 12( n − 2) + 8 = n3 b) 3(8) + 2( 12( n − 2) ) + 6(n − 2) 2 = 6n 2 c) Can the number 1734 ever appear in the 1-face column? 1734 = 6(n − 2) 2 DEAL OR NO DEAL NAME: _______________________________ For each round, find a) the expected value, b) the amount offered by the banker, and c) the percentage of the expected value offered by the banker. Round dollar amounts to the nearest dollar. ROUND CASES LEFT 20 15 11 8 6 5 4 3 2 1 EXPECTED VALUE BANKER’S OFFER % Round 1 Round 2 Round 3 Round 4 Round 5 Round 6 Round 7 Round 8 Round 9 Rnd 10 ROUND CASES LEFT 20 15 11 8 6 5 4 3 2 1 EXPECTED VALUE BANKER’S OFFER % Round 1 Round 2 Round 3 Round 4 Round 5 Round 6 Round 7 Round 8 Round 9 Round 10 GOOD PROBLEMS 1. A bus travels up a one mile hill at an average speed of 30 mph. At what average speed would it take to travel down the hill (one mile) to average 60 mph for the entire trip? 2. Is there a temperature that has the same numerical value in both Fahrenheit and Celsius? 3. On the “scifi scale” water freezes at 40 degrees and boils at 148 degrees. a. Calculate the corresponding temperature on the Fahrenheit scale for a temperature of 58 degrees on the “scifi” scale. b. Develop a formula for converting from the Fahrenheit scale to the Scifi scale. 4. You have 2 glass jars. One contains 1000 blue beads and the other 500 red. Take 50 blue and put them in the red jar, mix thoroughly. Next, randomly select 50 beads from the red jar and put them in the blue jar. Are there more blue beads in the red jar than there are red beads in the blue jar? 5. In a warehouse you obtain a 20% discount but you must pay a 15% sales tax. Which would you prefer to have calculated first, discount or tax? 6. Shade all of the squares blue around the outside rows (all the way around) of an n x n checkerboard. Write an expression describing the numbers of squares shaded blue. 7. Pizza Problem a. If the price is the same and the thickness is the same, which is the better buy, a 10-inch round pizza or a 9-inch square pizza? b. Let’s assume you don’t like the crust and are going to throw away one inch of crust from around the pizza. Which pizza is the better deal? 8. Find the next three numbers in each sequence: a. 1, 1, 2, 3, 5, 8, 13, 21, . . . b. 1, 11, 21, 1211, 111221, . . . THE PAINT PROBLEM WHAT’S THE PROBLEM? The gym floor needs to be repainted. The green is starting to look brown and the white is starting to look green. Both areas must be repainted. Your task is to determine the amount and the cost of the paint needed to cover the green and white markings on the gym floor. STEP 1 - DATA GATHERING In small groups, you will head to the gym to gather data. This data must include the measurements necessary to determine the areas of the regions that need to be painted. Since your time is limited, each group will be assigned only 3 of the 18 letters in WILLIAMSTON HORNETS. Record all measurements to the nearest quarter of an inch or half of a centimeter. DO NOT SPEAK TO PEOPLE OUTSIDE OF YOUR GROUP WHILE COLLECTING DATA. STEP 2 - AREA COMPUTATIONS Based on the measurements gathered in Step 1, determine the areas of the following regions: GREEN AREAS border around the court (1) 3 second lanes (2) 3 point “lines” (2) semicircular rings at the top of the keys (2) half court line (1) mid court lines (4) small ring at center court (1) large ring at center court (1) WHITE AREAS Semicircles at top of keys (2) Small circle at center court (1) Large circle at center court (1) Small blocks around lanes (12) Large blocks around lanes (4) Volleyball court lines — shorter lines (14) — longer lines (6) Letters W (1) I (2) L (2) A (1) M (1) S (2) T (2) O (2) N (2) H (1) R (1) E (1) Note - You are responsible for calculating the areas of three letters. You will be using a class average for the areas of the other letters. STEP 3 - PRESENTATION OF RESULTS A. Rectangular and Circular Areas - Explain how you gathered the data and determined the areas of all rectangular and circular regions. Include charts, diagrams and all steps used in calculating these areas. B. Estimated Areas - Some regions may require you to estimate areas. Provide a written explanation of your estimation process. Include a diagram to support your written explanation. C. Final Cost - For both colors (green and white), state: 1) the amount of area to be painted, 2) the amount of paint needed (see below), 3) the total cost of paint needed (see below). Based on the area information provided from parts A and B, include a brief explanation of how determined the amount of paint needed and the total cost of the paint. you PAINT INFORMATION Price - Green and white paint have the same price. Gallon - $36.00 Quart - $10.00 Coverage - Same for green and white paints. First Coat - 300 square feet per gallon Second Coat - 400 square feet per gallon THIS JOB REQUIRES 2 COATS OF PAINT! GRADING RUBRIC FOR THE PAINT PROJECT Three areas will be evaluated: displays (charts, diagrams, tables, etc.), computations, and the written work. They will be evaluated on a 5-point scale using the indicators below. The final score will be computed by doubling the scores for the displays and computations and adding the score for the written work so that the total points possible are 25. Your percentage grade on this project will be based on the total points earned out of the 25 points possible. Displays 1 - more than 3 mistakes, unreadable in places, neither neat nor organized, contains major errors 2 3 - no more than 3 mistakes, legible, neat or organized but not both, minor errors 4 5 - accurate, easy to read and understand, neat and organized, publishable quality Computations 