Internet Resources for Mathematics Students and Teachers Dr. Sandy Wagner Menlo Park, CA papasandy@comcast.net Internet Resources for Math Students and Teachers USING YOUR SEARCH ENGINE ................................................................................. 3 EDUCATION DIRECTORIES ....................................................................................... 4 NEWSPAPER RESOURCES .................................................................................................. 4 TEACHER PRODUCTIVITY ................................................................................................. 5 GRANTS AND FREE MATERIALS ....................................................................................... 5 TALKING TO OTHER TEACHERS (DISCUSSION BOARDS) ................................................... 5 MATH EDUCATION RESOURCES ............................................................................. 6 CALIFORNIA STATE SITES ........................................................................................ 7 STAYING UP TO DATE - NEWS ......................................................................................... 7 PROFESSIONAL ORGANIZATIONS AND CONFERENCES ................................. 8 EDUCATIONAL TECHNOLOGY ORGANIZATIONS ............................................................... 8 MATHEMATICS EDUCATION ORGANIZATIONS.................................................................. 8 MATHEMATICS LESSONS ON THE INTERNET .................................................... 9 K-12 LESSONS ................................................................................................................. 9 MATH LESSONS USING SPREADSHEETS .......................................................................... 10 WEBQUESTS ................................................................................................................... 10 MATH SOFTWARE ...................................................................................................... 11 SOFTWARE THAT IS AVAILABLE ONLINE ....................................................................... 11 THE MATH TOOLS SITE.................................................................................................. 13 GRAPHING SOFTWARE ................................................................................................... 14 GEOMETRY SOFTWARE .................................................................................................. 15 SYMBOL MANIPULATION SOFTWARE ............................................................................. 15 DATA SOURCES ON THE INTERNET ..................................................................... 16 CALCULATORS IN THE CLASSROOM .................................................................. 18 CALCULATOR-BASED LABORATORY (CBL) .................................................................. 19 SAMPLES RESULTS FROM CBL EXPERIMENTS ................................................................ 19 SAMPLE EXPERIMENT: THE MATHEMATICS OF PROJECTILE MOTION ............................ 20 Download a Word file with the complete text at http://home.comcast.net/~papasandy/CMC_wagner_2006.doc Sandy Wagner Asilomar/CMC-N 2006 2 Internet Resources for Math Students and Teachers Using Your Search Engine The best and least commercial is www.Google.com . Most search engines provide links to a wealth of information classified by subject. Try a search on Math. You‟ll get a huge number of hits. Change the search to „math lesson plans‟ (quotes not needed), and you‟ll see some very useful sites. Google‟s Directory: Organizing the Internet into Categories To reach Google‟s directory, click on more » above the Google Search field, then even more » then Directory. The directory is shown here. You will find mathematics under „Science‟ and mathematics education under Mathematics. Ask Jeeves for Kids www.Ajkids.com Type in „math help‟ and see what happens. Sandy Wagner Asilomar/CMC-N 2006 3 Internet Resources for Math Students and Teachers Education Directories Teachers‟ Domain – National Multimedia Library http://www.teachersdomain.org/ Search for Science, then choose “Young Inventors” Teachers First http://www.teachersfirst.com/ Provided by The Network for Instructional TV, Inc., a not-for- profit educational-technologies corporation. Kathy Schrock's Guide for Educators http://school.discovery.com/schrockguide/ Sponsored by The Discovery Channel. Flashcard Exchange http://www.flashcardexchange.com/ “This is your place to create, study, print, download and share flashcards.” Classroom Flyer (from Riverdeep) http://64.73.138.82/riverdeep/ed/ClassroomFlyerSignup.atl Daily or weekly email filled with classroom ideas. The advertising is minimal. Teachers Helping Teachers http://www.pacificnet.net/~mandel/ Developed by Dr. Scott Mandel, a teacher at Pacoima Middle School Television, Theatre &Fine Arts Magnet in Los Angeles Unified School District Proteacher: "The Web's Little Secret for Elementary School Teachers" http://www.proteacher.com Now including Teacher blogs Newspaper resources New York Times Learning Network http://www.nytimes.com/learning/index.html Type “mathematics” in the lesson plan search field Los Angeles