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.
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.
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.
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 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?
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  and 256,000,000 citizens with health care coverage . 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. 
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 .
 Plato Republic, Translated by Benjamin Jowett, Barnes and Noble Classic, 2004.
 Edward Maziarz and Thomas Greenwood, Greek Mathematical Philosophy, Mathematics People, Problem, Results Volume I, p 22-23.
 Dick Stanley, Proportionality, SUMMAC FORUM , September 1993, pages 5,6.
A Content Attainment Lesson Plan for Ratios for Ratios ©
Published by Lloyd Gavin