1 - frequent misuse (or no use) of units, more than 2 uses of the wrong formula, more than 3 computational errors, more than 2 errors in measurement 2 3 - occasional errors in units, occasional missing units, one or two uses of the wrong formula, no more than 3 computational errors, 1 or 2 errors in measurement 4 5 - appropriate units for all numbers, correct choice of formulas, no computational errors, accurate measurements Written Work 1 - no explanation of any estimates used, no description of the process of gathering data, frequent misspelling or misuse of mathematical terms, major errors in spelling or grammar, statement of final conclusion unclear and inaccurate 2 3 - incomplete explanation of any estimates used, incomplete description of the process of gathering data, occasional misspelling or misuse of mathematical terms, several errors in spelling and grammar, statement of final conclusion unclear or inaccurate 4 5 - complete explanation of any estimates used, description of gathering data included, all mathematical terms used and spelled correctly, minor errors in spelling and grammar, clear and accurate statement of the final conclusion Miniature Golf Project - The Design Phase Your design team has been hired by a local real estate developer to create plans for a miniature golf course. Each member of your team must submit designs for four holes. The developer is asking for a detailed sketch of each hole. Listed below are the developer's requirements. 1. He wants variety in the course, not the same design for each hole (keeps things exciting for the players). 2. If correctly played, there should be an opportunity for a hole-in-one on each hole (always a chance for excitement). 3. Every great course has a great finishing hole. Make sure your final hole is your best design (makes for an exciting finish, keeps them coming back). 4. The owner loves bank shots. Each hole must require at least two banks (challenge them). 5. The owner needs a catchy name for each hole. He is also requesting that you come up with a name for the course, something that will bring in customers. Include a name for each hole and a name for the course. The plan for each hole should include: 1. 2. 3. 4. 5. 6. 7. The name and par for each hole. The actual size of the hole and the scale used in your drawing. For each hole, find the area of the actual hole and the area of the scale drawing. Where the tee is located (T) and where the hole is located (H). All boundaries and obstacles. The correct path to be taken for a hole-in-one (including arrows). The reflections used to determine the correct path (numbered). IMPORTANT - All plans must be neat and accurate or they will be unacceptable to the developer. Examples of quality work will be shown in class. PRESENTATION - Each design team will need to make a presentation to the developer and interested community members. The goal of your presentation is to convince the panel of evaluators to choose your designs. Presentation guidelines will be discussed in class. The presentation rubric is attached. GRADES Your grade on this project will be based on how well you follow the developer’s requirements. Listed below are the point values for each requirement. Your percentage grade on this project will be equal to the number of points earned. Remember the developer is very picky and will only accept quality work! Individual Grade Golf Holes - 21 points per hole (84 total points) a) b) c) d) e) f) g) h) Name and par - 1 points Tee (T) and Hole (H) labeled - 1 point Actual size listed, scale shown, accuracy of scale drawing - 3 points Path for hole-in-one drawn with arrows - 1 points Reflections shown and numbered - 2 points Is the path accurate and mathematically correct - 5 points Area of actual hole and area of scale drawing is listed and accurate - 4 points Neatness/professional appearance - 4 points Group Grade - 16 points a) Name for the course - 2 points b) Variety in the holes - 4 points c) Presentation to the class - 10 points TOTAL POINTS POSSIBLE - 100 Dan Schab Williamston High School 3939 Vanneter Road Williamston, MI 48895 schabd@wmston.k12.mi.us Group Presentation Guidelines PART I - each presentation must contain the items below. Each item is worth one point. _____ _____ _____ _____ _____ _____ 1. 2. 3. 4. 5. 6. Introduction of group member(s) to the class. Name and general description of the golf course. Name and general description of the hole. Accurate geometric description of how to achieve a hole-in-one. Ideas for future revisions and improvements (at least two). Ask for questions. PART II - You will also receive a score from 0 to 4 based on the criteria below. __________ 4 Excellent conceptual understanding of the topic Uses visual aid to explain topic Uses mathematical language Neat, legible, and well-organized Communication is logical, clear and easily understood by everyone Answers peer questions appropriately Some conceptual understanding of the topic Uses visual aid to explain topic Uses mathematical language Neatness and organization need improvement Communication is logical, but unclear and not easily understood Attempts to answer peer questions Limited conceptual understanding of the topic Uses visual aid to explain topic Limited use of mathematical language Communication skills are fair Attempts to answer peer questions No conceptual understanding of the topic Very limited use of visual aid There is little, if any, mathematical understanding illustrated Communication skills are poor A very poor attempt has been made to answer peer questions Totally inappropriate response No response TOTAL: _________/ 10 3 2 1 0