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

TO THE TEACHER: This note provides an enriching participant experience through four activities. Completing an activity builds confidence to take on further mathematical challenges and lessens anxiety to work with mathematics objects.

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 R₁, R₁R₂ , 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

An Introduction to Healthcare: The Affordable Care Act

Like a Mardi Gras parade, the Affordable Care Act began rolling before us March 23, 2010. By 2015, the majority of its benefit will have been presented to the American public, and the floor of our country’ s commitment to universal health care will have largely been outlined. But unlike a Mardi Gras parade, the public is not scrambling for this parade takeaways.

The crowd of a Mardi Gras parade is festive and expectant. It responds to the snappy drum cadences announcing the floats to follow and it exuberantly jostle for positions to obtain the goods to be tossed. The ACA parade is rolling, the country’s health care goods and services are being announced, but the healthy seem disinterested.

Prevention, a foundation upon which the health care law rests, has been discussed and its strategies decided. If you missed the discussion, you can learn about the strategies destined to maintain your health. An easy to read government description on prevention describes free services, cancer screenings, free counseling, and other wellness services. Further, various health plans and where to go to eliminate confusion are given.

A much talked about topic in health care debates was the pre-existing conditions. Cancer, asthma, and diabetes are pre-existing conditions. The entire list of pre-existing conditions would surprise most Americans. They were the conditions disqualifying new applicants from insurability. ACA broke the preexisting conditions power with the birth of the Pre-Existing Condition Insurance Plan (PCIP).

PCIP is a federal insurance program giving American citizens with previously defined pre-existing conditions access to health care at affordable prices. These prices will be eliminated by 2015, at which time Americans will be insurable without restriction.

PCIP are available through state programs backed by the Affordable Care Act or through another federal program. Many of these programs offer substantial deductions in health care premiums to the once uninsurable.

The donut hole of Medicare Part D was another dark cloud in the health care debate.

Affordable Care Act (ACA) gives most Medicare clients with a prescription drug totals in the donut hole, a 50% discount on covered brand-name drugs at the time of purchase. From the present to 2020, ACA guarantees these clients continuous Medicare coverage for their prescription drugs. By 2020, ACA will have eliminated the donut hole.

All Americans should know The Constitution. It lists each citizen’s rights. Every American should know the Affordable Care Act. It is the gift each American extends to other American; the floor to the best health for that person, at that time. What a lovely gift! Take advantage of it!

Farewell to A Special One: Roger William Leezer

Again another special one departs. Within the last 365 days, there have been three. But none I held in higher esteem that Roger William Leezer. His personality, fair and patient; always understanding and ever respectful of others. I mourn the loss of this special person.

More.

A Contribution for Women History: Pioneer, Saint, and Example

The life of Jane Elizabeth Manning James abounded in gender, racial and religious struggles. Most would have become cynical, but Jane lived her life obedient to the dictates of her spirit; always eschewing the pressures to conform or embrace acceptance, and to viewing struggles as tests of worthiness for her life goal. Such a matured outlook must be trumpeted during Women History Month.

Jane Elizabeth Manning was born free into a poor family in Wilton Connecticut on May 11, 1819. When her father died while she was at an early age, Jane lived with a prosperous white couple and serve as their domestic. In that household she was Christianized, impregnated by a Presbyterian minister and gave birth to a son.

After hearing a Mormon missionary preach, Jane Manning renounced her previous faith and chose baptism into a Mormon covenant. Desiring to live among members of her new faith, she led her family on an eventful 800 miles journey to the Mormon community at Nauvoo IL.

On arrival, Jane Manning was directed to the home of Joseph Smith. He was impressed by the events of this family’s 800-mile journey and by Manning’s determination to live in the Illinois Mormon community. He employed her as his domestic and offered to adopt her into his family. Ignorant of the spiritual benefits of his gesture, Jane declined adoption into his family.

Life in Illinois was a mixed blessing. Jane Manning married Isaac James, another Mormon, grieved the murder of Joseph Smith and lived through the expulsion of the Mormon community from the state. Determined to rebuild their faith community, the James family and the Mormons journeyed westward. They were among the first wave of Mormon immigrants to enter the Salt Lake Valley.

After many years in the marriage, Jane’s husband left the marriage. As a single mother, life became very difficult. But Jane demanded a strong family work ethic and becoming a washerwoman to supplement the family income. The family suffered many trials but it persevered. Through the trying times, Jane never wavered in her faith. She wrote in her autobiography, “ I “[paid my] tithes . .. [kept] the word of wisdom and … set a good example to all.”

