SUPPORTING ALL STUDENTS USING PROJECT/PROBLEM-BASED LEARNING Dan Schab Williamston High School Williamston, MI schabd@wmston.k12.mi.us

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.

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.

WHY INTEGRATE PROJECTS INTO THE CURRICULUM? APPLICATION OF COURSE CONTENT DEVELOP PROBLEM-SOLVING SKILLS DEVELOP CAREER AND EMPLOYABILITY SKILLS INCREASE STUDENT MOTIVATION

THE GOLF COURSE PROJECT STEP 1: Students Placed Into Design Teams

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.

STUDENT WORK

STEP 3 – Three Dimensional Models are Built

EXAMPLES OF 3-D MODELS

3-D MODEL

3-D MODEL

3-D MODEL

3-D MODELS

3-D MODELS

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.

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

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

INCREASE STUDENT MOTIVATION

Rigor & Relevance Framework KNOWLEDGE Evaluation 6 Synthesis 5 Analysis 4 Application 3 Comprehension 2 Awareness 1 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

Construction Project Geometry 3rd hour

Sidewalk is 36 inches wide and 4 inches thick. House 18 ft 20 ft Sidewalk is 36 inches wide and 4 inches thick. Driveway is 15 feet wide, 60 feet long, and 6 inches thick. D R I V E W A Y

Breakdown of Costs Removal of Debris - $55 per cubic yard New Concrete - $77 per cubic yard Forms - $0.50 per linear foot Spreading the New Concrete – requires 1 minute per square yard, costs $25 per hour Finishing the Concrete - $0.13 per square foot Profit Margin = 15%

Our Company Our many workers here at J&M Co. work hard to get your landscaping done quickly, efficiently, and leave you with quality work. J&M 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.

Excavation Sidewalk Driveway Total Excavation Price 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

Forms Perimeter of driveway and sidewalk Cost of forms 60 + 15 + 57 + 18 + 20 + 15 + 15 +17 + 3 = 220 ft. Cost of forms 220 ft. x $ .50/ft. = $ 110 Total cost: $ 110

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

Profit $ 3485.09 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

The World of Geometry

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”

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.

The Geometric Challenge of Packaging

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.

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 box’s 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 brother’s room.

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.

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 can’s diameter. The box’s 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.

The container is a right cylinder, which holds long matches The container is a right cylinder, which holds long matches. The cylinder’s 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.

THE PROBLEM: what is the most efficient way to package spherical objects? 2 1/2 in 2 1/4 in 5 in

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.

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.

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.

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

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.

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?

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, . .