Feet Planted, Arms Reaching Skyward, Strong Hands

Feet firmly planted,

Arms reaching 746 feet skyward,

And hands gripping my raison d’étre.

I am a sight to behold.

Up close: not easily apprehended,

Afar: easily recognized, although seldom the focus.

Absent: chaos and inconvenience.

My twin is 4200 feet away.

Together we enable commerce,

support security and

provide convenience to transportation.

To the spiritual man,  

I am evidence that

The Creator now creates through humanity.

A Tower of the Golden Gate Bridge

A Mathematical Enrichment: Computing Areas and Circumferences Using Circle and Rectangle Formulas

TO THE TEACHER: This is a mathematical enrichment  experienced through four activities. Each activity builds confidence to take on further mathematical challenges and lessens anxiety to work with mathematics objects.

To properly assess this note, have  pencil and paper handy and follow the activity instructions.

The activities escalate in difficulty. The first activity introduces the definition of a racetrack, a racing lane and the greens of a racetrack: the objects of the enrichment. The activity closes asking the student to compute the length of the racing lane and the area of the track’s “greens”. Each computation is an easy application of the area and circumference formulas for circles and rectangles.

The second activity gives a method of how to construct a second racetrack with a longer racing lane while having the area of its greens exactly equal to the area of the greens of the standard racetrack. The most advanced concept in this construction is “a perpendicular segment to a given line at a given point.”. Experience supports illustrating each step of the construction process with a drawing of the instructions. It accelerates student understanding and discovery of the construction secrets.

The third activity presents a method to construct a sequence of racetracks whose racing lanes increase but the areas of greens are all equal. Although this activity is the most challenging, it is doable.

The fourth activity is optional. It is for the mathematically curious student who wishes to go deeper. Mathematical tools and hints are give to aid a deeper investigation of the enrichment topics. The note closes with an invitation to apply the lessons learned in this enrichment to situations one faces when dealing with people.

PEDAGOGY: Small learning groups have richer learning experiences than learning in lectures. Thus, the author encourages the teacher to use small groups with diverse mathematical abilities when using this enrichment.

A student who has completed geometry will find the ideas in the enrichment within reach. For such a student this enrichment will be an ideal independent study.

ACTIVITY 1: A Standard Race Track

A racetrack can be viewed as a closed figure of two parallel line segments of equal length capped on each end by a semicircle. The length of the diameters of the semicircles is equal to the separation between the line segments.

Labeling the racetrack: Select one of the line segments. In a counterclockwise fashion, label the endpoints A and B. Counterclockwise; label the endpoints of the parallel segment, C and D. This closed figure with the counterclockwise labeling is called the standard racetrack.

The racing lane and the greens: The racing lane is the path traced out by the parallel line segments and the two semicircles. The length of the racing lane is called the length of the racetrack. The region enclosed by the race lane is the greens.

The track dimensions: Let h denote the separation between the parallel line segments. And let l be the length of a line segment.

Give the formulas to compute the length of the racing lane and the area of the greens.

ACTIVITY 2: Constructing a new racetrack from a standard racetrack

A standard racetrack is given.

Let P1 be a point on the segment AB between A and B .

Let R1 be the endpoint of a perpendicular segment of length h/4 at P1.

Draw the segments AR1 and R1B.

The linear path through the points A, R1 and B is one side of the “new” racetrack. It replaces the old segment, AB, of the standard racetrack.

Without rotation, slide triangle AR1B so that AB lies on the segment CD so that point A lays on the pointD and the point B lays on the point C.

The point R1 covers a point. Call it, S1.

Draw the segments DS1 and S1C.

The path along the segments DS1 and S1C is the second side of the “new” racetrack.

The new racetrack is the path composed of the two semicircular ends; the linear segments through the points A, R1, and B; and the linear segments through the points D, S1 and C .

Compute the area of the greens in the standard racetrack and the area of the greens in the new racetrack. Compare the length of the standard racetrack to the length of the new racetrack? Give reasons for your answers.

A thought question

In the construction of the new racetrack, there were two choices for the point R1. Suppose the other choice for R1 were selected. Correspondingly, a different choice would have to be made for S1. What effect would these choices have on your answers to the previous questions?

