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Bridging Mathematics and Science

Bridging Mathematics and Science. Dr. Anthony S. Pyzdrowski Dr.. Laura J. Pyzdrowski (Mrs. Patricia Neel) WVCTM Conference, March 2003. Contents. Note to Users Unit Outline Assessment Web Sites Day 1: Structure of Bridges Girder Bridges Truss Bridges Arch Bridges

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Bridging Mathematics and Science

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  1. Bridging Mathematicsand Science Dr. Anthony S. Pyzdrowski Dr.. Laura J. Pyzdrowski (Mrs. Patricia Neel) WVCTM Conference, March 2003

  2. Contents • Note to Users • Unit Outline • Assessment • Web Sites • Day 1: Structure of Bridges • Girder Bridges • Truss Bridges • Arch Bridges • Cantilever Bridges • Suspension Bridges • Cable-Stayed Bridges • Vocabulary Words

  3. Contents Day 2: Structure of Bridges Day 4: Suspension and Cable-Stayed Bridges Suspension Bridges Cable-Stayed Bridges Day 5: Beams and Trusses Deflection of a Cantilevered Beam Activity Deflection of a Cantilevered Beam Data Table Teachers Notes Day 6 : Deflection of a Cantilevered Beam Activity Wrap-up/extension Sample Results

  4. Contents Day 9: Distribution of the Load in the Supports of a Beam Bridge Activity on the Distribution of the Load About this Activity Data Collection Distribution of the Load Data Table Assignment Use graphing paper to make a complete graph. Day 10: Distribution of the Load in the Supports of a Beam Bridge Use a graphing Calculator to Graph your data. Extension Exercise Distribution of the Suspended Load Data Table Questions to Think About

  5. Contents Day 11 : Distribution of the Load Activity Wrap-up Sample Results Day 12: Student Truss Bridges Day 13: Deflection of a Bridge About this Activity Data Collection Deflection of a Beam Bridge Data Table Deflection of a Truss Bridge Data Table Deflection of an Inverted Truss Bridge Data Table Use a graphing Calculator to Graph your data. Day 14 : Deflection of a Bridge Wrap-up Sample Results

  6. Contents Day 15: Unit Test Day 16: Test Rubric Proposed Academic Standards for Science, Technology and Mathematics Grade 7 Resources

  7. Note to Participants *The activities in the handouts were developed to be used in the unit on bridges. Only those activities which were developed by the authors are included in the handout.

  8. Unit Outline Pre-assignment – Students will do research by collecting pictures and information on at least three types of bridges. Day 1:* Introduction to Bridges – Short lecture and picture presentation about bridges. Lecture will include new vocabulary words. Students will assist in classifying the six main types of bridges. Students will be given a list of websites to be used during the unit. Day 2:* Students will be asked to read about and discuss the structure of bridges and in particular discuss the Tacoma Narrows Bridge, a structure that failed.

  9. Unit Outline Day 3: Student Activity: Students will construct a model bridge and explore how the construction affects safety and efficiency of materials. Day 4:* Activity recap and short lecture about Suspension and Cable-Stayed Bridges and three kinds of Forces – Tensile, Compressive and Shear Force. Day 5:* Short lecture on beams and trusses. Group Hands-on-activity introducing the concept of deflection. Students will set up the apparatus, collect and record data during class. As an assignment, students will make a graph. They will model the data in a table, then on a graph.

  10. Unit Outline Day 6:* Activity wrap-up. Students will discuss the mathematics (linear relationship) between defection and force. Day 7: Short lecture on strength of materials. Student activity to test for strong shapes, and lamination. Students will set up the apparatus, collect and record data during class. As an assignment, students will make a graph by plotting their results. Day 8: Activity wrap-up. Students will discuss the mathematics involving the lamination activity.

  11. Unit Outline Day 9:* Activity on the distribution of the load in the supports of a beam bridge. Students will set up the apparatus, collect and record data during class. As an assignment, students will make a graph by plotting their results. Day 10:* Students will use the graphing calculator to graph and analyze their data. Students will perform an extension exercise. Day 11:* Activity wrap-up. Students will discuss the mathematics involved in the Beam activity. Day 12:* Students will construct a Truss Bridge.

