SUPPORTING ALL STUDENTS USING PROJECT/PROBLEM-BASED LEARNING

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SUPPORTING ALL STUDENTS USING PROJECT/PROBLEM-BASE
D LEARNING

• Dan Schab
• Williamston High School
• Williamston, MI
• schabd_at_wmston.k12.mi.us

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What is Project-based Learning?

• Allowing students a degree of choice on topic,
  product, or presentation.
• Resulting in an end product such as a
  presentation or report.
• Involving multiple disciplines.
• Varying in duration from one period to a whole
  semester.

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New Role for the Teacher

• Featuring the teacher in the role of facilitator
  rather than leader.
• More coaching and modeling, less telling.
• More learning with students, less being the
  expert.
• More cross disciplinary thinking, less
  specialization.
• More performance-based assessment, less
  paper-and-pencil assessment.

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WHY INTEGRATE PROJECTS INTO THE CURRICULUM?

• APPLICATION OF COURSE CONTENT
• DEVELOP PROBLEM-SOLVING SKILLS
• DEVELOP CAREER AND EMPLOYABILITY SKILLS
• INCREASE STUDENT MOTIVATION

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THE GOLF COURSE PROJECT

• STEP 1 Students Placed Into Design Teams

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STEP 2 Plans are developed

• Student design teams brainstorm ideas for their
  golf course.
• Each student must design four golf holes.
• Detailed, two-dimensional drawings of each hole
  are completed.

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STUDENT WORK
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STEP 3 Three Dimensional
Models are Built
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EXAMPLES OF 3-D MODELS
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3-D MODEL
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3-D MODEL
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3-D MODEL
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3-D MODELS
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3-D MODELS
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STEP 4 PRESENTATION TO CLASS AND EVALUATION
PANEL

• Each student design team gives an oral
  presentation to a panel of evaluators.
• The purpose of the presentation is to share your
  golf course plans and to sell your idea to the
  panel.

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APPLICATION OF COURSE CONTENT

• Drawing/building 2 and 3 dimensional figures
• Measurement of angles, lengths, areas, and
  perimeters
• Ratios and proportions
• Parallel and perpendicular lines
• Properties of reflections

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Development of Career and Employability Skills

• Apply mathematical processes in work-related
  situations
• Present information in a variety of formats
• Plan and transform ideas into a concept or
  product
• Exhibit teamwork and take responsibility for
  influencing and accomplishing group goals
• Solve problems, make decisions and meet deadlines
  with minimum supervision

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INCREASE STUDENT MOTIVATION
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Rigor Relevance Framework
Evaluation 6 Synthesis 5 Analysis
4 Application 3 Comprehension 2 Awareness
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KNOWLEDGE
C Assimilation D Adaptation
A Acquisition B Application
1 2 3 4 5
Knowledge in one discipline Apply knowledge in one discipline Apply knowledge across disciplines Apply knowledge to real-world predictable situations Apply knowledge to real-world unpredictable situations
International Center for Leadership in Education
APPLICATION
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Construction Project

• Geometry
• 3rd hour

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Breakdown of Costs

1. Removal of Debris - 55 per cubic yard
2. New Concrete - 77 per cubic yard
3. Forms - 0.50 per linear foot
4. Spreading the New Concrete requires 1 minute
   per square yard, costs 25 per hour
5. Finishing the Concrete - 0.13 per square foot
6. Profit Margin 15

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Our Company

• Our many workers here at JM Co. work hard to get
  your landscaping done quickly, efficiently, and
  leave you with quality work.
• JM co. specializes in excavating old concrete
  and removing it, laying down new concrete with
  forms, and putting a nice and long lasting finish
  coat on your new concrete.

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Excavation

• Sidewalk
• Volume of sidewalk 420in. x 36in. x 4in
  60,480cubic in.
• Convert 60,480 x (1/46,656) (35/27)cubic yd.
• Driveway
• Volume of driveway 720in. x 180in. x 6in.
  777,600cubic in.
• Convert 777,600 x (1/46,656) (50/3)cubic yd.
• Total Excavation Price
• Add the volumes of the sidewalk and driveway
  (35/27) (50/3) (485/27) cubic yd.
• Price (485/27) x 55 987.96

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Forms

• Perimeter of driveway and sidewalk
• 60 15 57 18 20 15 15 17 3 220
  ft.
• Cost of forms
• 220 ft. x .50/ft. 110
• Total cost 110

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New Concrete

• Delivered concrete
• (485/27) x 77 1383.15
• Spreading the Concrete
• (60 x 15) (3 x 15) (3 x 20) 1005 square
  ft.
• Time 1005 minutes or 16.75 hours
• Labor is 25/hour
• 16.75 x 25 418.75
• Finishing coat
• Cost is .13/ square ft.
• 1005 x .13 130.65 Total 1932.55

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Profit

• Profit margin of 15
• Costs
• Excavation 987.96
• Forms 110 Total 3030.51
• New Concrete 1932.55
• Profit
• 3030.51 x .15 454.58
• Grand total
• 3485.09

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The World of Geometry
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THE WORLD OF GEOMETRY You will work in
groups of four students. Each group is
responsible for taking photographs of 12 items
that can be described/explored mathematically.
The 12 items must share some characteristic that
allows you to group them together. This common
theme should be included in the title of your
project. All group members must appear together
in at least one of the photos. Listed below are
ideas that may get your creative juices
flowing. The Mathematics of
Architecture Symmetry
Around Us The Art of
Geometry/The Geometry of Art
The Geometry of Sports
Tessellations of Williamston
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• FINAL PRODUCT
• The 12 pictures must be neatly displayed on a
  poster (or as slides in a PowerPoint
  presentation). A one-paragraph technical
  description of the photo should be included with
  each picture. At a minimum, this description
  must include a mathematical explanation of the
  photo, an estimation of the size of important
  dimensions, and a listing of where the picture
  was taken.
• Each group member must choose one photo that
  suggests a mathematical question. In writing,
  state the question and explore the solution. Two
  students may not use the same photo.

