The life of a PROPORTION

Every year I would stand in front of my Algebra I class with a problem behind me that looked like the following:

X ÷ 3 = (X + 1) ÷ 4

I would then ask them to solve for X.  Many if not most would be staring at me like deer in the headlights.  A few of them would  start, but eventually the hands would rise to ask.
"WHERE DO I BEGIN?".  My answer was always the same.  "WHAT DOES IT LOOK LIKE?"

("If it looks like a duck, walks like a duck, and quacks like a duck, it is probably a duck")

There it was, 2 equal fractions, as big as life, and they had not a clue on how to begin.  This has always bothered me.  That something that you learn in the 3rd grade, and continue to learn every year would confound my students.  However, I have been mistaken.  This concept, that I thought went back to 3rd grade was not even being mentioned until the 7th in many schools.  When I asked one 3rd grade teacher about the concept, she said "Oh really, I had no idea".  She has been teaching for 13 years.
So, what am I talking about? Well, in the 3rd grade we start the idea of reducing fractions.

 6/8 = 3/4  Why is that true?

Most will tell you that 2 goes into 6, 3 times, and into 8, 4 times.  This is true, but it is not why.

This is only the process that gets us there, because we have removed a value of one, 2/2 = 1.  However, the only reason that we have the right to place an = sign between 6/8 and 3/4 is that 8 x 3 = 6 x 4.
The cross products are always = if it is a true PROPORTION.  This enables a student to always know if they have reduced the fraction correctly.
I have had some teachers ask me not to share this concept with the class as it would be to difficult for them to understand.  If they have just learned to multiply, what a great place to apply it.  Seams reasonable to me.  After all, in the 4th and 5th grade they are adding and subtracting fractions, and are asked to create PROPORTIONS so they can add or subtract.
             Example:             3/5 =  x/20     Now x = 12
                             +  1/4 =  y/20               y = 5
                                           solution 17/20
How do we get the values of  x and y?
To get x we divide 5 into 20 and we get 4, then we multiply 4 x 3 = 12.
To get y we divide 4 into 20 and we get 5, then we multiply 5 x 1 = 5.

This process of dividing around and then multiplying on the diagonal is of great importance and should not be forgotten as we grow in our understanding of Math.


We have just given our students a great way to solve proportions.  You are dealing with a concept that is truly big.  Its fingers extend into all parts of Math, and the earlier their awareness the better.

• CHEMISTRY
• SIMILAR GEOMETRIC FIGURES
• UNIT PRICING
• CHANGING MILES PER HOUR TO FEET PER SECOND
• PERCENT PROBLEMS
• SCALE WORD PROBLEMS

These are just a few of the places where PROPORTIONS are used .  By knowing how to think in proportions it makes these types of problems easier.  Thus the concepts I talk about above will take your students a long way.  I have many more examples to explain these uses on my page called Proportions.  Simply click on the word on my home blog page.