A Content Attainment Lesson Plan for Ratios Part 3

Mathematics Teacher and Classroom

Mathematics Teacher and Classroom

This is third post in the series A Content Attainment Lesson Plan for Ratios. It gives suggestions to an alternative approach to teach proportions.

I give two guided presentations (situations). Each has everything students need to attain (discovery) the definition of a proportion. Encourage students to study both situations before attempting to give a final definition of proportion.

For this study adopt, the definition: A proportion is a statement that says two ratios are equal.

Situation 1

We wish to make predictions. Good predictions require knowledge of the world where the predictions apply. Our first task is to seek condition(s) that ensures good predictions.

Let’s begin.

Assume a world in which only two conditions affect events. Call the conditions, controlling conditions. If the behavior of these conditions are completely understood, good predictions are possible.

Below choose responses to the numbered statements that makes a “good” prediction possible?

1. The controlling conditions do not change. ][ The controlling conditions change.

2. Controlling conditions that are not changing can indicate world events are not changing. ][

Changing controlling conditions indicate world events are not changing.

Assume a number on a common scale summarizes each controlling conditions. Ratios are ideal measure to compare two things of the same type.

So choose a ratio of the controlling conditions numerical summaries. Call the ratio, the controlling ratio, since it is a numerical measure of the controlling conditions at a particular time. The only restriction place on the ratio is that its denominator cannot be zero.

If world conditions are not changing, which choice in the following do you expect to be true.

3. The controlling ratio is probably not changing. ][

The controlling ratio is changing.

Have students to discuss of this statement and then write a summary of the discussion.


ABC Manufacturing Company reported the following information:

On June 30, 2013, free cash flow was $284,867 and dividends paid were $99,851.

On March 31, 2013, free cash flow was $206,207 and dividends paid were $99,812.

On June 30, 2012, free cash flow were $175,837 and dividends paid were $100,054.

Your task is to predict whether this company can pay a good dividend. Free cash flow and dividends paid are the only information available.

Take the ratio of dividends paid to free cash flow to be your controlling ratio. Is it a good predictor? If so why, and give a prediction. If not, why not.

Teachers: With this exercise experience, have students rewrite their summary statements to mention the controlling ratio and to say whether it can or can not give reasonable predictions.

Some students may need a hint.

Hint : If the controlling ratios are not changing over time, good predictions are likely.

Situation 2:

A red dot is 1/4 of the distance from the center of a DVD disk. On the outer edge of the same radius is a green dot. The disk spins about its center.

Ask students to compute:

Compute the distance the red dot travels in one revolution. (A)__________________

Hint: C = 2 π r, where r is the radius of the circle the point describes.

Compute the distance the green dots travels in one revolution. (B)_______________

Give the ratio of the distance that the red dot travels to the distance that the green dot travels in one revolution. (C)_____________________________

What is the ratio when the disc rotates half of a revolution? a quarter of a revolution?

What do you think is the ratio for a partial revolution?

Can you compute the ratio for any partial travel about the center? ___________________

Are the disk sizes relevant to the ratio? _________________

Is the speed of the disk relevant to the ratio? __________

In the final analyses what was(were) the important factor(s) needed to compute the ratio?_____________

How do these answers make you feel about ratio?

Give your definition of proportion.__________________

A Problem

A disk of radius of 5 feet has a red dot 5 inches from its center. On the outer edge of a second disk of diameter 4 yards is a green dot. Where on the second disk should the red dot be placed to have a proportion between these two disks?

Writing proportions

Symbolically a : b represents the ratio of a to b.

a : b : :  c : d  is the symbolic representation of a proportion.  The symbol “: : “ is read “as”.   a : b : :  c : d  reads , a is to b   as   c is to d.

If the terms of the ratios are numbers, write a / b = c / d.


1. Over a three months period, the insider of a company bought about 120,000 shares of stock, compared with insider sales of 96,495 shares. Last year, insider purchases had been two million shares of stock, compared with insider sales of around 800,000.

Using these numbers to form two ratios. Do they form a proportion?

2. In 2010 The United States boasted one of the best health care systems in the world. It had an estimated 246 000 primary care physicians [1] and 256,000,000 citizens with health care coverage [2]. The Affordable Health Care Act will add 30,000,000 who are not able to afford the cost of coverage. If the ratio of physicians to those covered must be maintained to have the system’s quality, How many new doctors must the system have to cover the 30,000,000 increase?

