Draw two sets of vertical and horizontal axes on the chalkboard using chalk and a ruler.
Draw a line for the equation "y=x" that starts at the origin of the first graph, point (0,0) with the chalk and the ruler.
Label two coordinates on this line, one at (0,0) and the other at (5,5). Draw a vertical line running from the second point (5,5) to where the line meets the X-axis at (5,0) to create a triangle. Label this curve "Curve 1."
Draw a curve that does not have a constant slope in quadrant 1 on the second graph. Imagine this curve representing the types of natural functions, such as population growth over time. Label this curve "Curve 2."
Label any two points on Curve 2 (X1,Y1) and (X2,Y2), respectively. Draw vertical lines running from each point to where those lines meet the X-axis at points (X1,0) and (X2,0).
Color in the area under Curve 1 with chalk so that it forms a triangle. Calculate the area underneath Curve 1 with the equation "A=½ b*h." Note that geometry can determine the area under a simple curve such as Curve 1 but that the area marked off under Curve 2 does not provide such a simple, easily solvable geometric shape.
Draw three vertical lines from points on Curve 2 to the X-axis; this will break the area under Curve 2 into four segments. Imagine that you could calculate the area under Curve 2 by creating a series of rectangles underneath the curve and then adding up the area of each rectangle. This is how the concept of "Integrals" works.
Draw horizontal lines across the tops of the segments from one point to the next to create rectangles. Note that these rectangles fail to measure some of the area under the curve between the curve and the tops of the rectangles. You could eliminate this problem by drawing even more rectangles underneath Curve 2, if you could draw each rectangle as having a width of 1 point.
Draw more vertical lines running from each successive point on Curve 2 to the X-axis until you have completely filled in the area under Curve 2 with vertical lines of chalk. Note that you could calculate all of these rectangles with the geometric equation "A=l*h" and then add the areas of all the rectangles together, but it would take a long time. The "Integrals" function of Calculus provides a tool to quickly find the sum of all these very small width rectangles, and hence, find the area under the curve.
Calculate the slope of Curve 1 by holding the ruler along Curve 1. Notice that for every point the line extends on the X-axis, it extends upward by 1 point on the Y-axis. You could also prove this using the geometric slope equation: "Slope = (Change in X) divided by (Change in Y)."
Determine the slope of Curve 2 by holding the ruler in line with the curve as you did in Step 1. Notice how you cannot determine the slope for the whole curve or for the curve between any two points as you did with the simple equation that produced Curve 1. Calculus provides a tool called "Derivatives" that enables you to determine the slope from one point to the next.
Hold the ruler against the board at point (X1,Y1) with the edge of the ruler facing the direction of the slope of the line at point (X1,Y1). Notice that if you take a single point in isolation, you can see the slope for this point.
Move the ruler forward to point (X2,Y2) and again align the ruler's edge with the slope of that point. Notice that (X1,Y1) and (X2,Y2) have different slopes, and that the slope changes continually from one point to the next.
Move the ruler along in a fluid motion, continually modifying the slope direction of the ruler. The concept of derivatives provides a mathematical tool to "freeze" the ruler at any given point so that you can calculate its slope, or rate of change at that instant. You could use this tool to track the growth in cell populations in medicine, for example.