'between-ness' in addition/subtraction

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?

Function Family Builder

This environment allows you to explore families of linear, quadratic and absolute value functions. [br][br]Can you make any linear function with this environment?[br]Can you make any quadratic function with this environment?[br]Can you make any absolute value functions with this environment?[br][br]For the cases examined in this environment[br][br]•Vertical sliding of f(x) leads to f(x) + a. [What happens if a>0, a<0][br][br]•Horizontal sliding of f(x) leads to f(x+a). [What happens if a>0, a<0][br][br]•Vertical stretching & squeezing of f(x) leads to af(x). [What happens if a<0, 01][br][br]•Horizontal stretching & squeezing of f(x) leads to f(x/a). [What happens if a<0, 01][br][br]Do you believe these statements are true for any function of one variable? If so, can you prove it? If not, can you find a counterexample?[br][color=#ff0000][i][b][br]What questions could/would you pose to your students based on this applet?[/b][/i][/color]

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