At age 72 James became concerned about the status of her afterlife. She began to write to the Presidency of the Church to grant her and her family adoption and sealing into the faith; at first to an African American Mormon priest then to Joseph Smith. Neither request was granted.

After many denials, the Church approved Jane James to be baptized for her kindred family. She was sealed and adopted by proxy into eternal servitude to the Joseph Smith Family. Joseph Smith’s son stood proxy for his father. James was not allowed to attend the proceedings.

Apparently not pleased, Jane James again followed the counsel of her spirit and continued to seek adoption and sealing for herself and her family.

She wrote to the Mormon Church leader, John Taylor:

“I realize my race and color and can’t expect my endowments as . . . white[s] . . . [but] God promised Abraham that in his seed all the nations of the earth should be blest… as this is the fullness of all dispensations is there no blessing for me? “

After an extended period, her husband returned ill. She accepted him, nursed him until he died – a year later, and buried him from her house.

The year prior to Jane’s death, the Church proscribed that none with African blood in their veins could to be elevated to gain adoption or sealing in the faith.

On April 16, 1908, Jane died. The Desert News , the local Mormon paper, printed:

Few persons were more noted for faith and faithfulness than was Jane Manning James, and though of the humble of the earth she numbered friends and acquaintances by the hundreds. Many persons will regret to learn that the kind and generous soul has passed from the earth.

This eulogy summarizes the accounts of Jane’s generosity and support of the Mormon Church, its community, and to her faith brothers and sisters.

In 1979, the Mormon Church reversed its proscription against African Americans. In this new Church, Jane Manning and her family were sealed and adopted to ordination.

Documents attest to Jane’s unswerving faith, her moral life and diligent commitment to the Mormon community. These are noble reasons to remember Jane James but the reason she deserves to be remembered in history is more profound.

Jane James made her critical life decisions subject only to her spiritual counsel. In particular, the decisions to embrace the Mormon faith and to travel 800 miles to live in a Mormon community, both required great courage and indescribable faith, especially when one consider the prevailing attitude on gender and race.

When abandoned in marriage, James continued to show the same courage and faith – never abandoning her covenants with herself or her faith. Finally, her ongoing battle with Church authorities for adoption and sealing of herself and her family gives another view of the same courage and faith.

Jane Manning James’s life path is an instructive guide from which to gain wisdom. It is an example of obedience to self-counsel. As this is the only path on which one never compromises self, it is a superior course over conformity or acceptance. Further on this path, the spirit is always available to counsel and it never counsels beyond one’s capabilities.  More importantly, when followed, the soul rests in peace.

Notes were taken from:

Karen A. Johnson, Undaunted Courage and Faith: The Lives of Three Black Women in the West and Hawaii in the Early 19th Century, The Journal of African American History, Vol. 91, No. 1. 2006.

Learn More about Jane Manning James:

http://www.ldswomenofgod.com/blog/?p=1327

http://www.ldsgenesisgroup.org/jane_ensign.html

http://www.ldsgenesisgroup.org/archivenews/morrison.html

http://www.blackpast.org/?q=aaw/james-jane-elizabeth-manning-1813-1908

http://www.hds.harvard.edu/news/bulletin_mag/articles/39-12/mueller.html

Teach Ratios

Do you have two things to compare?  If each can be summarized by a quantity, a ratio is quick way to compare them.

This discussion answers the why’s, the when’s and the how’s  learners asks when first introduced to ratios.

THE CONTENT

1.  Why a ratio?

2.  How to represent a ratio

3.  Comparing of unlike things

4.  Miscellaneous examples

5.  Ratio as fractions

6.  A teaching moment

7.  The arithmetic of ratios

8.  Whole numbers and ratios

9.  Teaching problem solving

10. A teaching device

11. Test understanding

12. An interesting ratio

1. Why a ratio?

A ratio is the number of times one quantity [1] is contained in the other.  Thus it is easy to tell which quantity is the larger or the lesser.

2. How to represent a ratio

The ratio of the quantities, a and b, is expressed by the symbolism a : b. It is read a to b. This language is consistent with the everyday language of comparisons; this to that.

a in the symbol, a : b, is called the first term of the ratio and b is called second term of the ratio.

If the ratio is greater than one, the first term is associated with the greater sanity. If the ratio is equal to one the first and second term are associated with the same quantity. It the ratio is leas than one; the first term is associated with the lesser quantity.

