You
can picture the GCF as the intersection of Number1 and
Number2 (the factors that are in both) and the LCD as the
union (numbers that are in either).This week, we will briefly review some of the main points of the course.
Polynomials can be added, multiplied, subtracted and divided, just like other algebraic expressions. These manipulations can help solve equations and they can also simplify calculations.
Often, we wind up calculating a value, such as body mass index, over and over with the same basic formula. (See http://www.annecollins.com/weight-loss/body-mass-index.htm)
BMI =
(weight in Kg)
(height
in meters)²
The calculation in metric units requires one multiplication and one division. If we are using data in inches and pounds and there are only a few BMIs to generate, we first convert American to metric and then do the calculation. This becomes
BMI = (weight in pounds)* 0.45
[(height in inches)*
.0254] ²
The calculation, now up to three multiplications and one division, is not too difficult for several repetitions; however, suppose we are confronted by hundreds of recruits and the only tool is a four function calculator. Even a small reduction in the work would be welcome. Suppose we combine the conversion constants. The formula becomes:
BMI = 704.5* (weight in
pounds)
(height in inches)²
The modified expression only requires two multiplications and one division, a 25% reduction in work. Of course, with computers available the only time saved seems to be in typing the formula. In practice, every algebraic formula gets looked at many times.
You may think that you will never write computer programs, but putting a formula into a spreadsheet is the first step (actually, a spreadsheet with a formula is a computer program). If the formula is useful; you will save the spreadsheet. If the formula is too complex, you may have trouble understanding it when you need to make changes in six months. Keeping the formulas simple will be worth the effort by reducing the purchase of aspirins and antacids
In the early days of programming, most data was punched into cards and then fed into the computer. Because the cards had space for only 80 characters, the people who designed the data layout worked to conserve space. (This drive to save space was one of the reasons that years in dates were entered without the “19”).
One inventory project in the late 1950's gave the card designers severe headaches. Each item in stock needed its weight (in pounds) to be punched into the card; the problem was the range of weights. A bulldozer weighed 23,000 pounds; a 5 mg capacitor weighed 0.000011 pounds. The designer calculated that the weight would occupy five digits to the left of the decimal point and six to the right, for a total of eleven columns. There were only eight columns left on the card layout. He then came up with a brilliant idea: the operator would punch in the four most significant digits of the weight and one column for the number of spaces to move the decimal point, with a negative entry for values under 1 pound (the minus sign fit in the same column as the single digit.) The idea was particularly brilliant because the designer had never heard of scientific notation; he was amazed to learn, in 1979, that base/exponent notation was built into COBOL. Back then, accountants and engineers didn't speak to each other very often.
Some students expressed confusion between the Lowest Common Denominator (LCD) and the Greatest Common Factor (GCF).
According to “Dr. Math” the greatest common factor of two (or more) numbers is the product of all the factors the numbers have in common. There is a classic procedure, attributed to Euclid, for obtaining the GCF of two numbers:
Divide the larger number by the smaller to get the remainder. On the MS Calculator, the key for getting the remainder is "MOD".
If you have a remainder, it becomes the new "smaller number" and the old smaller number becomes the new larger number. Go back to step 1.
If the remainder is zero, then the smaller number that you divided by is the GCF and you're done.
As an example, let's find the GCF of 32 and 76.
76/32 is 2, with a remainder of 12.
32/12 is 2, with a remainder of 8.
12/8 is 1, with a remainder of 4.
8/4 is 2, with no remainder, so the GCF of 32 and 76 is 4.
"Dr. Math" suggests expressing both numbers as products of their prime factors, and looking for factors common to both:
32
= 2 x 2 x 2 x 2 x 2;
76 = 2
x 2 x 19; so the GCF is 2x2 or 4.
This method provides the same result as Euclid's method, however, if one of the numbers is 26569, obtaining the prime factors could take a long time.
The Lowest Common Multiple (LCM) of a pair of numbers is the smallest number that can be evenly divided by both of them. If the two numbers are the denominators of fractions, the LCM is also the LCD and very helpful in adding and subtracting.
The textbook suggests (Dugopolski, P. 342) completely factoring both numbers, removing the common factors, and calculating the product of the remaining factors. Unfortunately, complete factoring can become cumbersome. A potential shortcut for calculating the LCD is to first calculate the GCF via Euclid's method, then calculate
LCD = (Number1 * Number2) / GCF
You
can picture the GCF as the intersection of Number1 and
Number2 (the factors that are in both) and the LCD as the
union (numbers that are in either).
In the illustration, the GCF is the small area in green and the LCD includes all the colored factors.
One of the ways to overcome fear is to envision the worst that can go wrong and accept it. After that, anything less is a relief.
In the late 1950's, VWs were still designed to minimize cost. Instead of a motor to pump the windshield fluid, the Bug used compressed air.
The instruction manuals were just barely translated from the German, and gave the recommended pressure as 60 kilo Pascals (kPa). Naturally, my neighbor had no conversion chart from kPa to PSI; however, he was willing to calculate the value. He knew that “kilo” is the prefix for 1000 and that pascals are based on kilograms and meters, so a pascal should be 1 kilogram per square meter. His calculation was:
PSI = 60000 kg * (.0254 meters)² * 2.2 pounds
meter² inch² kg
= 60000 * (.0254)² * 2.2 pounds = 85.2 PSI
inch²
He inflated the compressed air tank to 85 PSI, added the windshield wiper fluid, and gave a test squirt. To his surprise, the spray arched over the top of the VW, wetting the car parked behind it. He checked the calculations, which were correct and then reviewed the formula. Then he looked up “pascal” in the dictionary.
The error was using the wrong definition; a pascal is actually 1 newton per square meter. The American measurement system uses pounds for force and for mass but the metric system uses two separate units. When you are measuring force, a pound is .215 newtons. The correct formula is
PSI = 60000 newton * (.0254 meters)² * .215 pounds
meter² inch² newton
= 60000 * (.0254)² * .215 pounds = 8.3 PSI
inch²
When the pressure in the air tank was lowered to 8 PSI, the windshield wiper fluid landed on the window, just as it should.
Dr. Schumann, one of my fellow instructors at U of Phoenix, signs all her correspondence with the quote from Albert Einstein, “Anyone who has never made a mistake has never tried anything new”.
Do you expect to use algebra at work or in the home in the next few months? If so, what applications?
Can you give any suggestions for improvements to this class for future sessions?
Have you ever made an error in real life mathematics?