  12. Unit Outline Day 13:* Activity on the deflection of a beam and truss bridges. Students will set up the apparatus, collect and record data during class. As an assignment, students will make a graph by plotting their results. Day 14:* Activity wrap-up. Students will discuss the mathematics involved in the Truss Bridge. Unit Review. Day 15:* Unit Test. Day 16:* Test Rubric.

  13. Six Main Types of Bridges (1) girder bridges, (2) truss bridges, (3) arch bridges, (4) cantilever bridges, (5) suspension bridges, and (6) cable-stayed bridges.

  14. Girder Bridges which include many highway bridges, are made of beams called girders whose ends simply rest on piers or abutments. The span length of girder bridges ranges up to 1,000 feet (300 meters)

  15. Truss Bridges are supported by frameworks called TRUSSES. The parts of the trusses are arranged in the form of triangles. Such bridges are built over canyons, rivers, and other areas. A truss bridge may have a main span that extends more than 1,000 feet (300 meters). The simplest truss consists of three parts fastened together at their ends to form a triangle.

  16. Arch Bridges are structures in which each span forms an arch. The spans range up to about 1,700 feet (518 meters) long. The arch bridge is one of the oldest types of bridges. Early arch bridges consisted of large stone blocks wedged together to form an arch.

  17. Cantilever Bridges consist of two independent beams called CANTILEVERS that extend from opposite banks of a waterway. The two cantilevers are joined together above the middle of the waterway by a beam, girder, or truss. Cantilever bridges may have spans as along as about 1,800 feet (549 meters). Many cantilever bridges have truss frameworks. (Picture by Scott M. Kozel)

  18. Suspension Bridges are perhaps the most impressive type of bridge because of their long main span and especially attractive appearance. These bridges have a roadway that hangs from steel cables that are supported by two high towers. Most suspension bridges have a main span more than 1,000 feet (300 meters) long. Some have a main span longer than 4,000 feet (1,200 meters). (Picture by Scott M. Kozel)

  19. Cable-Stayed Bridges are different from suspension bridges, their cables are more taut than the flexible cables of suspension bridges. They offer greater stiffness over that of suspension bridges. Cable-stayed bridges are economical for medium span applications in the 500 feet (150 meters) to 3,000 feet (900 meters) range. (Photograph by Steven Richman)

  20. Day 12: Student Truss Bridges Students will be given a Ziploc bag with the following contents: 30 popsicle sticks (hole drilled at each end) 9 lollipop sticks – 4 in. 9 straws – 3 in. length 9 rubber bands 1 3 x 16 in piece of balsa wood

  21. Day 12: Student Truss Bridges Students will see a completed truss bridge and construct one like the model. When the bridges are completed, students will compare the models.

  22. Day 13: Deflection of a Bridge Activity About this Activity In this activity, you will learn how the addition of a truss structure affects the deflection. You will also explore trusses that are attached above and below a beam bridge. You will collect and then record data in tables in order help answer questions about truss bridges

  23. Day 13: Deflection of a Bridge Activity • Presentation Note • From the Day 9: Distribution of the Load in the Supports of a Beam Bridge Activity • “Begin your activity by setting up your beam bridge model. • Mark the balsa wood 2.8m. in from one end with a ball point pen. ( This is position “0" and will be placed on one support.) Continue to mark the wood in 4 cm. intervals labeling each position through position 10. • ( Position “10" will be placed over the second support.)”

  24. Day 13: Deflection of a Bridge Activity Data Collection Make sure you have the following: 1 - Truss Bridge Model from the Truss Bridge Activity 1 - Metric Ruler Several Mass Weights ( approximately 100g each) 2 - Pencils or Pens to use as supports between supports and balsa wood. 2 - Wooden Block Supports

  25. Day 13: Deflection of a Bridge Activity Begin your activity by using only the beam bridge model portion of the truss bridge. You need to span two block supports with the balsa wood. Make sure that you place position “0” of the balsa wood over one support and position “10” over the other.

  26. Day 13: Deflection of a Bridge Activity Place a mass weight in the center of the beam bridge and record the mass, height without the mass, and deflection in the table below. Continue collecting data by adding enough mass until you have entries for 6 positions in your table.

