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Using to Introduce Linear Equations

Using to Introduce Linear Equations. David Pugh dapugh@rochester.k12.mn.us. Shannon Essler -Petty SESSLERPETTY@csbsju.edu.

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Using to Introduce Linear Equations

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  1. Using to Introduce Linear Equations David Pugh dapugh@rochester.k12.mn.us Shannon Essler-Petty SESSLERPETTY@csbsju.edu This workshop is designed to be interactive. To get the most out of this workshop, you should be at a computer. If you don’t have a computer, we recommend you use one of ours, find someone to share with, or grab the notes and head to another workshop. MCTM Conference May 5, 2012 11:15 – 12:45

  2. Acknowledgements Most of the material introducing GeoGebra was taken from GeoGebra websites or used with permission from handouts created for a GeoGebra workshop at Grand Valley State University led by: John Golden, GVSU Mathematics and Michelle Bunton, GVSU Regional Math & Science Center

  3. What is GeoGebra? • Dynamic Mathematics Software in one easy-to-use package • For learning and teaching at all levels of education • Joins interactive geometry, algebra, tables, graphing, calculus and statistics • Open source software, freely available from www.geogebra.org Quick Facts • GeoGebra facilitates the creation of mathematical constructions and models by students that allow interactive explorations by dragging objects and changing parameters. • GeoGebra is also an authoring tool that allows teachers to create interactive web-pages. Find interactive classroom materials and share your own work on www.geogebratube.org.

  4. After starting GeoGebra, the following window appears: By means of construction tools in the toolbar you can do constructions on the graphics view by mouse. At the same time the corresponding coordinates and equations are displayed in the algebra view. The input bar is used to enter coordinates, equations, commands and functions directly; these are displayed in the graphics view and in the algebra view immediately after pressing the Enter key. In GeoGebra, geometry and algebra work side by side.

  5. Pointers & Arrows The select arrow, on the left of the toolbar, allows the user to select an object to move, resize, delete, etc. The graphics select arrow, on the right of the toolbar, allows the user to move the entire graphics window and to rescale the axes by dragging a tic mark.

  6. Algebra Fun! Using the input bar, enter: An ordered pair in parenthesis. A=(3,3) (pi, cos(Pi)) Drag one of your created points to move it. Line[A,B] Drag a point. Drag the line. Clear the objects from the windows. (Select and right click or delete key) Input a line in slope-intercept form. Input a line in standard form. Input a line in function form: f(x)= Enter: (a, f(a)), where a is any number. Right click on the equations in the algebra view to change their form. Drag one of your lines to move it.

  7. Geometry: Construct a triangle A, B, C and its circumcircle. Open the Perspectives menu and select Geometry. Choose the tool “Polygon” from the toolbar. Now click on the graphics view three times to create the vertices A, B, and C. Close the triangle by clicking on point A again. Next, choose the tool “Perpendicular Bisector” (click on the small arrow at the fourth icon from the left) and construct two line bisectors by clicking on two sides of the triangle. Using the tool “Intersect Two Objects” you can click on the intersection of both line bisectors to get the center of your triangle’s circumcircle. To name it “M”, right-click on it (Mac OS: ctrl-click) and choose “Rename” from the appearing menu. To finish your construction, choose the tool “Circle with center through point” and click first on the center, then on any vertex of the triangle. Using the “Move” tool you can now use the mouse to drag the triangle vertices around - your construction will change dynamically with them.

  8. Success with Sliders Choose the slider tool (2nd from right), then click anywhere on the grid to insert a slider. Name the slider “m” – choose whatever parameters you wish. Repeat for another slider, named “b”. Input y=m*x+b Move the sliders to change the equation of the line. While we are dealing with slopes: Place two points anywhere on the grid. Use the line tool (3rd from left) to create a line through these two points. Choose the slope tool – it’s on the 5th icon from the right, then click on the line. Drag the “m=“ from the algebra view (on the left) to the grid. Move the points around to see the slope change.

  9. Spreadsheet Stuff Use the view menu to activate the spreadsheet. Enter five points (choose ones that will fit on the grid, or scale the grid to fit). Highlight the points, then use the list menu (2nd from the right) to create a list of points. You can also choose this by using right-click. Input fitline[A, B, C] (points must be capitalized). Input fitpoly[A, B, C,2]. Input fitpoly[A, B, C, 3]. What happens? Try fitpoly[A, B, C, D, 3] Is there a pattern to the number of points needed and the degree of the equation? Change the form of the equations by using right-click. Move the points around and see what happens.

  10. More Geometry Tools Line Tool: draw a line. Switch line tools and draw a ray and line segment. For one of them, use an existing point as the first or second point. Use the point tool option to find the midpoint of two of your points. Use the perpendicular line tool and the parallel line tool to construct new lines. Use the polygon tool to make a polygon with new points, and another using some existing points. Construct a circle using the two point tool, and another with the circle with a center an radius tool. Measure an angle, an area and a distance. Try to figure out how to use the rotation tool, the reflection tool, and/or the translation tool.

  11. If You’re Bored . . . • Solve a system of linear equations. • Graph a line through two points. Plot a third point and use the parallel line tool to make a line through the 3m point parallel to the first line. • Graph a line and a point not on the line. Find the distance from the point to the line. • Solve the system and . How many solutions are possible in a system with a quadratic and exponential? Are three solutions possible? • Graph a parabola and a point not on the parabola. Can you find the distance from the point to the parabola? • When is a system of quadratics inconsistent? (Inconsistent meaning no solution.) • Find the roots of a cubic. When can you use that to give a factored form or approximation? • Use any of the discussed commands to make a picture of a funny face using algebra.

  12. Linear Equations • Model based learning. • Inquiry based learning. • Four representations. http://www.geogebratube.org/collection/view/id/579 www.geogebratube.org and search users for CorvairDave Workshop discussion: Ideas for more lessons. Tips to make the lessons better. Ways to use the lessons. Cool new lessons you made or found. https://docs.google.com/document/d/1YjLTHuKYhbg-9Kc3KxsNm39eqPGGnN_swRtnZmS9cK0/edit Link can be found at www.davepugh.org/geogebra

  13. Further Information You can find further information, materials and help on these web pages: • Software http://www.geogebra.org • Manual & Tutorials http://wiki.geogebra.org • Worksheets & Materials http://www.geogebratube.org • User Forum http://www.geogebra.org/forum • Dave’s first attempt, these notes, other links http://www.davepugh.org/geogebra

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