Stanford talk
This is a collection of applets that illustrate some mathematical habits of mind. These mathematical habits of mind are central to the [b][i]RE[/i][/b]awakening of teacher curiosity and creativity. They are part of an approach to teacher education built on posing problems & making & exploring conjectures.
Table of Contents
- challenging the canon
- Solving Linear Equations Graphically & Symbolically
- parameter spaces - a multiple representation strategy
- linear functions - parameter plane
- Conjectures about Complex roots of polynomials
- ISOSCELES TRIANGLES
- polygonal linkages
- Point in an Equilateral Triangle - revisited
- Quadrilateral factory ?
- Polygon Explorer
- Polygon Synthesizer
- formulating measures
- Squareness of Parallelograms
- Rhomboidicity of Kites
- a road up a mountain
- Fuel Consumption Calculator - length/volume ratio
- working backward
- Build a triangle from medians?
- Build a Triangle from Medians - II
- UNsolving Linear Equations & Inequalities
- "between-ness" - a problem posing strategy
- 'between-ness' in addition/subtraction
- 'between-ness' in multiplication/division
- Parabolas between parabolas
- scaling - a problem posing strategy
- The optimal cone -
- A Cube Removed From A Cube
- The most efficient can of vegetables
- ...and finally...
- Two Planets & A Sun - A Digital Triptych
Solving Linear Equations Graphically & Symbolically
A linear equation is always of the form [b]f(x) = g(x)[/b]. [br]For example, in the equation [b]2x - 1 = -2x + 5[/b] we can regard f(x) as 2x - 1 and g(x) as -2x +5.[br][br]Solving a linear equation means transforming the original equation in to a new equation that has the function x on one side of the equal sign and a number (which is a constant function) on the other side. [br]In this case the [u]'solution equation'[/u] is [b]x = 1.5[/b] (why is 1.5 a function?)[br][br]This applet allows you to enter a linear function [b]f(x) = mx + b[/b] by varying m and b sliders and a function [b]g(x) = Mx + B[/b] by varying M and B sliders.[br][br]You may solve your equation [size=100][size=150][i][b]graphically[/b][/i][/size][/size] by dragging the GREEN, BLUE and WHITE dots on the graph in order to produce a [u]'solution equation'[/u] of the form [b]x = {constant function}[/b].[br][br][b]CHALLENGE[/b] - Dragging the WHITE dot changes both functions, but dragging the [color=#00ff00][i][b]GREEN[/b][/i][/color] dot changes only the [color=#00ff00][i][b]GREEN[/b][/i][/color] function and dragging the [color=#1e84cc][i][b]BLUE[/b][/i][/color] dot changes only the [color=#1e84cc][i][b]BLUE[/b][/i][/color] function.[br][br][b]This means that when you drag either the [color=#00ff00][i]GREEN[/i][/color] dot or the [color=#1e84cc][i]BLUE[/i][/color] dot you are changing only one side of the equation!! Why is this legitimate? [br][br]Why are we taught that you must do the same thing to both sides of the equation?[/b][br][br]What is true about all the legitimate things you can do to a linear equation? [b][br]- What are the symbolic operations that correspond to dragging each of the dots?[/b][br][br]You may also solve your equation [size=150][i][b]symbolically[/b][/i][/size] but using sliders to change the linear and constant terms on each side of the equation. [b][br]- What are the graphical operations that correspond to each of the sliders?[br][br][/b][color=#ff0000][b]What other questions could/would you ask of your students based on this applet?[/b][/color]
linear functions - parameter plane
A linear function in the x,y plane can be written as y = mx + b. [br]For example, the linear function y = 2x +3, has the values m = 2 and b = 3. [br][i][b]We can plot this point (2,3) in the m,b plane. [/b][/i][br][br]Every linear function in the x,y plane corresponds to a point in the m,b plane. [br][br][u][b]Exploration: [/b][/u] check the EXPLORE box and experiment to see how this works.[br][br]What points in the m,b plane correspond to the functions [br]y = b, b = -1, 0, 1, 2, 3, ... in the x,y plane?[br][br]What points in the m,b plane correspond to the functions [br]y = mx, m = -1, 0, 1, 2, 3, ... in the x,y plane?[br][br][b][u]Challenge: [/u][/b] check the CHALLENGE box - a dotted line will appear in the m,b plane. [br]You can position this dotted line by dragging the two small black rings. [br]After you have placed the dotted line where you want it, [br]you can slide a large RED dot along the dotted line. [br]Each [i][b]position of the large RED dot[/b][/i] has a set of coordinates [i][b](m,b) [/b][/i][br]and therefore corresponds to a [i][b]linear function in the x,y plane[/b][/i].[br][br]As you slide the RED dot along the dotted line you will notice that [br]all the linear functions in the x,y plane pass through a single point. [br][br][u][b]YOUR CHALLENGE:[/b][/u] How is the position of the fixed point in the x,y plane [br]related to the way you placed the dotted line in the m,b plane?