Times online http://www.latimes.com/news/education/ Current events in the education field. Sandy Wagner Asilomar/CMC-N 2006 4 Internet Resources for Math Students and Teachers Teacher Productivity Microsoft Office Templates http://office.microsoft.com/en-us/templates/default.aspx Examples: choose “Calendars” or choose “More categories” then “Quizzes and tests” Grants and Free Materials Donors Choose http://www.donorschoose.org/ Teachers write small grants for their classroom and donors log on and find something they want to support. Regions supported: Chicago, New York City, North Carolina, San Francisco Bay Area FREE: The Federal Resources for Educational Excellence (FREE) Web site makes it easy to find teaching resources on federal government Web sites. http://www.ed.gov/free/index.html Talking to Other Teachers (Discussion boards) Discussion Boards for K-6 teachers http://www.proteacher.net/discussions/ Try “homework” in the Archive search box Teacher to Teacher http://mathforum.org/t2t/ Try “fractions” in the Keyword field Teacher2Teacher (T2T®) Math Tools Discussion http://mathforum.org/mathtools/discuss.html?context=all Another good discussion group from the Math Forum. Sandy Wagner Asilomar/CMC-N 2006 5 Internet Resources for Math Students and Teachers Math Education Resources (Just a few of many) Eisenhower National Clearinghouse for Mathematics and Science Education (ENC) http://www.goenc.com/ ENC's mission is to identify effective curriculum resources, create high-quality professional development materials, and disseminate useful information and products to improve K-12 mathematics and science teaching and learning. Enter “problem solving” (no quotes needed) in the “Search Keywords” field Ask Dr. Math http://mathforum.org/dr.math/ Teachers and students can submit questions about mathematics itself or about teaching mathematics. A Homepage for New (And Not So New) Math Teachers http://people.clarityconnect.com/webpages/terri/terri.html Terri Husted is a classroom teacher who gives an enormous amount to her profession. Kids‟ “Maths” Dictionary http://www.amathsdictionaryforkids.com/maths/dictionary.html “Maths” is Australian for “Math”, but the words and definition are familiar. MathMovesU http://www.mathmovesu.com/ Raytheon Corporation presents motivational stories and contests. One Teacher‟s Focus on Teaching Multiplication http://www.multiplication.com Activities, worksheets, lessons, games – all to help teach multiplication. Mathematics Archive http://archives.math.utk.edu/ The Mathematics Archives is located at the University of Tennessee and is supported by the National Science Foundation. You will find links to other sites that discuss mathematics topics and mathematics teaching. Click on K-12 Teaching Materials. The Math Forum @ Drexel http://mathforum.org/ Like the Mathematics Archive, it contains information about mathematics topics from K to University, and ideas for teaching. It differs from the Mathematics Archives in that much of the material is prepared specifically for the Math Forum. Sandy Wagner Asilomar/CMC-N 2006 6 Internet Resources for Math Students and Teachers California State Sites California – The SCORE site http://score.kings.k12.ca.us/standards.matrix.html This is a link to California's mathematical standards and some resources to assist teachers in understanding and implementing these standards. Lessons linked to Math Standards (K-12) http://score.kings.k12.ca.us/standards.matrix.html Official State Publication of California Mathematics Standards http://www.cde.ca.gov/be/st/ss/mthmain.asp California Learning Resources Network (CLRN) http://www.clrn.org/math/ From the web site: “The purpose of the California Learning Resource Network is to provide a one-stop information source that enables California educators to identify supplemental electronic learning resources that both meet local instructional needs and embody the implementation of California curriculum frameworks and standards.” Click “Content Standards Search” (on right) then pick a grade level. Staying Up to Date - News California Online Mathematics Education Times (COMET) http://csmp.ucop.edu/cmp/comet/ COMET is a news journal designed for mathematics educators and educational leaders. COMET features useful and timely information on current educational issues, web resources, professional events and opportunities, and news articles from California and across the nation. To subscribe to the COMET listserv, send the following message to listserv@listserv.csufresno.edu : Subscribe COMET <your first name> <your last name> (Do not include anything in the subject field) National Math Newsletter: Math Forum Internet News http://mathforum.org/electronic.newsletter/ This service sends a weekly email message to subscribers. The message contains links to web sites of interest to mathematics educators. Scroll down to see this week‟s issue. Sandy Wagner Asilomar/CMC-N 2006 7 Internet Resources for Math Students and Teachers Professional Organizations and Conferences Professional organizations and conferences can be any teacher‟s introduction to finding information and meeting like-minded teachers who can be a great resource. With a local or national organization, you are no longer alone. Educational