CMC_wagner_2006

					       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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