3. Jackie bakes cupcakes for her church’s bake sales. For Sunday’s bake sale, Jackie will bake 45 cupcakes. After checking her available ingredients, she has only 3 tablespoons of the sugar. The recipe requires 23 tablespoons of sugar for 45 cupcakes. In her cupboard, Jackie has a 40-ounce bottle of agave, a sweetener. The agave manufacturer claims 4 tablespoons of agave has the same sweetening power as 5 tablespoons of sugar. How much agave must Jackie add to her sugar in order to bake 45 cupcakes?

4. The City of Sacramento CA has a sales tax. Assume all purchase items have the same tax rate – not true. Is the following true? Total price of any purchase in the City of Sacramento : the sum of the prices of the purchased : : the total purchase price of a nine-volt battery : the price of two nine-volt batteries. [5]

Describing concepts with proportions

The ancient Greeks believed numbers could describe natural concepts. To visualize a number they used line segments. This enabling them to visualize concepts with line segments.

To represent a given concept that is not as extreme as either of two given concepts,  a number between the assigned numbers of the extreme concepts is chosen.

A favored choice between two numbers is the midpoint of a line segment is the Arithmetic Mean of the two numbers.

A concept associated with the arithmetic mean has the balanced blend of the extreme concepts.

The Greeks used the proportion    (a-m) : (m-b) :: a : a  to describe the arithmetic mean, m, of a and b where a > b.

Another favored choice is the Geometric Mean, m, satisfying the proportion:     (a-m) : (m-b) :: (m-b) : b   where a > b.

A third choice to represent betweenness is the Harmonic Mean, m, it  satisfies   (a – m) : (m – b) :: a : b    for  a > b.

Problems. (Requires elementary algebra)

1. Express m, the means in the three proportions, in terms of a and b.

2. Given x, y > 0. What is x : y, if x2 – 6 y2 = xy ?

3. For what value(s) of x is the proportion (x+3) : (x +4) : : 5 : 6 true?

4. For a > x > 0, show that a2 – x2 : a2 + x2 > a-x : a+x.

5. If a : b = c : d prove a3/b + b3/a : c3/d + d3/c = ab : cd.

Hint: Let x = a : b = c : d .

6. If a : b = b : c , solve for b in terms of a and c.

7. Give numbers a, b, c such that b is the geometric mean of a and b.

Plato used the proportion in problem 6 to explain his divided line theory [3].


[1] http://www.cnn.com/2012/06/27/politics/btn-health-care

[2] http://www.kevinmd.com/blog/2012/03/onethird-hospitals-close-2020.htm

[3] Plato Republic, Translated by Benjamin Jowett, Barnes and Noble Classic, 2004.

[4] Edward Maziarz and Thomas Greenwood, Greek Mathematical Philosophy, Mathematics People, Problem, Results Volume I, p 22-23.

[5] Dick Stanley, Proportionality, SUMMAC FORUM , September 1993, pages 5,6.

A Content Attainment Lesson Plan for Ratios for Ratios ©

Published by Lloyd Gavin

Memories: Birth of a New Creation

To those who check this spot, I interpret your dilligence as a wish for my continued good health as well as  a hope that I have eventually said something of value.

Presently I am in a turbulent health environment   It has snatched from my unrelentless hold onto youth. I conceed.  My body cannot do what it once could. Thus I am constructing bunkers for future onsaults. Sideline about the timing of my stormsThey coincides with those widely spoken of in middle west United States.

This post chronicles events  May 2013 that have become treasured memories. They occurred before the onset of my turbulence.

The memories began  on a fifteen days crossing aboard the Nieuw Amsterdam. The ship departed Fort Lauderdale, FL  and I disembarked at Barcelona, Spain. Before reaching the destination, much drama passed. Click this link to read the details of the sea rescue.

The rescue was exciting. The cruisers gathered on her decks to witness.. When the first rescued was aboard, our humanitarian goose pimples spontaneously erupted into thunderous applauds.

A rescue on the high seas is something to witness. But for me, the skil it took to make this rescue successful was never overshadowed.

Picture an ‘oceangoing’ vessel on the high seas slowly but with much grace approaching a stranded 42-feet vessel. Coordinating the scales of these vessels on a constantly shifting sea that ended in successful rescue of stranded humans is what I believe we subsonciously recorded. Thereby triggering the spontaneous thunderous applauds to honor that skill to save lives. However exciting and memorable these events were not the reason for my trip .