3. Comparing unlike things is never permitted

Therefore the quantities used in a ratio must be of the same type. For example, a ratio cannot have one quantity in feet (a linear measure) and the other in ounces (a measure of weight or volume). Both quantities must be in feet or both must be in ounces.

4. Miscellaneous Examples

The Ratio of Forcible Rape to Violent Crime (2009)statistics:  88,097 : 1,318,398.

This is a ratio is less than one.

Ratio of 1955 assets: Exxon Mobil  to GM:

6,614.7 to 5,130.1  (measured in millions of dollars)

This is a ratio is greater than one.

Ratio of : S80 Turbo V6 engine to V70 5-cylinder engine: 2000 rpm: 2300 rpm.

The measurements were taken at 60 mph. The ratio is less than one, suggesting the S80 works less at that speed. Conclude the S80 has the more powerful. The author owns the test vehicles so the results cannot be generalized.

This is as a simplistic measurement of these vehicles’ power power plants.

5. Ratio as fractions

Ratios give the number of times one quantity contains the other. So do fractions. So a ratio can be expressed as a fraction.

a : b =  a / b.

6. A teaching moment

Are you a teacher? Do you wish your students the opportunity to learn about their school and at the same time use ratios? Then assign the following problems.

a) Give the ratios of men teachers to women teachers in the school; b) What is the ratio of female math teachers to the male math teachers;

c) Give the ratio of boys to girls in your math class.

Of course you will have to supply the data.

7. The Arithmetic of Ratios

To Create Equal Ratios: Ratios are fractions. Multiplying the numerator and denominator of a fraction by the same positive number does not change the value of a fraction.

So, multiplying the first and second term of a ratio by the same positive number will not change value of the original the ratio.

In symbols,  a : b = am    bm, (where m is a positive number).

Thus 3 : 4 =  21 : 28. Why?

To Increase The Size Of A Ratio: Add the same positive number to each term of the ratio. If the beginning ratio is less than one, the new ratio will be less than one and it will be greater than the beginning ratio. [2]

In symbols: If a : b < 1,   then for any positive number, m;

a : b < (a + m) : (b + m) < 1

3 : 4 <  8 : 9. Why?

8. Whole numbers and ratios

All ratios cannot be expressed with whole numbers.

For instance, the ratio of the length of the diagonal of a square to the length of a side of the square cannot be expressed with whole numbers.  Nor is it possible to express the ratio of the length of a circle (circumference) to the length of its diameter with whole numbers.

These ratios can be approximate to any degree of accuracy.

Numbers that cannot be expressed as a ratio of whole numbers are called incommensurable.

9. Problem solving with worked examples

ILLUSTRATIVE EXAMPLES

1. Arrange these ratios in decreasing order:

7 : 9,

3 to 4,

16 : 41,

3 : 8,

2 : 7.

Express these ratios as fractions. Then arrange the fractions in decreasing order. The ratios will have the same ordering.

Use cross-multiplication to determine the order of fractions.

2. For what value(s) of x will the ratio 3 + x : 4+x be equal 5 to 6?

Express the ratio as a fraction then solve. (3 + x) / (4+x ) = 5 / 6.

Get x = 2.

3. What number must be added to the terms of the ratio, 3 : 4, in order to get the ratio, 25 : 32?

Let x be the number that must be added to the numerator and denominator to get the ratio 25 : 32.

Using fractions, get   (3+x) / (4+x) = 25 / 32.

Multiply the previous equation by 32(4+x).

Get 32(3+x)= 25(x+4).

Now solve to x = 4 / 7.

4. Find two numbers whose ratio is 5: 6 and their sum is 121?

Let x, y be the two numbers so their ratio is  x : y

.

Use fractions for the ratios,

get   x / y = 5 / 6 and x + y   =121.

Multiply the last equation by 6y and

get: 5y=6x.

Substitute (121-x) for y in the equation on the prevuous line.

Get  5(121- x) = 6x.

Solve to get: x = 11*5 = 55.

So y = 121-55 = 66.

5. The ages of two people gives the ratio 3 : 4.  Thirty years ago the ratio of their ages was 1 :  3. What are their present ages?

Let x : y be the ratio of their present ages.

Write   x : y = 3 : 4 in fractions and get

x / y = 3 / 4.  Multiply by 4x, the equation becomes 3x=4y.

Now thirty years ago their ratio was (x-30) / (y-30) = 1 / 3.