Technology Organizations ISTE, the International Society for Technology in Education http://www.iste.org/ The slogan of this organization is „Teachers helping teachers use technology in the classroom.” Most issues of ISTE‟s journal, Learning and Leading with Technology, contain an article on using technology in mathematics or science instruction. ISTE has also taken the lead in developing national technology standards for teachers and students. Computer-Using Educators, the California affiliate of ISTE http://www.cue.org/ Mathematics Education Organizations National Council of Teachers of Mathematics (NCTM) http://www.nctm.org/ California Mathematics Council http://www.cmc-math.org/ School Science and Mathematics Association http://www.ssma.org/index.html Sandy Wagner Asilomar/CMC-N 2006 8 Internet Resources for Math Students and Teachers Mathematics Lessons On The Internet K-12 Lessons You can find lessons on the Internet that are technology-based or more traditional. With some effort you should be able to find lessons that fit your grade level and the degree to which you are ready to use technology in your teaching. K-12 Lesson Plans from the NCTM Illuminations site (hundreds!) http://illuminations.nctm.org/Lessons.aspx Lesson plans from the Math Archive (hundreds!) http://archives.math.utk.edu/k12.html#lessonplans Lessons plans from the Math Forum (thousands!) http://mathforum.org/library/resource_types/lesson_plans/ Mathematics lessons that are fun! http://math.rice.edu/~lanius/Lessons/ Some lessons are in Spanish The following two sets of lessons were produced during a summer workshop sponsored by the Math Forum and the Urban Systemic Initiative Exploring Data: K-8 Lessons on the Web http://mathforum.com/workshops/usi/dataproject/usi.elemlessons.html Exploring Data: 9-12 Lessons on the Web http://mathforum.com/workshops/usi/dataproject/usi.hslessons.html Sandy Wagner Asilomar/CMC-N 2006 9 Internet Resources for Math Students and Teachers Math Lessons using Spreadsheets Code Crackers Math Game http://www.microsoft.com/education/codecrak.mspx Algebraic Problem Solving Using Spreadsheets: A unit for grade 9 http://mathforum.com/workshops/sum98/participants/sinclair/problem/intro.html Linear Regression, a lesson using spreadsheet software and real world data http://www.mste.uiuc.edu/patel/amar430/intro.html A Set of Lessons That Use the Excel Spreadsheet http://www.microsoft.com/education/LessonPlans.mspx Webquests Main WebQuest Site http://webquest.org/ News and views about the WebQuest model, a constructivist lesson format used widely around the world. Click “Find WebQuests” (on the left) then go to the “Curriculum x Grade Level Matrix” and choose a subject and a grade level. Explore the California Missions: A Fourth Grade Webquest http://brittan.sancarlos.k12.ca.us/classrooms/library/missions/missions.html WebQuest Resources from Spartanburg School District (South Carolina) http://www.spa3.k12.sc.us/WebQuests.html This is a huge collection of teacher-created WebQuests, elementary grades through high school. Check out their WebQuest template: http://www.spa3.k12.sc.us/WebQuestTemplate/webquesttemp.htm . Sandy Wagner Asilomar/CMC-N 2006 10 Internet Resources for Math Students and Teachers Math Software Software That Is Available Online Illuminations of the NCTM Principles and Standards http://illuminations.nctm.org/ Excellent ideas and resources here for K-12 mathematics teachers. Click “Activities” to see a wide range of interactive software applications for grades K-12. National Library of Virtual Manipulatives http://nlvm.usu.edu/en/nav/vlibrary.html The Shodor Institute Middle School activities mapped to NCTM standards http://shodor.org/interactivate/activities Click “Probability” then select “Adjustable Spinner” Grades 3-5 activities mapped to NCTM standards http://www.shodor.org/interactivate/elementary/index.html Two Lemonade Stand simulations: http://www.ae4rv.com/games/lemonade.htm (this is a relatively simple version) http://www.lemonadegame.com/ (this version has more variables to deal with) Sandy Wagner Asilomar/CMC-N 2006 11 Internet Resources for Math Students and Teachers Rainfrest Maths http://www.rainforestmaths.com/ Another Australian site by the same author as the Kids‟ Maths Dictionary, this one has over 800 math activities covering grades K-6. Explore Math Relationships with Online Interactive Multimedia Software (Middle and High School Level) http://www.explorelearning.com/ These are interactive online graphical explorations that are similar to those that are available with some graphing software. Using the horizontal sliding bars, the student can change the values of the coefficients of a function and see the change in the graph. These activities require Shockwave, which is available at no charge from the site. Note: This site has recently changed to a fee-based service, but if you use the software and lesson plans it will be well-worth the price. According to the site, : “This site is dedicated to listings, descriptions and reviews of the best free educational software for kids and adults. Some of the software is web based (see the online section), and some is Windows software that you can download (see the downloadable section). http://www.educational-freeware.com/ Sandy Wagner Asilomar/CMC-N 2006 12 Internet Resources for Math Students and Teachers The Math Tools Site http://mathforum.org/mathtools/index.html This site is administered by the Math Forum at Drexel University, which is also responsible for such favorites as “Ask Dr. Math.”