The purpose of this trip was to see the church, la Sagrada Familia. It has been under construction for more than 100 years. Many reasons can be offered but in today’s let-do-it-quickly world that’s a long time to construct anything. The principle human character is Antoni Gaudi. His influence is massive! It invades a huge part of the city.

My hotel is in the Horta-Guinardo district. It was a leisurely 1/3 mile downhill stroll to Sagrada Familia.  Along my downward path, the spires of Sagrada were always in view.

In Neighborhood

Walk to Sagrada Familia

The neighborhood was busy: a relaxed busy.

Within one block

Within one block

It houses hospital complexes,

San Pau Hospital

Within one block

commercial and residential buildings.

Cafes, restuarants were inexpensive.

Speciality shops were everywhere.

Within one block of hotel: A water cascade

Within one block of hotel: A CANDY STORE

The people?  I experienced only friendliness, kindness and desire to be helpful. Now remember my background as we turn in another direction.

La Sagrada Familia is a church built of stone and knowledge but something beyond nature and man directs it. For me, it was a mathematical marvel unfolding before my eyes.

If students of calculus are able to visit the Sagrada Familia, I believe they would experience the same marvel, would desire to learn how it is being constructed and gain a deeper  appreciation for calculus, art and architecture, while witnessing the unfolding of this new creation. Further, they would  learn how la Sagrada Familia was conceived and  how basic mathematical models from their calculus classes are used to accomplish all of this.

La Sagrada Familia is a a new creation. It affirms that spirituality is an integral part of our being. It supports the value we place on art. It attests mathematics’ major role to man in the creation process and it makes clear the power of an economic stimulus in a waning economy.

Enjoy a few photos.

Is this how Gaudi sees himself: a slave to his call?

Is this how Gaudi sees himself?

How Gaudi sees himself?

The church’s exterior.

The entry is a marble slab honoring the Jerusalem pilgrimage.

Prigrimmage to Jerusalem

Prigrimage to Jerusalem

Inside the church: A shot from the rear of the church, looking upwards, a staircase: sanctuary right.

By Faith not by Sight


This picture is a composite of two images: an eye and a collection of five intersecting lines. The eye reflects the view’s momentary interest. It appears blurred. The blurred reflection is the reason I am reminded of the phrase – live by faith not by sight.

The prevalence of iPod™, iPad™, computers, books, movies, and driving automobiles raises our eye- the organ of sight- to an exalted position in making decisions. More clearly, after seeing another live a certain lifestyle, many desire to live the same life style; seeing a certain car race by, many desire to drive that model or a better one; seeing another sporting a sculptured body, many are driven to so sculpture their body, etc. Some actually get what they seek. A time later, the appeal of their  acquisition fades and what appeared to be a sure entrance to happiness proves to be an illusion. Hopeful to find happiness, the seeker continues the search but meets the same failed fate. After many  futile recurrences, he learns the senses reveal only the appearance of happiness – never happiness. To avoid the endless search for happiness that sensual input promises, live by faith.

Living by faith

To live by faith, adopt a well defined life plan, a plan that is not dependent on sight or worldly things. Each day, determined to live all aspects of the plan perfectly. Each step towards perfection serves you a bit of the sought after happiness.

Isaiah 28 reveals an invitation to embrace a life plan free of chases leading in dissatisfaction. Click to read. Properly interpreted, the lines in the second image illustrate the invitation. Below is my interpretation of of the lines.

The second theme: the collection of five lines

Three of the five lines in the collection intersect in a point to the left of the eye. One of the three lines is horizontal. A vertical line intersects the horizontal line to the right of the eye. The vertical line is the fourth line in the collection. The fifth line rises from the picture’s center-bottom to intersect the vertical line near the top of the picture.

Each line represents  the trace of a person’s life journey. Call it a life path. All possible life paths cannot possibly be shown, so only those having a certain defining qualities are shown. The actual trace is of lesser importance than its direction.

The horizontal line describes balance. It represents temperance, self-control, among other things, treating everyone with respect etc. It’s the life path of a just person. The vertical line has no deviations. It is a plumb line. It represents the life path of a righteous person. This is the good life promised in Isaiah 28. (A righteous person is also a just person)

The Isaiah theme

Begin the interpretation with a person positioned on the intersection of the three lines (life paths). Movement along a life path to the right advances the person towards the good life. Remaining in the same position or movement to the left is a failure to advance towards the good life. A person who moves on the horizontal line to the left is a just person but is not a righteous person. But a righteous person is a just person.