Simplify the last equation and get

3(x  – 30) = y – 30 or  3x  -  90 = y  –  30

Multiply the last equation by 3.

9x – 270 = 3y  – 90.

Replace the 3y in the last equation with 4x (see 6 lines above) get

9x – 270 = 4x – 90 or   5x =  180

So x = 36 and y = 48.

6. Show that when x is positive and a > x then a²-x² to a²+x² will be greater that  a-  x  to a+x.

Attack: Change the ratios to fractions. Then express the first fraction as a product of the second fraction and a number called, F.

If the size of F >  1, the first fraction/ratio is the greater.

If the size of F =  1, the fractions/ratios are equal.

If the size of F <  1, the first fraction/ratio is the smaller.

Factor the numerator of the first fraction. Then multiply the numerator and denominator by the positive number (a+x) and get:

(a²-x² ) / (a²+x² ) = (a -x)(a+x) (a+x) / (a+x ) (a²+x²) =

(a -x)/(a+x) * [ (a +x)² / (a²+x²) ]

F is the fraction in the brackets.  Since (a +x)² > ( a²+x² ), F >1.

Therefore the first ratio is the greater.

7. Find x : y,  if  x² + 15y² = 2xy.

Attack: Divide the equation by y^{2. .} Then arrange to

(x/y)² - 2(x/y) + 15 = 0.

Factor this quadratic.  Get, x/y =5.

8. What is  x to y, if (4x + 5y)/(3x – y)  = 2 ?

Attack: Multiply the equation by (3x-y).

Get (4x+5y)= 2(3x-y).  Solve to

2 x = 7 y .

Divide the equation by 7y.

x / y  =  7 / 2 ,  the desired ratio.

10. A teaching device

Having delivered an excellent explanation of a new concept and now certain the learner has mastered the language of the concept, problem solving skills are quickly grasped after learners have studied a set of worked examples that illustrate the problem solving techniques.

Try it. Below are ILLUSTRATIVE EXAMPLES for learners to study; alone or in groups. Inform them, they will be tested on these examples before the end of the class period. Then test them. Experiment with the amount of study time given the student. Start with 20 minutes.

During the following class meeting, test learners on the same problems. But this time, change the variable names. Keep the greater of the two grades.

11. Test understanding

Use these EXTRA PROBLEMS to determine the level of the learners’ understanding on solving ratio problems.

EXTRA PROBLEMS

1.  For what value of x will (14+x) : (16 + x)  equal  8 : 9?

2.  Find x : y;   if x and y satisfy 9x² +y² = 6xy.

3.  Arrange in ascending order:

3 ; 4

1 : 2 ,

13: 19,

5 : 8,

13 : 14

4.  Find  x : y ;  if  (4x+5y)/(3x-y) = 4.

5.  Show that, if from each term of a ratio the  multiplicative inverse of the other term is subtracted, the new ratio is the original ratio.  Multiplicative inverse of  the number, w is   1 / w.

6. The ages of two persons have ratio 3 : 4. Thirty years ago the ratio of their ages was 1: 3, what are their present ages?

[1] Quantity is a basic property of a thing. It is measurable. Quantities can be divisible or indivisible. Divisible quantities are a collection of objects like a flock, or a herd, a group of people or a company of soldiers. When a quantity is indivisible, it refers to something continuous; for instance liquid, heat, or time. Quantities expressed by numbers can be ranked by size.

[2] Treat the ratios as fractions. Subtract both ratios from one. Note they have the same numerator.

1 – a / b = (b-a) / b

1 – (a+m) / (b+m)  = (b-a) / (b+m)

But the denominator of the second difference is greater than the denominator of the first difference. This means it is the lesser of the two differences. Hence the ratio of this subtraction is closer to one than the other. It makes it larger since both ratios are less than one. So the second difference is a smaller number.  Hence the second ratio is closer to one than the first ratio.

12. An Interesting Ratio

On the line segment AB, let C be the mean proportional between the points A and B.  The ratio BC/AB is incommensurate. The ancients knew this ratio. They used it in their architecture, to describe beauty and balance. Today it is used in modern art design. It’s helpful in the explanations of many phenomena, as well as in designs in the human body.

A simple procedure shows the ratio value is between .6175 and .6182. Many presentations approximating this value exist. A simple one using only basic rules of fractions and elementary terminology of line segments is provided here.

