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Lesson 3: Proton: Who knew skateboarders were doing math & physics! Math, Measurement, and Scale What does a skateboarder, a pole vaulter, and NASA’s Vomit Comet have in common? First, we can step back in time to the 17th century when mathemetician and physicist, Galileo, was rolling balls on a flat board that was leaning on something. He found that the balls would travel up and then down in the same shape and eventually proved his theory mathematically.

Just as Galileo found the rolling balls followed the same shape when rolled on an incline plane; this bouncing ball, captured with a stroboscopic flash at 25 images per second, demonstrates a similar shape.

Galileo Galilei by Giusto Sustermans

The shape that Galileo discovered is called a parabola. In mathematics, the parabola is used in algebra, physics, and geometry. The parabola is seen in many places in our physical world from athletics to satellites.

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Can you think anything else in our physical world that moves in this pattern? How about a baseball? When the player throws the baseball into the air, this is called a vertical trajectory, and the baseball actually flies up and then down in about the same shape.

Baseball and Ballistics A baseball will fly in a somewhat uniform shape allowing for some air resistance or air friction. The lower the speed of the object, the less distorted the shape. A baseball player understands the flight pattern of a baseball and where it will land.

An object at higher speeds will follow a more distorted path.

For example, in ballistics, scientists and mathematicians have to take into account the higher speeds of a missile because the parabola shape is highly distorted due to air resistance.

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Parabolas and Skateboarding There are 2 examples of parabolas in skateboarding.

EXAMPLE #1: When a skateboarder is traveling down a ramp, he or she is getting the momentum to travel back up the other side. If you skate down an 8 ft. half pipe, you will most likely be able to make it up the other side of an 8 ft. ramp. Hopefully! This shape, or line of descent and ascent, forms a parabola. At the bottom of the ramp, there is the focus point of the parabola. At this point the skateboarder hunches down and uses his or her body to push off and increase the momentum to make it up the other side of the ramp EXAMPLE #2: Another parabola in skateboarding is the vertical parabola. The skateboarder is a vertical projectile. With momentum, he or she flies up to a point and then, what comes up must come down.

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When a diver jumps off a diving board, his body is an object extended in space. The diver’s body is a center of mass, and as it moves, it forms a parabola. At the top of the parabola is the point of free fall and the focus point of the parabola. Just like the diver, the skateboarder’s body is the center of mass and travels to a focal point being suspended in space for a moment.

The FOCUS Point Whether the skateboarder is rolling down or up to the center of the parabola it is called the focus. The focus is an important part of the parabola, because this is the place where the main action occurs. On a descending parabola, the focus is just above the vertex or center point of the parabola. When the skateboarder ascends vertically, the focus is just below the vertex. Notice how the focus is aligned with the vertex and the axis or line of symmetry!

Do you recognize the parabolas in the Proton film lesson?

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A skateboarder becomes a projectile. What happens to the projectile (skateboarder) when he hits the focus of a vertical parabola? Zero Gravity! Skateboarder + Vertical Momentum = Zero Gravity Pilots in NASA’s Vomit Comet take the aircraft up to practice weightlessness for their future astronauts. They travel up in the shape of a parabola and when they reach the top center, or the focus spot, they achieve zero gravity, weightlessness. This is the moment of free fall.

This is the commercial plane Zero-G. A normal airplane takes off at a 20 degree angle. This plane begins the parabola at a 50 degree angle.

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Measurement and Symmetry Building a Skate Park Skateboard ramps are made with precision measurements and symmetry. A parabola has symmetry. That means that one side of the parabola is a mirror image of the other side. It’s reflective.

In history, Archimedes, a 3rd century geometer, may have used a parabolic reflector to fight the Romans. The parabolic reflector, is a double reflecting device that can concentrate a light or other forms of electromagnetic radiation to a common focal point or focus. It is legend that he used the sun’s rays to set fire to the decks of the Romans’ ships. This reflective symmetry is used in many areas of our world which began with 17th century telescopes leading to microwave and satellite dish antennas, solar power, car headlights, and a simple flash light.

The legend of Archimedes

Archimedes’ knowledge being used today.

