area
1) The size of circle O can be changed by dragging point A horizontally. Drag point A and observe the effect on how the circle looks. Also observe how the length of radius OA changes as you drag the point.
2) Make a conjecture on how you think the area of circle O will be affected as you drag point A to the left. How do you think the area will be affected as you drag point A to the right?
3) Click on the button, “Show Area”. The area of circle O will appear. Notice how the area changes as you move point A left and right. Does this verify your conjecture?
4) Imagine you plotted a point where you let the x-value be represented by the radius OA and let the y-value be equal to the area of circle O. Describe below how you think that point would behave as you changed the radius length of circle O.
5) Click on the buttons, “Show Point (radius, area)” and “Show Calculation”. As you drag point A horizontally, observe how the plotted point changes coordinates. Also observe the given statistic. Is this how you expected the point to behave? Describe any similarities or differences between this point and the point from the last activity with circumference. Explain why those similarities or differences arise in this situation.
6) While observing the plotted point react to your dragging of point A, make additional conjectures below about how the graph that the collection of these plotted points would produce will look. How does the shown statistic help you make new conjectures?
7) Click on the button, “Show Traced Point”. You will see how your plotted point will pass through all of the “traced points”, giving you a better idea of what this graph would look like. Does the graph look how you expected it to look? Explain anything you noticed about the graph that surprised you.
8) Now that you know a bit more about the collection of plotted points (radius, area), can you make any conjectures as to why the graph looks this way? Describe them below. Make note of things like the intercepts, how the graph was created, and anything you know about circles and area when making your conjectures.
9) Write the formula for the area of a circle below? How does the graph that was created connect with that formula? What are the main differences between this activity and the one that involved circumference?