PreCalculus Animations with GeoGebra

Dr. Kevin Hopkinskhopkins@sbuniv.edu

Southwest Baptist UniversityBolivar, MO 65613

GeoGebra

• Free, downloadable software• http://www.geogebra.org• This entire presentation, with working links, is

available by following the link in the list of presentations at https://sites.google.com/site/drhopkinsbackground/

GeoGebra

• How many have used GeoGebra quite a bit?• How many have used it just a little?• If you haven’t seen it before, some of what we

do may be overwhelming.• I plan to adjust the pace of the talk based on

the interest of the attendees.• As teachers, you understand the difficulty of

teaching to a class of diverse learners.

Searching

• Can do some searching on GeoGebraTube (http://www.geogebratube.org/) just to see what is out there.

• As with any type of searching, sometimes you get good results without much effort.

• Other times you look at lots of so-so stuff before finding (hopefully) what you were really hoping for.

• We’ll look at a few things I have produced first to give you some ideas of what GeoGebra can do.

• Geometry– Area of Annulus (like a contest problem) at http://

ggbtu.be/b1560533 – Similar Triangles http://ggbtu.be/b1560533

• How does missed HW, participation affect grades: http://tube.geogebra.org/student/b92036#material/46508 ?

• Internet Explorer doesn’t go directly to intended page, Google Chrome does.

• Line of Best Fit: http://tube.geogebra.org/student/b92036#material/48459

Two Purposes

• Help students visualize what happens as we change one variable– This used to be my only purpose

• I’ve come to realize a second purpose– Can ask a lot more “what if ... “ questions– Demonstrate the richness of mathematics

• Makes the problem more of a “treasure hunt”• What can be learned in this complex

situation?• Most real life problems do NOT have just one

variable• Textbook problems almost always have ONLY

one variable • Other “variables” have been arbitrarily fixed

to make the problem more doable (but students don’t realize this)

• Students may ask, “Who cares about this ONE problem?”

• Students can realize there are many problems associated with a particular scenerio (the textbook has decided to focus on just one)

• Allows students to get a fuller picture of what mathematics is all about

• A POWERFUL way to study complicated and interrelated patterns

Caution—Be sure of your purpose• Just because you can do an interactive

presentation, doesn’t mean that is the best way for a beginning student to grasp it.

• On the other hand, even if students don’t grasp everything, hopefully they catch the complexity of the problem and how math helps in quantifying that complexity.

• I enjoy the challenge of how to order the operations to get an animation to show the features I want to highlight.

• Lines-how do m and b affect the shape: http://tube.geogebra.org/student/b92036#material/46676

• Exponentials-how do C and a affect the shape: http://tube.geogebra.org/student/b92036#material/50432

• Functions-f(x)+c vs f(x+c): http://tube.geogebra.org/student/b92036#material/52304

Worksheets• Geogebra allows you to store and organize your

figures in worksheets at GeoGebraTube.• I have used this figure to create interactive

worksheets, like– Contracting log expressions: http://

tube.geogebra.org/student/b92036#material/50428 – Intersection of Lines: http://

tube.geogebra.org/student/b92036#material/85901 – http://tube.geogebra.org/student/b92036 has all my

current worksheets for College Algebra– http://tube.geogebra.org/b/85255 has my

PreCalculus review problems

• Let’s brainstorm together and you, the audience, suggest ideas

• Then I’ll try to build a demonstration• While building the demonstration, I’ll try to

show some features of GeoGebra• Hopefully you’ll see the process is reasonable

to accomplish• I hope you see how to do it

• There is a Help function in GeoGebra• You could do a google search on the topic

and GeoGebra to see if others have posted help on your topic

• Enter in the function name only and GeoGebra will tell you what form the input should be.

• A few slides follow to illustrate some of the features of GeoGebra (from an older version)

Menus

• As with most Windows programs, there are various menus and icons (and somewhat hidden icons)

• Also commands to build things• Finding them all can be a challenge• Allows for many different mathematical

functions• Variety of options of each object you create,

including setting of conditions on when the objects actually appear.

Geometry

• Can do most typical constructions• Once made, you can move things around (the

dynamic part of the program)• Wherever the symbol or is, more

options exist.

• Most typical measurements

• Can have two graphics windows open at once

• Variety of options on the properties of each object you create (such as color, what shows, the style it shows), including setting of conditions on when the objects actually appear.

• If copying pictures to put into a document to print, may take some experimentation to decide appropriate style to use.

Algebra

• Can enter as functions (help is obtained with small triangle to right of input line

• Will take a variety of forms of equations

• Has an autocomplete feature that helps in deciding form of input

• Can be useful if you have some idea of what command might be

• Slider feature is quite helpful in showing the impact of parameters.• This is where most of my “animations” come from• Can use it live, or export picture as an animated gif• Define it first, then can use it• Or introduce an unused variable in your expression and GeoGebra

will ask if you wish to create a slider• Go back to selection mode after defining a slider

• Can display text, including a variety of symbols and the values of variables.

• Some formulas “pretty print” with a different command than just printing them in text.

Spreadsheets• Will work as a

spreadsheet as well• This doesn’t always

seem to work exactly as I think it should

• Can send things from spreadsheet to graph and vice versa

Helpful Hints

• Under Options, Advanced, Graphics preferences is where you find you can set your window dimensions, whether axis and grid shows up.

• Zoom is accomplished with Shift-Click (hold) on an Axis and then moving mouse

• Can save files in different formats and export the graph itself as a picture (even an animated gif)

• “Keeping All the Trains on the Tracks”-from Mathematics Teaching in the Middle School, Vol. 16, No.2 , September 2010, p. 116-120.

• They talk about using the TI-83/84 to produce table, graph, and animated parametric graph as a way to explore d = r*t problems.

• GeoGebra can also do this, http://ggbtu.be/b1560533 • Someone has done it MUCH better than I did at

http://www.mathcasts.org/gg/enliven/af/lines/slopes/car_race_anim.html .

• Can even extend this to do something like http://smartskies.nasa.gov/lineup/index.html , to simulate air traffic control.

• From Mathematics Teaching in the Middle School, Vol. 16, No. 1, August 2010, p. 19 in Palette of Problems, #12.

• A square with sides that measure 10 cm and an isosceles right triangle with legs of 20 cm overlap so that the vertex of the right angle of the triangle is at the center of the square. What is the area of the overlap? (Hint: If you rotate the triangle, the overlapping area remains the same). http://ggbtu.be/b1560533

• Could extend this to PROVE why area remains the same.

• More from Mathematics Teaching in the Middle School http://ggbtu.be/b1560533

• Average of averages• Pizza problem• Resale problem

• Area: http://tube.geogebra.org/student/b92036#material/52073 , volume: http://tube.geogebra.org/student/b92036#material/52149 , as we double input, triple input, etc.

• Power Functions: http://tube.geogebra.org/student/b92036#material/52295 y=kx^p, how do k, p affect the shape

• Graph of sine function as circular function: http://tube.geogebra.org/b/85255#material/146619

• Graph of tangent: http://tube.geogebra.org/b/85255#material/146622

• Many have already contributed to http://www.geogebra.org/en/wiki/index.php/English & http://www.geogebratube.org/

• Can see what others have developed, such as http://www.geogebratube.org/student/m279

• http://www.mathcasts.org/ also has many GeoGebra activities developed

• Could give more ideas on ways it could be used

• Mathematica demonstrations-http://demonstrations.wolfram.com/

• I hope you have some ideas on how PreCalculus animations can be used to illustrate PreCalculus topics and further students’ understanding

• I hope you have some ideas on how Geogebra can be used

• Thank you for coming