. The goal of the Math Tools site is “to create a community digital library that supports the use and development of software for mathematics education.” Here is a sample of how to view tools for a grade level and subject area. First enter your search criteria – the example shows 3rd grade number sense, and within that, place value. The result of the search is shown below: The information with the yellow background is displayed when you pass the cursor over the title. Clicking on the title will cause the tool to run. Sandy Wagner Asilomar/CMC-N 2006 13 Internet Resources for Math Students and Teachers Graphing Software Software that can quickly produce the graph of a function allows the student or teacher to explore mathematical relationships. For example, the accompanying figure shows a screen from a free program named Winplot. The picture shows how a simple function y x 2 (the red curve) is affected by the addition or subtraction of a constant term. The screen shows the graphs of the functions y x 2 1 (blue) and y x 2 2 (green) so that the student can compare them with the original red curve. When using graphing software the teacher might lead a class discussion or design an exploratory exercise for students to do individually and then write a report of the findings. Peanut Software is the name of the site that offers programs of interest to high school math teachers, including Winplot, which is shown above. They are all written by Rick Parris of Phillips Exeter Academy, and offered free of charge. If you like any of these programs it would be nice to thank Rick for his donations to all of us. http://math.exeter.edu/rparris/ e-Tutor - Online Graphing Calculator According to Education Freeware, “This simple and neat online graphing software draws graphs from mathematical equations. You can draw several graphs at once and see how they interact, and you can zoom in and out and move the origin.” E-Tutor is in business to provide online instructional materials across the curriculum. The graphing calculator is a promotional tool to attract customers. But it‟s still a good graphing tool. http://www.e-tutor.com/et2/graphing Graphing Calculator is a commercial graphing program that allows the user to type in the equation without solving for one of the variables. In another example, the picture shows the use of a variable parameter n in the function y x 2 n and provides a slider bar to change the value of n . http://www.pacifict.com/Products.html The program is available for Macintosh and Windows computers and is a newer version of NuCalc, which was formerly shipped free on all Macs. Sandy Wagner Asilomar/CMC-N 2006 14 Internet Resources for Math Students and Teachers Geometry Software Geometer‟s Sketchpad Key Curriculum Press‟s widely-used dynamic geometry tool for high school geometry from has inspired a wealth of classroom activities in classes form Middle School to Calculus. Sketchpad Information http://www.keypress.com/sketchpad/ Resources that support users of Sketchpad and users of Key Curriculum Press texts: http://www.keypress.com/sketchpad/general_resources/ Online Java Sketchpad demonstrations: http://www.keypress.com/sketchpad/javasketchpad/gallery/index.php Symbol Manipulation Software It is amazing to realize that there are computer programs and calculators that can perform all the symbolic procedures of high school and college mathematics, up to and including first year calculus. That is, they can solve equations, factor polynomials, and simplify expressions. If there are computers and calculators that can do the mathematics that is taught in grades K-12, then the key question becomes, “What should the mathematics curriculum look like as these programs become widely available to students?” Here are two examples of symbol manipulation software: TI Interactive is mathematics software from Texas Instruments. It includes the following features: Word Processor with integrated math system TI graphing calculator functionality Symbolic Computer Algebra System Integrated Web browser Data editor with spreadsheet Graphing Technology Connectivity http://education.ti.com/product/software/tii/features/features.html Live Math http://www.livemath.com/ In addition to graphing, symbol manipulation, exact symbolic equation solving, and other features of advanced mathematics, Live Math allows students to solve equations step-by- step, and allows teachers to create exploration environments. Sandy Wagner Asilomar/CMC-N 2006 15 Internet Resources for Math Students and Teachers Data Sources On The Internet Real data can be the source of interesting mathematics lessons. You can find data from social science, science, and other sources on the web that can be incorporated into lessons. Data on Individual States http://quickfacts.census.gov/qfd/index.html This site contains a map of the United States on which you can click on any state to reveal the counties, then on individual counties to get data on