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.

# Monthly Archives: March 2011

# 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, whereas Jane lived her life obedient to the dictates of her spirit; always eschewing the pressures to embrace acceptance, of the status quo. She viewed her life 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 visiting Presbyterian minister and gave birth to her first born, 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 from Wilton to the Mormon community at Nauvoo IL.

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

Life in Illinois was a mixed blessing. Jane Manning married Isaac James, another Mormon; grieved the murder of Joseph Smith and experienced the expulsion of the Mormon community from the Illinois. Determined to be a part of the rebuilding of her faith community, the James’ family and expelled Mormons journeyed westward. They were to be among the first 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 was very difficult. But Jane demanded of her family a strong work ethic. She became a washerwoman to supplement the family income. Many trials befell her family but they persevered. Through the trying times, she never wavered in her faith. Jane 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, and by proxy she was sealed and adopted 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.

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, Isaac James returned to the family ill. She accepted him, nursed him until he died – one 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, to her moral life and to her 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 far more profound.

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

Abandoned in marriage, Jane’s courage and faith became her refuge. Never did she waiver from her covenants nor did she become bitter towards the servings of life. Her ongoing battle for spiritual adoption and sealing of herself and her family shows the measure of this courage and faith. This petite black woman stood against the giant structure of her church with perseverance, her sling; and stones from the Word to eventually find a soft spot that slew the giant of her church’s resistance – a true reenactment of the David / Goliath story.

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

**is expressed by the symbolism**

*b,***It is read**

*a : b.*

*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*** t**he 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 *a ^{2}-x^{2} to*

*a*will be greater that

^{2}+x^{2}*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:

*(a ^{2}-x^{2 }) / (a^{2}+x^{2} ) = (a^{ }-x)(a+x) (a+x) / (a+x ) (a^{2}+x^{2}) = *

* *

*(a ^{ }-x)/(a+x) * [ (a^{ }+x)^{2 } / (a^{2}+x^{2})*

*]*

**F** is the fraction in the brackets. Since *(a ^{ }+x)^{2} > ( a^{2}+x^{2}* ),

**F >1**.

Therefore the first ratio is the greater.

7. Find x : y, if *x ^{2 }+ 15y^{2} = 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 *9x ^{2} +y^{2 }= 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

** **

** **

** **

** **

**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).