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matrix animation

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									                     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                           3                         2                        1
   Transformations       All 5 worked successfully        Only 4 worked            Only 3 worked           2 or less worked
                             Superbly effective          Highly effective             Effective               Ineffective
  Quality & Creativity        Stays on screen            Stays on screen        Not always on screen     Most action off screen
                               Very creative                 Creative            Somewhat creative           Not creative
                                                       Grammar and vocab.
                          Correct grammar and                                    Quite a few errors in    Significant errors in
                                                       mostly correct, listed
  Written Description      vocabulary, listed all                               grammar/vocab., some     grammar/vocab., most
                                                      most matrices and their
                          matrices and their uses                                 matrices not listed    matrices not discussed

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                         WHERE TO LOOK                   COMMAND                          WHERE TO LOOK

 AxesOff/AxesON                  FORMAT                          ClrDraw                          DRAW

 For                             PRGM CTL                        END                              PRGM CTL

 Line                            DRAW                             →                               STO in lower left

 [ and ]                         above X and --                   ,                               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.

       AxesOff                                           Clears the axes from the screen.

       ClrDraw                                           Clears any previous drawing from the screen

       [A] → [B]                                         Places the shape matrix into matrix B.

       For (N, 1, 8, 1)                                  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.
       For (K, 1, 50, 1)                                       Beginning of a loop that causes the program to wait a
                                                         moment before changing the drawing.

       End                                               End of pause loop.

       ClrDraw                                           Clears drawing to get ready for the next one.

       End                                               End of loop (N, 1, 8, 1)
6)    Before running the program, you have to store the rotation matrix for R45o in matrix C and the shape matrix in matrix
                                              ⎡0 0 3 0 ⎤
     A. The shape matrix for this example is ⎢         ⎥ . 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.
    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

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

                   20 ft

           18 ft

                                   Sidewalk is 36 inches wide

                                   Driveway is 15 feet wide and
                                   60 feet long

                   Number of cubes Number of cubes Number of cubes Number of cubes Number of cubes                    Tota
      Size         needed to build  with exactly 3  with exactly 2  with exactly 1   with 0 faces                      fac
                      (volume)      faces painted   faces painted    face painted      painted                         (su





  10 x 10 x 10


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?


                  Number of       Number of         Number of        Number of                       Total number
                                                                                      Number of
                 cubes needed     cubes with        cubes with       cubes with                         of faces
    Size                                                                             cubes with 0
                    to build       exactly 3      exactly 2 faces   exactly 1 face                      painted
                                                                                     faces painted
                   (volume)      faces painted       painted           painted                       (surface area)

 2x2x2                8                8                0                 0                0               24

 3x3x3               27                8                12                6                1               54

 4x4x4               64                8                24               24                8               96
 5x5x5                 125                 8         36               54          27        150

 10 x 10 x
                      1000                 8         96              384         512        600

 nxnxn                  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           EXPECTED                   BANKER’S OFFER                 %
                   LEFT             VALUE
         Round 1     20

         Round 2     15

         Round 3     11

         Round 4     8

         Round 5     6

         Round 6     5

         Round 7     4

         Round 8     3

         Round 9     2

          Rnd 10     1

         ROUND     CASES           EXPECTED                   BANKER’S OFFER                 %
                   LEFT             VALUE
         Round 1     20

         Round 2     15

         Round 3     11

         Round 4     8

         Round 5     6

         Round 6     5

         Round 7     4

         Round 8     3

         Round 9     2


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 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.


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.


Based on the measurements gathered in Step 1, determine the areas of the following regions:

GREEN AREAS                                          WHITE AREAS

border around the court (1)                          Semicircles at top of keys (2)
3 second lanes (2)                                   Small circle at center court (1)
3 point “lines” (2)                                  Large circle at center court (1)
semicircular rings at the top of the keys (2)        Small blocks around lanes (12)
half court line (1)                                  Large blocks around lanes (4)
mid court lines (4)                                  Volleyball court lines
small ring at center court (1)                       — shorter lines (14)
large ring at center court (1)                       — longer lines (6)
                                                     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.


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        you
determined the amount of paint needed and the total cost of the paint.


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



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.


1 - more than 3 mistakes, unreadable in places, neither neat nor organized, contains major errors
3 - no more than 3 mistakes, legible, neat or organized but not both, minor errors
5 - accurate, easy to read and understand, neat and organized, publishable quality


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
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
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
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
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
      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.   The name and par for each hole.
2.   The actual size of the hole and the scale used in your drawing.
3.   For each hole, find the area of the actual hole and the area of the scale drawing.
4.   Where the tee is located (T) and where the hole is located (H).
5.   All boundaries and obstacles.
6.   The correct path to be taken for a hole-in-one (including arrows).
7.   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.

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)   Name and par - 1 points
b)   Tee (T) and Hole (H) labeled - 1 point
c)   Actual size listed, scale shown, accuracy of scale drawing - 3 points
d)    Path for hole-in-one drawn with arrows - 1 points
e)   Reflections shown and numbered - 2 points
f)   Is the path accurate and mathematically correct - 5 points
g)    Area of actual hole and area of scale drawing is listed and accurate - 4 points
h)    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


Dan Schab
Williamston High School
3939 Vanneter Road
Williamston, MI 48895

Group Presentation Guidelines

PART I - each presentation must contain the items below. Each item is worth one point.

_____   1.   Introduction of group member(s) to the class.
_____   2.   Name and general description of the golf course.
_____   3.   Name and general description of the hole.
_____   4.   Accurate geometric description of how to achieve a hole-in-one.
_____   5.   Ideas for future revisions and improvements (at least two).
_____   6.   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

 3     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

 2     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

 1     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

 0     Totally inappropriate response
       No response

                                                             TOTAL: _________/ 10

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