Topic #2: Mathematical Relationships
1. Solving Equations Using Algebra
2. Units and Equations
3. Tables and Graphs
4. Linear Relationships
5. Quadratic Relationships
6. Inverse Variation
7. Geometry
8. Trigonometry*
Notes should include:
Vocabulary: Any vocabulary used by the instructor to explain the topic or illustrate
concepts and solutions to problems should always be written out and defined in your
notes. Also, any vocabulary you encounter in assigned readings should also be included
in your vocabulary list. Of course anything that belongs in your notes also belongs in
your "head". That means that you should get right down to business and learn the
vocabulary so you can discuss the information intelligently.
Solving Equations Using Algebra: Single variable, Algebra I equations are the primary
form of equations that you will use to solve for unknown quantities. You need to be able
to solve such equations using variable symbols and substitute measurement values into
them for the purpose of deriving correct answers to problems assigned.
Units and Equations: Units of measurement are extremely important when solving
problems. Not only are units necessary to the reporting of any measurement, they are
necessary to any calculation to prove that the answer you arrived at is correctly derived
from the measurements that you chose to begin with.
Tables and Graphs: To be able to draw correct conclusions from data gathered by you
or someone else requires that this data be organized into tables, charts, and graphs.
Graphs are very important because they both allow you to draw conclusions about
relationships between the variables and to derive new measurements. Slopes of graphs
and areas under graphs represent powerful tools for deriving new and often important
measurements not obtained by the direct use of measuring instruments.
Variables: Three very important terms associated with the discussion of and the
representation of measurements defined as variables are the independent variable, the
dependent variable, and the controlled variable. During an experiment, the independent
variable is the measurement that is manipulated or changed to observe what will happen
to another measurement value called the dependent variable. In Algebra I, you often used
the expression "y is a function of x". That is exactly what we are describing here. The
dependent variable's value “y” depends on the value of the independent variable “x” as
determined by the experimenter. Finally, the controlled variable or variables are all the
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variable measurement(s) that might influence the outcome of the experiment, if
they are not held constant (controlled). For example, in an experiment where a student is
testing the effect of plant fertilizer (the independent variable, x) on plant growth (the
dependent variable, y), variables like the amount of sunlight, the amount of water, and the
temperature of the environment need to be controlled. Explain why this is true in your
notes.
Linear Relationships: In Algebra I, you would have worked with linear equations such
as the one frequently represented by the statement y = m x + b. This equation and others
like it, represent a relationship between the x and the y variable called a direct variation.
This means that in such situations involving the two variables the y-variable value is
proportional to the x-variable value. That is, as the x value increases (or decreases) the y
value increases (or decreases) proportionally. If you had chemistry, you might recall that
Charles' Law is an example of this type of relationship.
Inverse Variation: In Algebra I, you probably worked with this type of relationship.
The equation for this situation is x y = k, where k is a numerical constant based on the
multiplication product of x and y. This represents a relationship between x and y called
an inverse variation. This means that in such situations involving the two variables the yvariable value is inversely proportional to the x-variable value. That is, as the x value
increases (or decreases) the y value decreases (or increases) as an inverse proportionality.
If you had chemistry, you might recall that Boyle's Law is an example of this type of
relationship.
Quadratic Relationships: This is not necessarily a relationship that you encountered in
Algebra I. A graph of this kind of function has the shape of a parabola, thus its name. The
equation for this relationship is y = ax2 + bx + c. The parabolic relationship exist when
one variable depends on the square of the other. Most often we’ll see the equation in the
form of y = kx2, where k is a numerical constant based on the y to x2 ratio. This
relationship is often expressed by saying that the y-variable varies directly with the
square of the x-variable.
Geometry: The universe is made up of dimensions. The concept of dimensionality is one
every physics student must comprehend. One dimension (x) is described as a straight
line. A straight line has length. Two dimensions (x, y) is described as the surface area of
an object. Surface area has two length measurements. One is usually called length while
the other for convenience is called width instead of length 2. Three dimensions (x, y, z) is
described as the volume of an object. Volume involves three lengths. In a simple cube or
rectangular solid shape we often call the three lengths by the names length, for length 1,
width, for length 2, and height, thickness, or depth for length 3. Mentally visualizing
dimensions is an important both in physics and in engineering.
Trigonometry: It is often necessary when solving certain kinds of problems that involve
what is called vectors that certain trigonometric functions be used to find out information
based on triangles. The Trigonometry functions of Sine, Cosine, and Tangent as well as
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geometry theorem called the Pythagorean theorem are important tools when working
with right angle triangles. For non-right angle triangles the Trig functions called “Law of
Sines” and “Law of Cosines” are often used as well as “Adding Component Vectors”.
(*Trig functions are studied later in the course when the topic of vectors is introduced.)
Vocabulary: Equation, solution, variable, dependent variable, independent variable,
controlled variable, data table, graph, curve, function, x-axis, y-axis, plot, data point,
slope, y-intercept, relationship, direct variation, inverse variation, direct proportion,
inverse proportion, linear relationship, inverse relationship, quadratic relationship
Skills to be learned:
1. Solving Equations Using Algebra
2. Using Units in Equations (Factor Labeling, also called Dimensional Analysis)
3. Plotting Graphs
4. Working with Linear Relationships: Calculating slopes and Areas under graphs.
5. Working with Inverse Variation Relationships
6. Working with Parabolic Relationships
7. Working with 1, 2 and 3 dimensions (Geometry)
8. Applied Data Analysis – Assigned Lab Experiment(s)
9. Writing Lab Reports – Hands on experience based on assigned lab activities
Assignments:
Textbook: Read / Study / Learn the data analysis done in Section 2.3
Workbook Exercise(s): PS#1-1a (no. 10 is a quadratic equation), PS#1-1b,
PS#1-13, PS#1-13b, PS#1-14, PS#1-15, PS#1-16, PS#1-17
PS#2-0, PS#2-1, PS#2-2, PS#2-3, PS#2-4; PS#1-21,
Activities: TBA
Resources:
This Handout and the Overhead and Board Notes discussed in class
Textbook: Chapter 2, Section 2.3
Workbook Lessons & Problem Sets:
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