From Distances to the Equation for a Circle

A series of activities and explorations starting with distance (using Pythagorean Theorem) and ending up with the equation for a circle.
[1] rt tri - Used to discuss warm-up with Pythagorean Theorem. Drag or control-click on B to change coordinates. [2] our pts - enter coordinates of location with radius of 5 into myx and myy columns (points will move to locations. All quadrants? Symmetry? Using Integers ONLY? Using only ONE integer? NEITHER is an integer? >What Shape is Emerging?< [3] trace - leave points visible and drag around circle. Trace leaves trail of points. Fills in more points on this shape. Can we get all the points? ([4]). To erase: Deselect trace and then to erase trace points do File to Refresh Views or Control F. [4] circle - With or without (see erase above) trace points visible, select circle and drag around. Note how some values change as the center is moved or the radius is changed. Now try and move the circle to approximately match the values from our points. Write the equation. What do you notice? Values are close to …? What might ‘exact’ be? [5] see pts - note how red points appear where blue points (our points) are. [6] see slider - sliders appear. What happens when you change values of Xcenter an Ycenter? [7] Hide by deselecting (and refreshing with Control F) everything except the red points (and sliders). Move the points to the right and up (e.g., 3 and 2). Now repeat [3], [4] and [5] … Repeat [7] by moving points to another location. [8] With pattern in equation, create circles that: (a) are in the first quadrant, (b) are in the third quadrant, (c) straddle two quadrants, etc.