Point in an Equilateral Triangle - revisited
The GOLD point in the left panel is chosen at random inside the equilateral triangle. The red, blue and green segments are lines drawn from the GOLD point and perpendicular to each of the sides. Their lengths vary in size as you move the GOLD point from place to place inside the triangle. However, the sum of their lengths is constant. What is the sum equal to? Why? Can you prove it?[br] [[i]You can change the size of the equilateral triangle by dragging the BLACK dots.[/i]][br][br]Would a similar thing be true in a square? Why or why not? [br][br]What about other regular polygons with an odd number of sides? with an even number of sides?[br][br]Under what circumstances can the red, green and blue lengths can form a triangle?[br][br]What is the probability that a point chosen at random inside the triangle will determine three lengths that can form a triangle?[br][br]Can you prove your conjecture?[br][size=85][i][b][br][see further challenge below][/b][/i][/size]
[b][color=#1551b5]- GOING FURTHER[br][br]What positions of the GOLD dot determine an isosceles triangle?[br][br]Can you use the fact that the sum of the distances from any point in the interior of an equilateral triangle to its sides is constant to prove that -[br][br][i]knowing the area and the perimeter of a triangle does NOT uniquely determine the triangle[/i][/color][/b]
Squareness of Parallelograms
|
A rhombus is a parallelogram with perpendicular diagonals that in certain circumstances is a square. A rectangle is a parallelogram with interior right angles that in certain circumstances is a square. Can you devise a measure of squareness of parallelograms that allows you to tell how much "squarer" one parallelogram is than another? Can you devise a second and different measure of squareness of parallelograms? Under what circumstances might one such measure be preferable to the other? |
|
|
Build a triangle from medians?
Three line segments ([b][i]dotted lines[/i][/b]) intersect at a point that divides the lengths of each of the segments in the ratio of 2 to 1.[br][br]These three line segments could be the medians of a triangle.[br][br]You can use the sliders to set the size of these three line segments. The changed lengths will maintain the 2:1 ratio. Try to set them so that points A and B, B and C, and C and A are joined by straight line segments.[br][br]Alternatively, you can try to drag the points A,B and C to where you think the vertices of the target triangle are. When A, B and C lie on the vertices of the target triangle the angle between two segments of the same color will be 180 degrees.[br][br]Can a triangle always be made this way? Can you prove it?
'between-ness' in addition/subtraction
'between-ness' in addition/subtraction
The optimal cone -
For a fixed slant length, what is the maximum volume under the cone?
Two Planets & A Sun - A Digital Triptych
The motions of a Sun and two planets as seen in each of their coordinate systems.[br][br]You can animate the motion by clicking on the lower left hand corner of the screen.[br][br]You can show the trajectories by checking the trajectories checkbox.[br][br]You can show the distances between the bodies by checking the distance checkbox.[br][br]Does this help you understand the retrograde motion of Mars?[br][br]Why does the triangle of distances look the same in all three coordinate systems?[br][br][b][i]What are some questions you would pose to your students that could be explored with this applet?[/i][/b][br][br]