





Chapter 7. Relationships 









A system of abstract things contains things that have common parts or related dimensions. Above we have an abstract system of things including of a line between two opposite corner points (A and B) of a rectangle that lie on a circle. Even though we may describe a few initial things in a system, there usually exist many other interrelated things and subthings. In this abstract system, we can further identify two adjacent right triangles and various arc segments. We will also say "abstract system" to refer to a system of abstract things.




The quantities a, b, and c of the dimensions SideA, SideB, and SideC of the thing TriangleABC relate to each other through the Pythagorean theorem for right triangles. If these three quantities were to appear in a word problem, we can then use this relationship to relate on of these sides to the other two. For instance, we may relate the third side entered into the meaning table with the other two sides.




Other relationships come from the definition of new dimensions. In such cases, a given word (like diameter, perimeter, and circumference) determines appropriate relationships. Notice how the relationships entered into the above meaning table come from definition formulas such as:
p=2*a+2*b q= a+b+c C= π * d r= d/2 A= a*b X= a*b/2 Y=p*r^2
Each of the entries in the meaning table illustrates an additional dimension of a system consisting of a rectangle inscribed by a circle introduced by relating it to previously defined dimension. In essence, the expressions placed in the Relationship column of the meaning table come form the definition of the that dimension. When formulating word problems using Unified Math®, we will take this general approach of introducing formulas as definitions of the meaning associated with the quantities. 

















A fundamental relationship discovered by the Greek mathematician Archimedes of Syracuse (around 250 BC) describes the relationship between things on a balance placed at specific distances away from the center of the balance. The following unified quantities and their relationship characterize this system:




Many words in the English language refer to comparisons, many of which have opposites: thinnersmaller, colderhotter, heavierlighter, topbottom, etc. What does it mean to be on "top"? The answer "a greater vertical length from the earth" implies mathematical measurements and comparisons. These words translate into the abstract concepts of "greater than" (>) and "less than" (<). Their use in word problems usually invokes a measure of how much greater or how much less than, words that indicate adding and subtracting. We consider all such comparative words as mathematical vocabulary words. Such words usually lead to relationships involving either an addition or a subtraction. Consider the following phrases:
The following unified quantities and their relationship characterize these phrases:












The word "dynamic" implies that things are in motion. Sir Isaac Newton, one of the inventors of calculus, explored the relationships between the dimensions associated with things attracted to each other by gravity. Today we use his name to specific a unit of force; so 20 N (read twenty newtons) specifies an amount of force. By definition N=kg*m/s^{2}. He found that the acceleration (a) that pulls a thing toward the earth is a constant of proportionality between the Force (F) exerted on the thing and the mass (m) of the thing. So for gravity, F=m*a. This and other relationships characterize a dynamic system:






Activities:
Explorations:
Exercises:


Copyright © 2004 Dr. Ranel E. Erickson 