This ratio reveals an interesting aspect of mathematics. It casts light on our understanding of many things, even though it was not studied for this information. This is not uncommon for a mathematical study to reveal useful information that was not the target of its creation. This reality alone gives mathematics its power and beauty.

12. Challenging Problem

Take C to be the mean proportional between the points, A and B.

Compute the 4^th approximate to the ratio AC : AB. Use simple continued fraction techniques.

G. Chrysral, Textbook of Algebra Volume II  7^th Edition Chelsea Publishing Company

Charles Smith, Elementary Algebra, The Macmillian Company

An Interesting Ratio

Mean Proportional

 

An Interesting Ratio

On the line segment BA, when C is the mean proportional between the points B and A, the  ratio BC/BA is incommensurate. This ratio appears in ancient architecture, in modern art design, in many designs in the universe and in designs in the human body.

There are many treatments that show the value of the ratio is between .6175 and .6182. A simple explanation that uses basic rules of fractions and elementary terminology of line segments awaits your examination. Click, Ratio , to view.

An Inspiring Nugget from American History: Black Migration to Kansas

After Reconstruction drew to a close, Southern white citizens moved to retake the economic, political and social reins of the region. Intent to this end, no method was rejected. And none was more effective than the use of fear. Thus the Ku Klux Klan, The White Citizen Council and other groups that used fear to wrestle control of the South were thrust to leading positions in American’s post-Civil war apartheid.

Without education, in a complex society, in a land foreign to him and with little experience of travel, the newly emancipated slave would have been hopeless. But seeds to help the newly emancipated regain his dignity had been broadcasted by Benjamin “Pap” Singleton.

Pap Singleton had been a slave in Kentucky. Like many slaves, white blood flowed in his veins. He learned the skills of a cabinet making in slavery and he also learned from his experience in  slavery to detest his master’s treatment of slaves. After many trials Singleton escaped to Canada.

After Emancipation, Singleton returned to Tennessee to help newly freed slaves gain economic and political power. A noble dream indeed! but Tennessee proved resistant of this dream. So Pap sought to realize the dream elsewhere.

Eventually Congress declared the lands of Kansas and Nebraska to be free. Using his personal resources, Singleton disseminated flyers by mail and personal travel to lure newly freed slaves to settle in one of eleven colonies he planted in the State of Kansas. So powerful was black immigration at this time, the United States Senate conducted hearings to uncover the source of the mass movement.

Singleton never lived in any of his colonies. But he earned the moniker “Black Moses”. The blacks who responded to his call and left Louisiana, Mississippi, Alabama, and the Carolinas were called the Exodusters. These brave people ventured from the lands of their comfort to build a life of dignity for themselves and their children. None of Singleton’s colonies exist today. But they were the promised lands to many. For others they were way stations for their children to a better life.

There were many western colonies to which the Exoduster flew. Only one is in existence today, Nicodemus, Kansas.

Nicodemus is another testimony of a people who traveled an unknown world in search of a better life. Today, it is a tiny uncorporated town. Descendants of the original arrivals continue to honor the town’s celebration of Emancipation. The celebration began in 1878; the year after the town was founded and it has never missed a year. Today it is called Homecoming. From all parts of the country, descendants of the original of Nicodemus pioneers return to refuel their spirit, to remember the struggle of their ancestors, to celebrate the town’s victory over adversity, and to keep alive their heritage.

The Nicodemus colony was born in the minds of newly freed blacks when W. R. Hill, a white land developer, spoke to the black Georgetown Kentucky church and told them about government land available for homesteading. He told about of  acres of land  free for the claiming and  he painted a picture of a settlement with soil ready to yield anything planted, a picture of wild animals available for food, a picture of herds of wild horses waiting to be caught, tamed and ready for work. Finally he painted them a picture of an established town with streets, a church and a general store.

W. R. Hill and the first settlers arrived in Nicodemus (July 30, 1877). There was no town. The pioneers wanted to hang Hill. He hid. Tempers abated and the Kansas winds destroyed the pioneer’s first attempt at shelter. With winter approaching, the Nicodemus pioneers burrowed in the ground for protection against the harsh winter environment, the hot Kansas summer and the Kansas wind storms. The Dust Storms hit. Neighbors and the Osage Indians helped the Nicodemus pioneers through the early trials. The townspeople kept the faith and Nicodemus survived.

Nicodemus is now a part of the National Park Service It is perpetuated as a part of our heritage. It is the story that survival trumps cultural differences.

