Investigations in Baseball Construction

Objectives:

1. Students will apply the Pythagorean Theorem in a practical setting.
2. Students will evaluate distances generated by their individual constructions using the distance formula.
3. Students will identify and demonstrate proper construction of isosceles right triangles.
4. Students will explore the definition of circles.
5. 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 1st and 3rd bases.

1.  If either 1st or 3rd 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 1st or 3rd.  Is there an alternate way to construct 1st and 3rd bases?  If so, explain why or why not?

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

4.  In the figure below, 2nd 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 2nd 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 2nd base?  Similarly, what is the distance from 1st to 3rd?  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 1st base of a major league field to your field?

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

12.  The 1st and 3rd 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!