The top figure shows that if a ball traveled 14. 7 cm in one time period,
adding the acceleration constant ( 9. 8 cm) will yield the distance traveled
in the next time period.
In the bottom figure, the initial white strip is followed by blue strips (n- 1),
visually expressing the algorithm Distance n = 4. 9 + (n- 1) × 9. 8.
bottom left on page 10) to provide a visual record of how much
farther the ball traveled in each successive 10th of a second.
The distance traveled in each 10th of a second is its speed,
often computed as meters per second. Your students can
explore this further by investigating the rate of acceleration, which appears to be constant. In our example, a series
of blue rectangles highlights the difference between successive strips. If students align the blue rectangles, it becomes
clear that they are the same height. This illustrates that the
rate of acceleration due to gravity is constant ( 9. 8 meters
per second per second, or per second squared).
Giving students the opportunity to determine this
through an activity that initially involves no numbers can
lay the groundwork for extensions that add numeric values.
Velocity can be expressed as the rate of change during
a given time period. Acceleration is the rate of change in
velocity. Adding the acceleration constant to the distance
traveled in the previous time period predicts the distance
that the ball will travel in the next time period. Teachers
can illustrate this graphically, verbally, or numerically, as
shown in the figure at the top of the page.
A number of sources of error can affect the figures that
a class records. However, if the sources of error are consistent, the overall relationship should be similar.
This pattern can serve as an entry point to algorithmic
thinking. You can write the overall pattern for each successive time period this way:
Distance 1 = 4. 9 + (0 × 9. 8)
Distance 2 = 4. 9 + ( 1 × 9. 8)
Distance 3 = 4. 9 + ( 2 × 9. 8)
Teachers can present this more generally with the symbol
n replacing the actual number (Distance n = 4. 9 + (n- 1) ×
9. 8) or in a visual format, as shown in the table at the left.
By scaffolding the introduction to this general concept
with a mixed-reality graphical depiction, teachers can
introduce the same concept across a broad range of grade
levels. Once students understand the underlying concept
through a graphical introduction, teachers can introduce
increasing levels of abstraction at appropriate developmental ages.
A slow-motion video of a ball drop that students can
use to make their own measurements is available at www.
maketolearn.org/balldrop.
There you can also print out a centimeter ruler if you
would like to experiment with creating your own video.
This exercise is feasible with almost any conventional video
camera. The image of the ball may blur in later frames as
the speed increases, but measurements made at the bottom
of the ball will provide a reasonable approximation of the
distance traveled.
You can extend this activity to a number of other scenarios. For example, will the pattern be the same if students
toss the ball upward instead of dropping it? What type of
pattern would students observe if they throw the ball laterally? What pattern would they record if they rolled the ball
down an inclined plane?
An initial graphical exploration of these relationships
without numbers provides a way to make concepts more
accessible. Once students understand the underlying pattern, teachers can use numerical explanations to provide
deeper understanding of the underlying algorithms.
Glen Bull ( gbull@virginia.edu) is co-director of the Center
for Technology and Teacher Education in the Curry School
of Education at the University of Virginia. Willy Kjellstrom
( willyk@virginia.edu) is a professor of instructional technology at Black Hills State University in South Dakota. Yash
Patel ( YashPatel@virginia.edu) is a graduate fellow in the
Center for Technology and Teacher Education.