References

Daniel Chu & Bill Shaw, Going Home to Nicodemus, Silver Burdett Press, Morristown New Jersey.

Lisa Scheller, Returning to hallowed grounds, KU GIVING, Volume 41 Number 2.

Roy Garvin, Benjamin, or “Pap,” Singleton and His Followers,

The Journal of Negro History, Vol. 33, No. 1 (Jan., 1948).

Sacramento Ballet: Romeo and Juliet

Last Thursday night, Sacramento Ballet staged its opening performance of Romeo and Juliet at the Sacramento Community Center. The fresh Sacramento evening air failed to stay the attendees as they filled the auditorium with a festive but expectant mood.

At the customary signal, attendees scurried to their places and the auditorium came to silence and then darkness. The red velvet theater curtains became the screen for floor-to-ceiling project images of the Capulet and the Montague Families courts of arms. Each projection garnered equal area – a clue that we were about to witness this spirited family rivalry. With this opening, the ballet had begun.

Silence broke with the blast of the opening music. The telltale lighted musician’s stands were not visible. The stage was flanked with two larger-than-life speakers that filled the auditorium with the ballet’s opening sounds. Despite the generosity of the Raley Family support, the economy exerted it presence at the event. I was disappointed by the absence of an orchestra.

The curtains parted to a joyous market scene. My disappointment gently faded as this company’s assiduously told the timeless story of “love at first sight” with exquisite dancing, well-choreographed sword fighting, and dramatic acting.

The ballet’s light moments – the antics of Mercutio –  evoked laughter from the audience. The happy moments of the wedding undoubtedly parted smiles on many women’s faces. And the serious moments transported each attendee to witness a senseless death and eventually unfortunate tragedy. It wasn’t long before the economic peril  - no orchestra  -  faded from my concern.

I had seen this company perform Romeo and Juliet accompanied by an orchestra. Of course, it was great!  But this performance ranks as one of its best as the dancers more than made up for the piped music.

Amanda Peet’s danced a lyrical Juliet. She convincingly displayed the emotions of strong defiance to marriage custom; to complete surrender to love. Her scenes with Romeo: tranquil grace filled with emotion. This veteran dancer is a pleasure to watch as she used body to articulate Juliet.

Stefan Calka danced Romeo. His beautifully sculptured body attested attention to trade. His disciplined dance highlighted o the Juliet character from their stealth observation of each other in the their first scene to the violently passionate expression in their last scene. They were an electric pair full of emotion and of spirit.

As the performance progressed, an opening distraction during the market scene faded to my region of opening night forgiveness. My expectation had been met. The audience voiced its satisfaction with a standing ovation to thunderous applaud. Apparently its expectation had been met also.

This evening was a delightful February gift to Sacramento’s ballet aficionados. And to the newly initiated, this company was a great ambassador.

A Letter To A Good Friend

Thanks for your letter. Receiving a letter from a special one is a blessing. It rejuvenates the spirit and it lifts the ego.

I rejoice with you the approach of your formalized study. I had no doubt that you would prevail. Your heart, and your mind seek right and right always triumphant in the long run.

It has been a long time since have been together. Absence tends to fog our memory of the subtle things and even the defining things of a person. We may remember that we like a person but forget the things that attracted us to him.

Writing is good, but it is not like face to face contact. A person’s movement, his speech patterns, his laughter or lack of laughter initiates undetected various comforts within us. So it is with you and with me.

That thought gives me the opportunity to resurrect a statement from your last missile and to respond.

I know just like my son and you say, just deal with what I have at hand to deal with. I am a worker and I feel like the GOD I serve has given me a vision and I just want to work toward making this vision a reality. Right now I have many more things on my agenda like PhD, writing books, getting a job in the medical field and doing many things while my mind is still alert and eager.

The things exciting your mind excite me. I encourage you to continue to entertain them. They will distill what you need for future success. I see many options open to you. A PhD in medicine and a position on a medical school faculty is a great life. You will have writing and teaching as your prime directive.

I have met some self-centered doctors and have lamented the waste.  As a teacher on a medical faculty,  you are positioned to heal these souls from this disease of misplaced priority. If teaching at a medical school doesn’t excite you but a medical practice aiding women spiritually and physically does, these are options also. This is worthy work.  Jesus used this approach to introduce humanity to his valued ministry. Disseminating the ideas and finding of your practice would be life enhancing for those with ears.

My friend, I am happy about your progress and your focus.

Best wishes and prayers are always on your behalf.

Always your friend.