  27. Day 13: Deflection of a Bridge Activity Deflection of a Beam Bridge Data Table

  28. Day 13: Deflection of a Bridge Activity Repeat the data collection for the truss bridge. Use the same mass entries as those in the first table.

  29. Day 13: Deflection of a Bridge Activity Place a mass weight in the center of the truss bridge and record the mass, height without the mass, and deflection in the table below. Continue collecting data by adding enough mass until you have entries for 6 positions in your table.

  30. Day 13: Deflection of a Bridge Activity Deflection of a Truss Bridge Data Table

  31. Day 13: Deflection of a Bridge Activity Repeat the data collection for the inverted truss bridge. Use the same mass entries as those in the first table.

  32. Day 13: Deflection of a Bridge Activity Place a mass weight in the center of the inverted truss bridge and record the mass, height without the mass, and deflection in the table below. Continue collecting data by adding enough mass until you have entries for 6 positions in your table.

  33. Day 13: Deflection of a Bridge Activity Deflection of a Truss Bridge Data Table

  34. Day 13: Deflection of a Bridge Activity Questions Explain how a truss structure affects the deflection of a bridge. Does there seem to be a difference in the deflection between the non-inverted and inverted truss bridge?

  35. Day 13: Deflection of a Bridge Activity Use a graphics calculator to graph your data 1.Turn on your graphics calculator, Select MODE arrow down to Float and use the right arrow to highlight 5 and ENTER press 2nd QUIT when done.

  36. Day 13: Deflection of a Bridge Activity To Enter your Data from a Table: 2. Select LIST 3. Use the arrow keys to highlight L1 and select CLEAR, ENTER 4. Use the arrow keys to highlight L2 and select CLEAR, ENTER 5. Move to the first cell under L1 and enter your mass data. 6. Select the right arrow key to move to L2, and enter your data for deflection of a beam bridge . Press 2nd QUIT when done.

  37. Day 13: Deflection of a Bridge Activity To Turn on a Plot: 7. Select 2nd PLOT and use the arrow keys to highlight 1. 8. Select ENTER 9. Using the arrow keys and the ENTER key, make sure that Plot 1 is turned On, and choose to make a Scatter Plot. By default, Xlist is L1 and Ylist is L2. Choose any Mark type that you wish. 10. Select and make sure all other plots are turned off ( use steps 8 and 9 above if necessary.) Press 2nd QUIT when done.

  38. Day 13: Deflection of a Bridge Activity To See a Plot: 11. Press Y= and use CLEAR and the arrow keys to clear old equations. 12. Press GRAPH, ZOOM and the select 7 to see all of the points from your table.

  39. Day 13: Deflection of a Bridge Activity To Make a Manual Fit Line: 13. Select 2nd STAT, CALC and select 3 for Manual-Fit . 14. Move the cursor to the upper right to begin your fit line and select ENTER to tie down a point on the line. 15. Using the arrow keys continue the line segment and then select ENTER to tie down the second point; your manual-fit line is drawn. 16. What is the equation of the line of manual-fit? y = (as it appears on your calculator screen.) 17. You may move the arrow keys to move the line until you select ENTER .

  40. Day 13: Deflection of a Bridge Activity To Make a Fit Line Using a Linear Regression: 18. Select 2nd STAT,CALC and select 5 for a Linear Regression. 19. Select 2nd VARS select 2, 1, ENTER to place your fit equation into the Y1 equation. 20. a = b = 21. Select Y=

  41. Day 13: Deflection of a Bridge Activity 22. What is the equation of the line of best fit? Y1 = (as it appears on your calculator screen.) 23. What is the y-intercept of the line of best fit? 24. What is the slope of the line of best fit? 25. What does x represent in this problem? 26. Which number values make sense for x and why? 27. What does y represent in this problem? 28. Which number values make sense for y and why? 30. Select GRAPH

  42. Day 13: Deflection of a Bridge Activity Repeat the process for the truss bridge data using Lists 1 and 3 for StatPlot2 and the inverted truss bridge data using Lists 1 and 4 and StatPlot 3. ( Skip the manual fit.) If you look carefully, you should see three scatter plots in your viewing window. Use them to answer the following: A) If the mass is constant, is there a significant difference in deflection between a truss that is above or one that is below the beam bridge? Explain. B)How does the use of a truss affect the deflection of a beam bridge? Explain.

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