http://web.hku.hk/~amslee/gsp.pdf

Geometer's Sketchpad

prepared by Arthur Leeamslee@hku.hk

Jun 2008

please download the Sketchpad file for workshop:http://web.hku.hk/~amslee/jun08.gsp

Locus Exploration

Find B so that BC to AB is 2.Construct & Explore

Make a table to record positions of such points.

Plot the points stored in the table.

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http://screencast.com/t/DlZ4cyql

Explore the locus with parametric coloring.

Locus Exploration

http://screencast.com/t/7FSrs6WQn

Two parabolic chords, difference of slopes

Mark 3 points on a quadratic graph.

Construct & Explore

Measure the slopes of 2 chords.Calculate the difference of the slopes.What happens if the points are changed?

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http://screencast.com/t/Zf7pPcBORM

http://screencast.com/t/Mo9SLI7q

Construct a quadratic graphs with 3 given points

http://screencast.com/t/w4MHDevQ

Antiderivative of a quadratic function

Quadrilateral, Sum of Squares of Opposite Sides

Construct tangents with custom tools

http://screencast.com/t/IdpCDc9AQH

Another construction of tangents

http://screencast.com/t/dGuewyDqf

equation of tangent is created from the derivative calculated and coordinates of the given point on the graph

Exponential Function

http://screencast.com/t/W7RHdPxmf

Perimeter and Area of a Regular Polygon

http://screencast.com/t/h1gfQNgvNJg

Line Pattern

http://screencast.com/t/SeD51wyB4

Some readings about use of images and videos with dynamic tools

Oldknow, A. (2005)ICT - bringing maths to life and vice versaMicromath; Summer 2005; 21, 2; Academic Research Library. pg. 16

Oldknow, A. (2003)Geometric and Algebraic Modelling with Dynamic Geometry SoftwareMicromath (Summer, 2003, pp.16-19)

http://cripe03.ugent.be/Vidshell/Vidshell.htm

Vidshell

http://www.atm.org.uk/mt/micromath/mm192oldknowa.pdf

Oldknow, A. (2003)Mathematics from still and video imagesMicromath (Summer, 2003, pp.30-34)

Sharp, B.D. (2007)Making the Most of Digital ImageryMathematics Teacher 100-9, May 2007, pp.590-593free preview at

http://nctm.org/publications/mt.aspx?id=8594

a free program for video analysis is mentioned in Oldknow (2003)

http://www.atm.org.uk/mt/micromath.htmlFrom

Students do not make the Cartesian connection

van Dyke, F. & White, A. (2004) Making Graphs Count. Mathematics Teaching (188) pp. 42-45

Students simply do not see the connection and so do not take advantage of graphs to solve an equation.

Only 35% of the students tested saw that (12,1) provided them with a solution to the equation. Overall, students on average correctly answered only one out of four questions concerning the Cartesian connection. Many students, when interviewed, said that when they saw the word solution they immediately felt they had to do some algebraic process in order to discover some unknown quantity. We need to be aware that students often associate terms in mathematics with set procedures and in general feel that doing mathematics always entails algebraic manipulation.

Students do not make the Cartesian connection

van Dyke, F. & White, A. (2004) Making Graphs Count. Mathematics Teaching (188) pp. 42-45

Students have difficulty using functional notation with graphs

van Dyke, F. & White, A. (2004) Making Graphs Count. Mathematics Teaching (188) pp. 42-45

Knuth, E. (2000). Student Understanding of the Cartesian Connection: An Exploratory Study. Journal for Research in Mathematics Education. 31(4), 500-508.

http://www.jstor.org/view/00218251/ap020152/02a00060/0

http://math.buffalostate.edu/~mcmillen/Kunth.pdf

Knuth, E. (2000). Understanding connections between equations and graphs. The Mathematics Teacher. 93(1), 48-53.

Only 1/5 of the 35 students responded that finding a solution without the missing coefficient was possible, although the most obvious response was that the coordinates of any point on the graph would yield a solution.

... the majority of students did not perceive that the graph had any relevant information that would directly contribute to finding a solution.

http://math.buffalostate.edu/~mcmillen/Kunth.pdf

Knuth, E. (2000). Understanding connections between equations and graphs. The Mathematics Teacher. 93(1), 48-53.

Students solved part (b) primarily in two ways: (1) rewriting the equation in slope-intercept form and using the slope found from the graph to find the missing coefficient; or (2) substituting an x-value and its corresponding y-value, both found from the graph, into the equation and solving for the missing coefficient.

75% of the students selected the former method, ... only 3 students used the latter method.

http://math.buffalostate.edu/~mcmillen/Kunth.pdf

Knuth, E. (2000). Understanding connections between equations and graphs. The Mathematics Teacher. 93(1), 48-53.

Ironically, a number of students responded to part (a) by stating that finding a solution was impossible without the missing coefficient; however, these same students responded to part (b) by stating that using the graph would allow them to find the missing coefficient.

... these students did not recognize that the procedures that they used relied on the fact that any point on a line is a solution to the equation of the line.