population, housing, business, and other categories. Illuminations site‟s version: http://illuminations.nctm.org/ActivityDetail.asp x?ID=151 Click Data and select Gasoline Use. Why are California and New York so low? This site allows students to enter their own state data. Real Data from Whatcom Community College (Bellingham, WA) Online Math Center http://math.whatcom.ctc.edu/content/Links.phtml?cat=18 Although the site is from a community college, the data is relevant to grades 8-12 also. Create A Graph Create 5 different kinds of graphs from your own data. http://nces.ed.gov/nceskids/createagraph/ Sandy Wagner Asilomar/CMC-N 2006 16 Internet Resources for Math Students and Teachers Sunrise and Sunset http://aa.usno.navy.mil/AA/data/docs/RS_OneYear.html Enter a city anywhere in the world and you will get Number of Hours of Daylight in Two US Cities the time of sunrise and sunset for every day in the current year. Subtract those numbers and you get the 23 Key West 21 number of hours of daylight each day. 19 Fairbanks # of daylight hours 17 We found the data for two cities at the extreme ends 15 13 of the United States: Key West, FL and Fairbanks, 11 9 AK and imported the data into a spreadsheet. The 7 5 spreadsheet graph shows that the number of hours 3 0 50 100 150 200 250 300 350 peaks at the middle of the year and is at a minimum at Day of the year the end and beginning of the year. The shape of the curves is approximately sinusoidal (that is, it can be modeled by a trigonometric sine or cosine function). How to incorporate Internet data into a spreadsheet: The set of data described above is displayed on the Internet in columns in a non- proportional font, which maintains a constant space between the columns. Here are the steps for taking a table from such an Internet source into a spreadsheet: 1. Make sure the data you want is in a non-proportional font, for example the Courier font. If it is not, change to such a font before importing the data into the spreadsheet. 2. On the Internet, click and drag to highlight the numbers that you wish to use. Do not include the text headings unless there is exactly one heading per column. 3. Copy the selection to the clipboard by clicking Edit-Copy 4. Open the program NotePad (Windows) or SimpleText (Mac). Make sure that Word Wrap is off. 5. Click Edit-Paste. You should see all the data, in columns. If it is not in columns, turn off Word Wrap. 6. Save this file. 7. In the spreadsheet program open the file you just saved. The spreadsheet program will ask some questions but soon will have your data in the appropriate columns. If each column of data is not in its own separate spreadsheet column, you cannot proceed. Return to step 4. To make our graph we used Excel from Microsoft Office ‟95. 8. For the sunrise and sunset data you need to make three long columns: the numbers 1 through 366 (or 365), the sunrise time, and the sunset time. 9. Make a fourth column, the length of the day, by subtracting the sunrise time from the sunset time. Note: There is a special challenge with data like this that measures time. Since 7:34 is written 734 in the table, in order to subtract these times, you need to convert 734 to a decimal number 7 34/60 = 7.57. The formula we used was: INT(B2/100)+(B2-INT(B2/100)*100)/60, where cell B2 was a time. 10. Graph column 4 against column 1 using the spreadsheet graphing facility. Sandy Wagner Asilomar/CMC-N 2006 17 Internet Resources for Math Students and Teachers Calculators in the Classroom Why Use Calculators in the Math Classroom? The following quote is from an article entitled “The Role of Calculators in Math Education.” http://education.ti.com/sites/US/downloads/pdf/therole.pdf “Calculators allow students access to mathematical concepts and experiences from which they were previously limited with only paper and pencil. Because calculators make possible mathematical exploration, experimentation, and enhancement of learning mathematical concepts, the National Council of Teachers of Mathematics (NCTM) and various other organizations and individuals recommend that appropriate calculators be made available for use by students at every grade level from kindergarten through college. Despite the extensive research documenting the benefits of calculator use, there are still many skeptics who worry that calculator use will impair students‟ mathematical ability and result in increased mathematical illiteracy.” -- Introduction This article describes research and exemplary uses of calculators in K-12 mathematics classes, and it also addresses specific concerns that parents and teachers have voiced about the possible negative effects of calculator use. NCTM Position on Calculators in the Classroom NCTM Position (summary) School mathematics programs should provide students with a range of knowledge, skills, and tools. Students need an understanding of number and operations, including the use of computational procedures, estimation, mental mathematics, and the appropriate use of the calculator. A balanced mathematics program develops students‟ confidence and understanding of when and how to use these skills and tools. Students need to develop their basic mathematical understandings to solve problems both in and out of