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Skate Ramp Construction In the Learning Center, you will find Skate Ramp Kits that include the directions you will need to build all of these ramps: FUNBOX: WITH OR WITHOUT RAIL

4 FOOT QUARTER PIPE

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VERT HALF PIPE

MINI VERT RAMP

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Making a Skate Park to Scale As we said before, the measurements must be accurate and symmetrical. You could build any type of ramp at home or you could build one to scale. What is scale? When an architect or engineer plans to build a skyscraper or a bridge they don’t just go out and start building on a piece of land. First, they make a model to scale using different materials than they would use in the real project. In some instances, graphic engineers are using computer technology to build virtual models of buildings, bridges, dams, and housing projects. In transportation engineers will build a model for the jet, car, or boat. This is a model of the Chrysler building in New York City. It was made to scale. Building a Quarter Pipe to Scale Some people think a scale model is just smaller, but it’s not. Scale is using math to find a proportion. For example, if you had the plans for a quarter pipe, which is 4 feet high, 8 feet wide, and 8 feet deep, and you wanted to make a smaller version for a younger child or a fingerboard, you would need to build the quarter pipe to scale using proportions. Quarter Pipe:4 ft. high, 8 ft. wide, 8 ft. deep (4’ x 8’ x 8’)

Half Scale: 2 ft. high, 4 ft. wide, 4 ft. deep (2’ x 4’ x 4’) (Mathematically, there are 3 ways to do it: All of the dimensions are cut in half,or divided by 2, or multiplied by .5.) Quarter or One-Fourth Scale: 1 ft. high, 2 ft. wide, 2 ft. deep (1’ x 2’ x 2’) (All of the dimensions are cut in fourths, or divided by 4, or multiplied by .25) *One-Eighth Scale : 6 in. high, 1 ft. wide, 1 ft. deep (All of the dimensions are cut in eighths, or divided by 8, or multiplied by .125) (*One-Eighth Scale is used for Finger Boards and Table Top Skate Parks)

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One-Eighth Scale Can you figure out what dimensions you would use to construct this building to one-eighth scale? This building is 32 feet in height, 24 feet in width, and 40 feet in depth. 32’ x 24’x 40’ What would the dimensions be if you built it to one-eighth scale? Take all three dimensions and cut into eighths, or divide by 8, or multiply by .125 Answer: 4’ x 3’ x 5’ The building is now a great size for a dog house! Math Counts! A quick review.... Just in case your didn’t know where I got the .125 for multiplying... It’s the fraction 1 over 8 and you divide the one (numerator) by the 8 (denominator) and you get a decimal. Then you multiply the dimensions by your scaled decimal and you have your new dimensions. You may need to use a decimal for some scale models to be very accurate. If you could do this problem, then you are ready to make some ramps to scale for fingerboards. Fingerboard Ramp Kits are also available in the Learning Center. Scale: The Bigger Picture: Earth When Proton did his gargantuan ramp drop from planet Zein through the universe and down to planet Earth, the size of planet Earth became larger and larger as he approached. This is another example of scale. There is a great video on scale called “Powers of Ten” by Charles and Ray Eames. Check it out on Youtube. You can fly like Proton did from the Universe to planet Earth with GoogleEarth software. Download it free at http://google.earth.com Type in your home address and fly from the universe to your own house!

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Vocabulary Words: parabola focus zero-gravity symmetry scale

Math Lesson for Eccentric Racing Network Created by Karen Stewart 5th Grade teacher Jamul-Dulzura Union School District Resources: http://en.wikipedia.org/wiki/Parabola http://www.xtremeskater.com/quarterpipeplans “Powers of Ten” by Charles and Ray Eames GoogleEarth

EDUCATION STANDARDS NATIONAL STANDARDS_MATH National Council of Teachers of Math Principals and Standards for School Mathematics Geometry Standards Grades 3-5. 6-8 and 9-12 Measurement Standards Grades 3-5, 6-8 and 9-12 Number and Operation Standards Grades 6-8 Problem Solving Standards Grades 6-8 and 9-12 CALIFORNIA STATE CONTENT STANDARDS: GRADE FIVE MATH (Standards for Grade 5-8 apply, however, only Grade Five Standards are shown here.) NUMBER SENSE 1.0 Students compute with very large and very small numbers, positive integers, decimals, and fractions and understand the relationship between decimals, fractions, and percents. They understand the relative magnitudes of numbers: 1.5 Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers. 2.0 Students perform calculations and solve problems involving addition, subtraction, and simple multiplication and division of fractions and decimals: 2.1 Add, subtract, multiply, and divide with decimals; add with negative integers; subtract positive integers from negative integers; and verify the reasonableness of the results. MEASUREMENT AND GEOMETRY 1.0 Students understand and compute the volumes and areas of simple objects: 1.1 Derive and use the formula for the area of a triangle and of a parallelogram by comparing it with the formula for the area of a rectangle (i.e., two of the same triangles make a parallelogram with twice the area; a parallelogram is compared