ACTIVITY 3: A sequence of racetracks: the greens of all the members have equal area but the lengths of the members increase

Given a standard racetrack. Let P1 and P2 be two points on AB that are strictly between the points, Aand B. Further assume P2 is between P1 and B.

On the perpendicular to AB at P1 choose a point at a distance h/41 from P1 and call it R1.

On the perpendicular to AB at P2 choose a point at a distance h/42 from P2 on the opposite side of ABas R1. Call the point R2

Draw the segments: A R1R1R2 , R2B. This path along these segments is one side of the new racetrack.

Without rotation, slide this new side so that the point A lies on D and the point B lays on C.

Call the point that R1 covers S1 and call the point that cover, R2 covers S2 .

Draw the segments: DS1, S1S2, , S2C . This path along these segments is the second side of the new racetrack.

We now have three racetracks: the standard racetrack; the racetrack obtained in Activity 2 from the standard race track by using only P1 ; and finally the racetrack constructed by using P1 and P2.

Compute the area of the greens in these three racetracks? Compare the lengths of these three racetracks? Give reasons for your answer.

Give a method to construct a sequence of new racetracks the lengths of all its members increase but the areas of all the greens are equal ?

ACTIVITY 4: Optional

The following facts can be used to demonstrate that the lengths of the linear segments in a constructed sequence can never be larger that a certain number. See geometric sums.

1. If x and y are real numbers then √(x2+y2) ≤ |x|+|y|.

2. Hint: Square each side of the inequality. Or interpret x and y to be the lengths of sides of a triangle.

The construction in this enrichment guarantees that the areas of all greens will be equal to the area of the standard track. Further this construction yields that the length of all of the racetracks has an upper limit regardless of the number of linear segments on a side.
It is predictable that the lengths of the racetracks increase. But that the greens all have equal areas is not evident. Only through further investigation is it evident. Use this observation to discuss the wisdom of judging others.

A Mathematical Enrichment: Analysis of a Two-horse Race.

teacher and mathematics class

TO THE TEACHER: This note can be a project assigned to small groups or it can be a student assignment to be given at math club. Written to stir up interest in mathematics and to become a foundation on which to develop group cohesion, the teacher’s role is limited to strategic questioning to keep participants actively engaged. For example: What do you think would happen if the lane confinement requirement were changed? How important is the requirement of equal speed throughout the race? And “Can you think of another application for these ideas?”

To master the material, the student must know elementary facts about circles, how to compute the circumference (length) of a circle, the definition of parallel line and the distance between parallel lines.

THE PROBLEM: Two horses race on a standard racing track. They are confined to specific racing lanes and they run the same speeds throughout the race.

THE STANDARD RACING TRACK: The center of the inner lane of a racing track can be viewed as a closed figure of two parallel line segments of equal length capped on each end by a semicircle. The length of the diameters of the semicircles is equal to the separation between the line segments. The centers of the second and third lanes are concentric paths. All lanes are separated by a distance ∆ from the adjacent lane.

THE ANALYSIS – TWO HORSES

The horses are confined to the inner and second racing lanes. Let D be the length of the second lane minus the length of the inner lane. And let r be the length of radius of the inner semicircle:  the distance separating the centers of adjacent lanes.

D = { 2π (r + ∆) + 2l } - { 2πr + 2l }

=2π ∆

This number is positive. Thus the second lane is longer.

Since horses run the same speed throughout the race, the horse in the inner lane must cross the finish line first, thereby winning the race.

My left to right hip measurement is 18 inches. Let’s take this as the measurement between the centers of the inner and second lanes.

So ∆ = 18/12 feet.

Hence D = 2π ∆ > 6 •18/12 feet.

D > 9 feet.

The horse in the inner lane wins by at least 9 feet if the horses run at the same speed and stay in prescribed lanes throughout the race.

Hardly a photo finish!

Suppose the two horses are confined to the first and third racing lanes. Where will the horse in the third racing lane be when the horse in the inner lane touches the finish line?

Discuss strategies for the second horse to win the race.


 

 CONCLUDING REMARK: As an assigned study to a small group, member interactions and communication have the potential to strengthen group bonding thereby lessening student fears to communicate mathematics. As an added benefit, students are lightly to put more situations to mathematical scrutiny.

 

 

 

http://en.wikipedia.org/wiki/Geometric_series#Sum