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The Geometric Challenge
of Packaging
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This replacement motor for our dishwasher was
packed and shipped to our house in a cube-shaped
box.
The dishwasher motor has a cylindrical shape, but
two styrofoam inserts make it fit snugly in the
box. Thus the motor is safely packaged for
shipping.
12 in
12 in
The cube-shape of the box makes it easy to stack
lots of these boxes in a warehouse.
12 in
12 in
PHOTO LOCATIONS in our kitchen.
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My brother got a new pair of shoes and the box
they came in is a rectangular prism.
The size and shape of the box is determined by
the shoes. The boxs length is determined by the
shoes length and the height is determined by the
shoes width. The width of the box is a little
less than 2 times the height of the shoes.
4 1/2 in
8 in
13 in
You can see that the shoes pack nicely in the box
because the two shoes are congruent and pack into
the box with rotational symmetry.
PHOTO LOCATIONS in my brothers room.
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This box, which contains breakfast cereal, is a
right, rectangular prism. The cereal is in a bag
and the volume of the bag is less than the volume
of the box.
3 1/4 in
13 in
8 in
There are many purposes to the packaging of
things like cereal. The inner plastic package
keeps the small cereal pieces together (and also
keeps it fresh and provides a pocket of air for
protection). Then, the outer rectangular box is
designed to protect the inner package and to
stack together well on the store shelves.
PHOTO LOCATIONS in our pantry.
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This box, which contains a dozen cans of Coke,
is a rectangular prism.
Size of the Coke can
The width of the box is just a little larger than
the length of the Coke can (a cylinder) its
height is twice the cans diameter. The boxs
length is just a little larger than 6 times the
diameter of the can.
15 3/4 in
Height 4 3/4 in Diameter 2 5/8 in
Size of 12 Coke cans
6 x 2 5/8 in 15 3/4 in 2 x 2 5/8 in 5 1/8 in
2 5/8 in
4 7/8 in
4 3/4 in
5 3/16 in
PHOTO LOCATION in our basement fridge.
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The container is a right cylinder, which holds
long matches. The cylinders diameter is large
enough to hold about 50 matches its height is
the length of the matches. When the container was
full of matches, the matches packed together in a
hexagonal pattern.
11 in
2 in
PHOTO LOCATIONS by our fireplace.
Other objects that are cylindrical also pack
in a hexagonal pattern inside a cylindrical
container. If you look at the cross section, the
cylinders look like circles. When the cylinders
pack hexagonally, the circles overlap and fill
in more space.
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THE PROBLEM what is the most efficient way to
package spherical objects?
2 1/2 in
2 1/4 in
5 in
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CONCLUSION A cylindrical container would use
less material to pack two balls than the smallest
rectangular box.
The volume of the smallest cylindrical container
is 12.5 r3 . The volume of the smallest
rectangular box is 16r3. The empty space in the
cylindrical container is 4.12 r3 and the empty
space in the rectangular box is 7.62 r3.
The surface area of the cylinder is 31.4 r2 and
the surface area of the box is 40 r2.
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The specific mathematical problem for this
assignment was to see the most efficient way to
package spheres. In this case, I compared the
material (surface area) required to package two
spherical objects using a rectangular prism or a
cylinder. In a rectangular prism, the height of
the prism is twice the diameter of the balls,
while the bases are each dXd. This means that
packing the balls would require 4 sides of 2dXd
and 2 ends at dxd. This gives a total of 10d2
surface area of the package. In the cylinder,
however, the bases would be 2 sides of p x r2
and the side of the cylinder would be p x d x 2d
for the height of the two balls. This means that
2.5 pd2 is the surface area needed for a
cylindrical package. With rounding, 2.5 x 3.14
7.85, which is more than 20 less than the 10 d2
needed for the rectangle. Obviously, the volume
of these two packages would also be different
(12.5 r3 for the cylinder or 16r3 for the
rectangular prism), meaning that less volume is
wasted in packaging the two balls (Vol 2 x 4/3
x 3.14 r3 8.38 r3). Even though packaging
can be made more efficiently using geometry, it
is also important to understand that how the
packages themselves are packed together will also
make a difference. It is easier to stack boxes
with straight edges on top of each other. Either
way you have to face turning rounded objects into
a more package-friendly thing.
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PROJECT INTRODUCTION

• 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.

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Matrix Animation Project GOALS

• To become familiar with programming on a graphing
  calculators.
• To utilize transformation matrices.
• To use creativity, technology, and mathematical
  problem-solving to create a product

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• You will use your knowledge of matrices and
  transformations to write a program that
• draws a unique shape,
• transforms the shape in at least 5 ways using at
  least 3 of the following types of
  transformations size change rotation
  reflection and translation.
• 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.

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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?
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GOOD PROBLEMS 3. In a store you obtain a 20
discount but you must pay a 15 sales tax. Which
would you prefer to have calculated first,
discount or tax? 4. Find the next three numbers
in each sequence a) 1, 1, 2, 3, 5, 8, 13, 21, .
. . b) 1, 11, 21, 1211, 111221, . .