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with a rectangle of the same area by cutting and pasting a right triangle on the parallelogram). 1.2 Construct a cube and rectangular box from two-dimensional patterns and use these patterns to compute the surface area for these objects. 1.3 Understand the concept of volume and use the appropriate units in common measuring systems (i.e., cubic centimeter [cm3], cubic meter [m3], cubic inch [in3], cubic yard [yd3]) to compute the volume of rectangular solids. 1.4 Differentiate between, and use appropriate units of measures for, two-and three-dimensional objects (i.e., find the perimeter, area, volume) 2.0 Students identify, describe, and classify the properties of, and the relationships between, plane and solid geometric figures: 2.1 Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and triangles by using appropriate tools (e.g., straightedge, ruler, compass, protractor, drawing software). MATHEMATICAL REASONING 1.0 Students make decisions about how to approach problems: 1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, sequencing and prioritizing information, and observing patterns. 1.2 Determine when and how to break a problem into simpler parts. 2.0 Students use strategies, skills, and concepts in finding solutions: 2.1 Use estimation to verify the reasonableness of calculated results. 2.2 Apply strategies and results from simpler problems to more complex problems. 2.3 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. 2.4 Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work. 2.5 Indicate the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy. 2.6 Make precise calculations and check the validity of the results from the context of the problem. 3.0 Students move beyond a particular problem by generalizing to other situations: 3.1 Evaluate the reasonableness of the solution in the context of the original situation. 3.2 Note the method of deriving the solution and demonstrate a conceptual understanding of the derivation by solving similar problems. 3.3 Develop generalizations of the results obtained and apply them in other circumstances.

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CHARACTER LESSON 3: Proton__ Math Integrity – Trustfulness Character Standard: A person of good character is honest and truthful. Activity 1: NSES Standard A: Science as Inquiry NCTE/IRA Literacy Standard 3: Comprehension Literacy Standard 5: Writing Q&A: Famous skateboarder, Proton – will he show up? “Proton always keeps his word.” In this film, how did Proton “keep his word?”

Activity 2:

NCTE/IRA Literacy Standard 3:Comprehension Literacy Standard 5: Writing Literacy Standard 11: Reflection

In Your Own Words What does “honesty” mean to you? What does it mean to be “truthful”? To be honest, you should Honest people know that

, are wrong.

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and

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Activity 3:

NCTE/IRA Literacy Standard 3: Comprehension Literacy Standard 5: Writing Literacy Standard 11: Reflection

What Do You Do? You see one of your friends cheating on a test. A friend tells you a rumor about one of your classmates. A friend wants you to “cover for him” because he didn’t do his homework. You go to a movie your parents do not want you to see. You find someone’s wallet with $20 in it. You promise to keep a secret but tell the secret to your close friend. Activity 4:

NCTE/IRA Literacy Standard 3:Comprehension Literacy Standard 5: Writing Literacy Standard 11: Reflection

Truth in Advertising Find an advertisement in a newspaper or magazine. Read it carefully. Notice the graphics and pictures used in the ad. Think about the “truthfulness” of the ad; that is, think critically about this and other ads. List questions that concern you about the ad(s):

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Now, look at some commercials you see on television. Pick one or two. Think critically about these commercials. List questions that concern you about the commercial(s):

Activity 5:

NCTE/IRA Literacy Standard 3: Comprehension Literacy Standard 5: Writing

Honesty and Truthfulness – Acrostic Poems Create an acrostic poem using the name “Proton.” Put it on a chart and post it at home or in your classroom at school. Here is an example to get you started: Play fair Right or wrong, be honest and truthful Others trust you Tell the truth Others care, so should you Never lie

Character Lesson written by Dr. Ed Deroche: Director of the University of San Diego's Character Development Center