Why start at the intersection of the three lines? 

Beginning the interpretation at the intersection of the three lines, means at any time a person is able to choose: to be just, to remain at the present life state, to move away for the good life or to move towards the good life. The choice is for the person to make. The fifth line in the picture is a reminder that all life paths can end in the good life.


Michael Pick presented this two images picture as a writing assignment in WordPress Daily Post, February 18,2013. The blurred reflection in the eye characterizes the nature of sense information, in particular sight. Information from the senses, (sight) is never complete. Life concurs this.

Isaiah 28 is a revelation. Revelations are better guides by which to live than responding to sight information. The lines in the second image represent the choices in Isaiah 28 hence they promise a good life, a life free of disappointment.

Please Excuse My Poor Manners

I know I should have registered a post by now. The Daily Post has warned us about the use of widgets to alert readers when post scheduling is erratic. I have not learned the sercet of widgets. I promise to make an effort.

The long interval between post is caused by a convergence of many events:

  • Extended visits while in Louisiana to a longtime friend and my dear uncle .
  • Taking extra time to see parts of this beautuful country along Interstate 40.
  • Resting from the long drive. I was alone.
  • Presently satisfying scheduled appointment with doctors.

Within a week I will post thoughts about this past GPchallenge picture.


Please remain patient and excuse my poor manners.


A New Order

This photo ushers forth three significant themes from my memory. The streetcar ascending the hill (I choose to interpret it to be ascending), the embracing couple, the passengers departing the streetcar and those allowing them to depart before boarding: each is a chapter in a dream-come-true story.

Let’s focus on the streetcar. It represents the train of life that slowly advances humanity to a higher awakening. The departing passengers are the old guard. They are at the end of their life journey. The passengers waiting to board represent the new order that is to be tried.

This scene symbolizes my patience to board the life train and ride it into a long awaited new order.

Soon I will depart for my home state to attend the investiture of a high school classmate to be Chief Justice of the Louisiana Supreme Court.  This is a historic event.  It begins a new order.

My classmate is not the first female to lead this court but she will be  the court’s first African-American Chief Justice. How can I describe the significance of this uphill trek?

When we graduated high school, Louisiana was among the segregated states in the South.  Its legislation and culture prohibited African-Americans from attending LSU and Tulane, the premier educational centers of the state. Education in Louisiana was separate but  seldom equal. Although Louisiana segregation laws have long been set aside, this investiture makes her one of the sworn to guard exclusion based on racial designations in Louisiana.

The streetcar is advancing  up the hill.

I see myself as a passenger patiently waiting to board the streetcar to this historic investiture.

My great grandparents were Louisiana residents. They dreamed of this day. They died hoping for a new order, but they never saw it. They died in hope. With my weaken but proud eyes, on February 28, I board the life train to witness what they hoped for.

The embracing couple symbolizes a new responsible humanity. It is educated. It knows humanity’s past fears. Education is its only defense against the  pitfalls and lies that ruled previous generations. More importantly, the couple’s education is its only protection against threatening things yet unseen. But they embrace. Their greeting is warm and they whisper into  each other’s ear a vow to never let the sun set without throughly addressing a division between them. They know, all too well, an untreated wound in their relationship is the beginning of a slow death to society.

Their embrace shouts the arrival of the new order, a dream come true – at last!

Sacramento Ballet Company Presents The Great Gatsby

February 10, 2013, the Sacramento Ballet Company presented the Great Gatsby  at its matinee performance. The performance educated, energized and delighted the audience.

Through dance the ballet company illustrated how the convergence of drama, intrigue and changing morals in F. Scott Fitzgerald’s third novel ended in senseless tragedy.

The famed author tells a story that show how youthful love, playful infatuation and moral restlessness within a wealthy society group can end in tragedy. The time is the rollicking jazz era of sex and alcohol  in the 1920’s.

The dancing was excellent.

The ballet company showcased the attitudes  and the mannerisms of the wealthy with  smooth foxy trots, class cohesion through group closed precision high kicking, their predatory prowling while avoiding to become prey and their hostility towards the less fortunate through tango-like movements; and finally their grace through smooth and exact swing techniques.

The music was superb.

