Our Answer

Potential connections in this activity:

• The relationship between distance covered in a given amount of time and speed
• How the rate of change between two points is related to the slope of a line through two nearby points
• The concept of instantaneous rate of change, which may be explored by checking the slope for several nearby points

This activity also prepares students for future work with instantaneous velocity/acceleration, including the first derivative, dy/dx, of a function.

This sequence is less common in classroom mathematics, and it prompts a deeper search for connections. Rather than first being given an equation, students work from real data and use algebraic techniques and their prior knowledge to write an equation for an existing curve.

This activity examines the velocity of a ball at various times. The change in velocity is estimated by finding the slope of a line through very nearby points, which should be the same as working with the differences in y and x values for the two points. It may be useful for the teacher to emphasize the role of very small intervals when estimating the velocity at a particular time, and point out the fact that what we mean by "speed" is often only an average, such as when we divide total distance by total time. It also good to point out that when used in a physics context, velocity has a positive or negative direction.

Students should be able to connect what they learned in this activity to acceleration and rate problems in physics, such as planetary orbits, the path of a baseball thrown in the air, or the acceleration and deceleration of a vehicle.