Assignment 8

"Mathematics is the handwriting on the human consciousness of the very Spirit of Life itself." -- (Claude Bragdon)

Math History Tidbit:
The period from roughly 300 B.C. to 300 A.D. was the golden age of geometry. Humans used the amazing Elements of Euclid to try to explain and understand the universe they knew. Eratosthenes (c. 230 B.C.) calculated the circumference of the earth with an error of less than 2%. Aristarchus (c. 280 B.C.) calculated diameters for the sun and the moon. While his calculated results were very inaccurate, his mathematical reasoning was sound. The Greeks were the first civilization in the western world who realized that the universe (as they knew it) had a mathematical design.

Herkimer's Corner

What did Herkimer say about the man who was frantically shouting about events that happened many years ago?

Answer: He said the man was historical.

Things Herky would like to know:

Why is there an expiration date on sour cream?

Why do they call it a TV set when you only get one?

Reading: Sections 2.2, 2.3, pages 75-85. (Basically, this is a review of Algebra I material. It is important.)

Written: Pages 86-87/Do at least ten problems in the set 37-57. Make sure you could do all of the problems if you had to do so.

Write neatly. Your homework papers should always represent something you would show to a college admission's representative and be willing to say "this is an example of my work." Written homework problems should be self-explanatory. That is, the reader should understand the PREMISE (information provided in the statement of the problem) and your CONCLUSION.


Items for reflection:

Mathematical word analysis:
SUBTRACTION: From the words sub (under) and traction (act or state of being drawn away). For instance
14 - 5 = 9. You "draw away" the 5 from the 14. And, the 5, with the subtraction symbol, would often be written "under" the 14.

Slope is change in y divided by change in x. If the slope of the line is 0, then there is no change in y (line is horizontal). If a line has an undefined slope, there is no change in x (line is vertical). Note very carefully that if a line has a slope of 0, it has a slope. This is different than saying the slope is undefined. A line with an undefined slope does have an equation of the form x = constant.

A line with equation y = 8 is horizontal and the slope is 0. You are simply describing all points in the plane with y-coordinate = 8.

A line with equation x = 12 is vertical and the slope is undefined. You are simply describing all points in the plane with x-coordinate = 12.

Parallel lines have equal slopes. (This should make sense.)

You should realize that a horizontal line is perpendicular to a vertical line. A horizontal line has slope = 0, and a vertical line has an undefined slope. Now, consider two lines, neither of which is horizontal or vertical. They will be perpendicular is the product of their slopes is -1. (Or, we can say if one slope is the negative reciprocal of the other.) The following statements should make sense:

  • The lines y = -3x + 876 and y = (1/3)x - 22 are perpendicular.
  • A line that is perpendicular to 2x + 4y = 55 must have a slope of 2.
  • The line through (2,9) that is perpendicular to y = -(1/5)x + 33 has equation y - 9 = 5(x-2).

Math is a language. If you will read it properly, and interpret it properly as you do your assignments, you will gain MATH POWER on a daily basis.

Problem: Find the value of K so that the line joining the points (K,6) and (9,2K) has a slope of -2.

Solution (with communication):

Premise ==> (2K - 6)/(9 -K) = 2

==> 2K - 6 = 18 - 2K

==> 4K - 6 = 18

==> 4K = 24

==> K = 6.

Check: The line joining (6,6) and (9,12) has slope (12 - 6)/(9 - 6) = 6/3 = 2.

Problem: Let L be the line joining (2,14) and (k,k). What is the value of K if L is perpendicular to the line y = -(1/3)x + 32.

Solution (with communication):

L perpendicular to y = (-1/3)x + 32 ==> slope of L is 3. Hence

(k - 14)/(k - 2) = 3

==> k - 14 = 3k - 6

==> -14 = 2k - 6

==> 2k = -8

==> k = -4.

Check: The line joining (2,14) and (-4,-4) has slope

[14 - (-4)]/[(2 - (-4)] = 18/6 = 3.