school. http://nctm.org/about/position_statements/computation.htm The Role of Calculators in Math Education - Dispelling the Myths Use these arguments to help convince parents and administrators about the value of calculators in the classroom . http://education.ti.com/sites/US/downloads/pdf/therole.pdf (page 6) Education Portal at Texas Instruments http://education.ti.com/educationportal/ Stokes Publishing Company Overhead projector calculators and other innovative products http://www.stokespublishing.com/ Sandy Wagner Asilomar/CMC-N 2006 18 Internet Resources for Math Students and Teachers Calculator-Based Laboratory (CBL) Texas Instruments CBL (pictured) http://education.ti.com/us/product/tech/datacollection/features/features.html The device that connects to the calculator is pictured above, along with the distance-measuring device. Vernier Systems Vernier develops its own CBL instruments and sells Texas Instruments products. http://www.vernier.com/legacy/cbl/index.html Samples Results from CBL experiments Experiments like the following are described in books developed by Vernier and Texas Instruments. http://www.vernier.com/cmat/index.html Bouncing Ball: Using the distance-measuring device (what TI calls a „motion detector‟) the motion of a bouncing ball can be captured and displayed on a graph. The separate parabolas can be analyzed individually and the heights of the bounces, which can be modeled with a negative exponential function, can be studied. Light Intensity: In this experiment the light intensity is measured at different distances and the data is stored and graphed in the calculator. The decrease in intensity (I) as distance (X) increases is K modeled by the equation I 2 . X Temperature: In this experiment the temperature probe was placed into boiling water, which was then removed from the heat. The cooling curve of Temperature (T) for any time (t) after the water is K removed from heat is modeled by the equation T R t e Microphone and Tuning Fork: The microphone measures the changes in air pressure that the ear perceives as sound. The sound of a pure note produced by a tuning fork produces a set of data that can be modeled by a trigonometric function, either sine or cosine. The difference in frequency of two tuning forks can be observed by looking at the frequencies of the two sine or Sandy Wagner Asilomar/CMC-N 2006 19 Internet Resources for Math Students and Teachers cosine waves, as shown in the two accompanying graphs. Frequency is defined as the number of waves per unit of time. The graphs show that the C note has a lower frequency (a little more than three waves in the graph window) than the G note (approximately five waves in the graph window). Trigonometry teachers can also have their students find the equations that describe the two sets of data and show the students how the frequency appears in the formula. Tuning Fork (C) Tuning Fork (G) Sample Experiment: The Mathematics of Projectile Motion The following example shows how the TI-83/84 graphing calculator can be used to find a mathematical model for real data. The experiment consisted of tossing a basketball straight up and measuring its distance from a motion detector that was placed on the floor directly below the ball. The first screen shows the graph of the distance vs. time. The parabola represents the up-and-down flight of the ball. The points to the left and right of the parabola show the motion of the ball in the hands of the experimenter before it was tossed and after it was caught. The calculator provides the capability of deleting the points that are not of interest, and this is shown in the second screen. Next, we shifted the parabola over to the left so that we could have the toss start at time = 0. (This screen is not shown here.) The calculator provides a method of determining an equation that best fits the points that form the parabola. The process uses the mathematical method called regression, which finds the equation that best fits the data points. The experimenter must tell the calculator what type of curve would probably be best. The choices include straight line, parabola, sine curve, and others. The equation that results is shown in the third screen. We interpret that screen by saying that the height in feet (h) of the ball t seconds after it was thrown is h 15.427t 2 12.861t 1.439 . Sandy Wagner Asilomar/CMC-N 2006 20 Internet Resources for Math Students and Teachers The final screen shows how well the curve fits the data points. The equation that describes the data can then be used to predict something about the flight of the ball. Each coefficient tells a different story: The coefficient a is half the gravitational constant, which on earth is 32 for flight measured in feet and seconds. The value of a is negative because gravity is pulling the ball down. The fact that twice a is about 30.8 means that the percent error of this prediction is about 4%. The coefficient b is the velocity with which the ball is thrown. So we conclude that the ball was thrown in the air at almost 13 feet per second. The coefficient c is the height above the distance-measuring device from which the ball was thrown: about 1.4 feet. Sandy Wagner Asilomar/CMC-N 2006 21

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