Topic #02 - Mathematical Relationships
Topic 2 - Concepts and Skills
Vocabulary, Variables and Equations
|Topic #02 Concepts and Skills
Link to "Concepts and Skills"
|Vocabulary, Variables and Equations
equation: an expression or a proposition, often algebraic, asserting the equality of two quantities
solution: the process of determining the answer to a problem
variable: a quantity or function that may assume any given value or set of values
dependent variable: a variable in an experiment that depends on the value of another variable. As in y depends on x.
independent variable: a variable in an experiemnt that is not dependent on another variable, but whode value is often manipulted during the experiment.
controlled variable: variables that are held constant so they do not influence the ourcome of an experiment
data: values, such as observations and measurements, derived from scientific experiments
data table: a table of data organized in such a way to reveal patterns among observations and measurements
graph: a series of points, discrete or continuous, as in forming a curve or surface, where each point represents a value of a given function.
function: a mathematical relationship expressed as an equation that describes a relationship between or among variable variable values
curve: a generic term used to mean the function / equation displayed on a graph
x-axis: the horizonatal line through the origin on a cartesian coordinate system where in 2 dimensions y = 0 and in 3 dimensions y = 0 and z = 0
y-axis: the vertical line through the origin on a cartesian coordinate system where in 2 dimensions x = 0 and in 3 dimensions x = 0 and z = 0
plot: to determine and mark (points), as on plotting paper (graph paper), by means of measurements or coordinates
data point: one point representing one piece of data on a graph. (On a two dimensional graph a data point is defined in terms of an x and y coordinate pair. On three dimensional graph a data point is represented by an x, y and z value.)
slope: the rate of change in a derived quantity found by dividing the change in the y value divided by the change in the x value on a two dimensional graph
y-intercept: the y value in a function that has been plotted when the x value is zero
relationship: the mathematical connection between to variable values as defined by a function (equation)
direct variation: a realtionship that says that one variable such as the y variable varies directly with another variable such as the x variable. Mathematicians say this as "y is a function of x".
direct proportion: a means of saying that the variable values in a direct variation relationship are proportional to one another. That is as one increases so does the other.
inverse variation: a relationship that says that one variable such as the y variable varies inversely with another variable such as the x variable. Mathematicians say this as "y is a function of 1/x".
inverse proportion: a means of saying that the variable values in an inverse variation relationship are inversely proportional to one another. That is as one increases the other decreases proportionally.
linear relationship: another way of describig that a relationship between two variables is one of being a direct variation. Linear (functions) relationships always appear as straight lines when plotted on a two dimensional cartesian coordinate system.
quadratic relationship: a relationship that says that one variable such
as the y variable varies directly with the square of another variable.
Mathematicians say this as "y varying directly with the square of x is
a parabolic relationship".
18. the density of a substance is its mass per unit volume.
Problem Set #2: Follow the written directions unless indicated otherwise by your instructor.
1. You know that 2 = 8 / 4 . Using the ideas that you have learned from studying algebra, rewrite this statement (rem: an equation is called a statement) such that the rest of the statement equals the number shown.
a. 8 b. 4
2. Solve the following equations for v:
a. d = v t b. t = d / v c. a = v2 / 2 s d. v /a = b / c
3. Solve the equation d = a t2 / 2 for:
a. t2 b. a c. 2
4. solve for the variable E for:
a. f = E / s b. m = 2E / v2 c. E / c2 = m
5. Solve the equation P = F v for:
a. v b. f
6. solve the equation v2 = 2 d a for:
a. d b. a
7. solve each of these equations for x:
a. w = f x b. g = f / x c. n = x / y d. d = a x2 /2
8. Find the answers to these problems using consistent units.
a. Find the area of a rectangle 2 mm by 30 cm.
b. Find the perimeter of a rectangle 25 cm by 2.00 mm
9. Substitute the suitable units into the following equations and state which are correct outcomes. Do write a brief explanation if an outcome is wrong, why it is wrong.
a. area = length x width x height [use cm for length, width, and height]
b. time = distance / speed [use m for distance, and m / s for speed]
c distance = speed x time2. [use m / s for speed and s for time]
Graphing functions and plotting data:
Graph 1: Linear graphs. A set of x, y data pairs starts
out at 0, 0 with y increasing at
Graph 2: Hyperbolic graphs. A set of data starts out
at 1, 50 with y decreasing by
Graph 3: Parabolic graphs. A set of data starts out
at 0, 0, with y increasing as the
The following section deals with triangles and trig functions.
10. Draw a triangle with a horizontal line (base line) labeled s. Construct a vertical line, the one at a right angle to the base line upwards from the base line at the right end of the base line. Label this vertical line r. Next, connect the far left end of the base line with the top of the vertical line. This third line is called the hypotenuse of the right triangle. Label this hypotenuse t. Finally, label the angle opposite side s as S, the angle opposite side r as R, and the angle opposite side t as T. Having done this, you should answer the following questions.
a. Which side of the triangle is opposite angle R?
11.a. Find the trignometric function values for the following angles:
a. sin A, for A = 10, 30, and 45 degrees
11.b Find the size of the angles associated with each trigonometric function below. Often the Greek letter theta, q , is used to designate the unknown angle. We are using the letter A, for angle, in this situation. [On the calculator use the inverse key or 2nd function key with the trig. function indicated to find the angle.]
a sin A = 0.500 b. sin A = 0.985 c. cos A = 0.707
12. One angle of a right triangle is 20.0 degrees. The length
of the Hypotenuse is 6 cm.
13. one angle of a right triangle is 35 degrees. the length of the opposite side is 14 cm. Use the tangent of 35 degrees to calculate the length of the side adjacent to the angle.
14. If a baseball is hit at an angle of 14o, how high will it be after covering a horizontal distance of 84 m? How far will it have traveled through the air? (You may assume a straight path.)