The orchestra energized the auditorium  with Ragtime and Dixieland sounds.  A bossy tuba and a snappy snare drum metered the classic jazz beat of the period. Not to be out done, a jubilant clarinet raced the scales with sounds of joy.  All within me danced. Applauds and calls flowed from attendees to appreciate the joyful swing  music.  During other tunes, the clarinet called forth feeling of pain with sustained sounds that whined and stretched into a wail.

The performance began ten minutes late. Management announced the late start accommodated the many last minute ticket purchasers. The inconvenience was worth the wait. In the language of modern competitive sports, the Sacramento Balllet Company left it all on stage. The exuberance and chatter of the departing crowd suggested a bright future for this dance company ‘s future performances.

Isaiah 28 For The Second Millennium

A true cornerstone

The Lord Almighty said:

I have placed a special one among you.

He is tested and has my authority.

For those who rely on him,

your concerns are covered.

If you follow him,

your interactions are just and

I will declare you righteous.

Those who rely on him:

Your ungodly alliances with associations and worldly things,

I void.

Your lies and falsehoods behind which you hid,

I annul.

In my special one,

there is rest, repose and peace for your soul.

His rest will completely wraps you in justice,

His righteousness and  His strength

will support you during troubled times.

Pay attention to His faithfulness, His proven faithfulness;

to do his work, his strange work,
    and perform his task, his alien task.

I  permitted his death on a cross.

I permitted the curse

to compensate for your return to the Divine.

Then I made him My throne of mercy,

through which I dispense grace.

My Fellow Travelers: In times of crisis,

The Lord Almighty wishes you to remember

the beauty and the mystery of His work –  His strange work;

and that His alien tasks create what seems alien to His nature

though, it is good,

all the time.



For an NIV translation of Isaiah 28, go to:


For a friend with an advancing degenerative condition

Coming to the end of a journey can be difficult.

During these moments, we are inclined to make decisions

that opens voids we feel needs to be filled.

Uncertainty pops up then  anxiety, and in some cases fear.

Scripture tell us that these conditions are the children of punishment,

the lack of knowledge and the  lack of  love.

But you ARE loved,

by Jesus, your family, your spiritual associates and by me.

My prayers for your heart

Remember your pools of love;

Rest in therm.

Each of us pray a quality of life in which you can prosper.

May Jesus continue to bless you in the way He deems the best.

Always for you.

A Dream Transformation Project

Five years ago my wife and I took the step to purge the  I wish we had …  and  I really wanted, but… feelings we had about certain aspects of our house.

There is no neighborhood better than ours. So ridding ourselves of these deficit feelings meant remodelling was our only option. We did. Now we gleam a sense of  This is the way it should be.

Remodelling is a daunting process. It requires:

  • Having complete knowledge of the desired end,
  • Selecting a good contractor and
  • Exercising patience to project completion.

During the remodelling process, normal living routines are disrupted; thus the fainthearted need not apply.

After much soul searching and discussions, we committed our dreams to paper. Each presented a dream transformation. We synthesized them into a mutually accepted dream plan; constrained by our modest pensions and the possibility that this was likely our last opportunity to live our dream. To make it all happen, we engaged a good contractor. This proved to be our most trying decision.

We interviewed four contractors. All were bore stellar recommendations from their professional association.

We chose the contractor who quieted our logic and emotional soul.  He was the one who never shot from the hip nor varied his story.

Our makeover commenced.

All bathrooms were updated, the house interior and exterior surfaces were painted, living room and dining room were given new wood floors, the common room got a new fireplace insert and each bedroom was given a new ceiling fan. Finally the rear landscapes were redesigned for senior care.

This post focuses only on the master bath remodel.

There we targeted three centers in the master bathroom remodel:

  • Replace the old shower with a larger shower stall,
  • Install a soaking tub to replace the  Roman tub and
  • Replace the five-foot-closet in the bath area with a second wash area.

The old shower, Roman tub and closet were demolished.

New plumbing, shower tile, floor tile, wainscot and a floor warming system were installed.

Removal of the five-foot-closet yielded space for the  second washing area and an antique vanity.

A claw foot slipper tub replaced the Roman tub.

The dream plan provided for more effective lighting and better window coverings. The remodeled bathroom is a well-lighted space that delivers an invigorating bath experience to each user.

At the completion of the project, unexpected avenues had opened to us. Having to reveal our desires for the new living space opened to us to a richer sharing. now we are even better friends.