__Investigations in
Baseball Construction__

** **

** **

**Objectives**:

- Students will apply the Pythagorean Theorem in a practical setting.
- Students will evaluate distances generated by their individual constructions using the distance formula.
- Students will identify and demonstrate proper construction of isosceles right triangles.
- Students will explore the definition of circles.
- Students will properly use proportions and ratios.

Many students tend to memorize different mathematical equations but seldom understand their actual applications outside the classroom. Students with low-socioeconomic status may not have the community resources to see geometric applications at work in everyday life. One geometric topic in particular is the various uses of the Pythagorean Theorem in its relation to the distance formula. Students are more apt to see these equations as two totally non-related items in geometry. With the use of Geometry Sketch Pad technology and a series of investigative questions, I believe I can increase my studentsÕ understanding and correlation of the distance formula and the Pythagorean Theorem.

In the following activity, I will have students construct a scale model of a baseball field. Some students may not have access to a baseball field, so I will provide them with a typical model of a regulation-sized field with dimensions:

At first glance, there is much complexity to constructing this field, but with GSP technology, the students will be able to construct accurate scale models of the infield and beyond. We will have a Construction and an Exploration section as we progress through our activity.

__Construction__:

Begin by having them construct two perpendicular lines in
the form of a V-shape to represent the direction of home plate (H) and the 1^{st}
and 3^{rd} bases.

1. If either 1^{st}
or 3^{rd} base is arbitrarily chosen, how would you construct the
opposite base using circles?
Without measurement explain why?

2. Here in this
construction, the use of circles and definition of radius can be useful in determining
the position of either 1^{st} or 3rd. Is there an alternate way to construct 1^{st} and 3^{rd}
bases? If so, explain why or why
not?

3. Without
measurement, is the construction of 2^{nd} base automatically
determined? If so, how? If not, construct and label.

4. In the
figure below, 2^{nd} base is constructed by using the intersection of
circles. Explain how and construct
a similar figure.

See here for alternate construction.

5. Is there an
alternate way to construct 2^{nd} base?

6. Based on **your** construction, how far apart are your bases from one
another? (**Use** your measuring
tool.)

7**. Without
using** the measuring tool, what is he
distance the catcher must throw to reach 2^{nd} base? Similarly, what is the distance from 1^{st}
to 3^{rd}? Are they the
same distance, if so, why? Use your measurement tool to check your
accuracy. *At this point, they
will have accurately constructed right angles between the bases. This will be my first opportunity to
introduce the Pythagorean Theorem with the use of the word distance. *

8. What is the total distance around the bases? What is another name for this distance?

*Most of their lengths will be different, which gives them
the opportunity to do individual calculations.*

10. What is the
ratio of the distance from home plate to 1^{st} base of a major league
field to your field?

11. What is the
ratio of the throw from home to 2^{nd} base of a major league field to
your field? Is it the same as #10?

__ __

12. The 1^{st}
and 3^{rd} base coachesÕ box must be near the base runner at the
perspective positions. This
box is positioned for the coach to have the perfect view of the runner rounding
the bases. The box is placed 15ft
from the base/foul line, 10ft wide and is 20ft long, parallel to the
corresponding baseline. Construct
a proper position for the coaches to stand.

13. Given the
distances in 12, what should be the proportional measurements of **your **coachesÕ boxes?

14. The dotted lines are the coachesÕ line of sight from the furthest point of the coachesÕ box. What is the horizontal distance from their position to home plate?

15. In a
regulation-sized field, home plate to the center field wall is 400ft. If the homerun wall is 8 feet high,
what is the minimum distance (horizontal distance) the ball would have to
travel from home plate, assuming the ball is hit **4ft off the ground**?
(Assume the ball is hit as a line drive.)

Possible finished infield. Click here. Attempt to make a better one!