Math Terms For Praxis Ii

Front 1

pictograph

Back 1

uses pictures or symbols to show data

Front 2

line plot

Back 2

uses symbols above a number line to show data

Front 3

line graph

Back 3

shows increases or decreases over time

Front 4

vertical axis

Back 4

up-and-down number line on a graph

Front 5

horizontal axis

Back 5

left-to-right number line on a graph

Front 6

stem-and-leaf plot

Back 6

stem - with 2-digit data, the part that shows tens

leaf - with 2-digit data, the part that shows ones

Front 7

pentomino

Back 7

a figure made of five congruent squares joined edge to edge

Front 8

Volume of a Prism

Back 8

V= Bh

Volume= Base x Height

Front 9

Surface Area of a Prism

Back 9

The surface area of a prism is the sum of the areas of all the sides of the prism. The formula for the surface area of a prism therefore depends on the type of prism you are dealing with. As with volume, we cover the specifics of calculating surface area as we cover each type of geometric solid.

Front 10

Surface Area of a Rectangular Solid

Back 10

Surface Area= 2lw+2lh+2wh

Front 11

Surface Area of a Cube

Back 11

Surface Area of a Cube=6s^2

s=side of cube

Front 12

Surface Area of a Cylinder

Back 12

Surface Area=2xpixr+2xpixh

Front 13

Surface Area and Volume of a Cone

Back 13

Surface Area=pixrxs+pixr^2

Volume=1/3xpixr^2xh

Front 14

Surface Area of a Sphere

Back 14

surface area = 4pir^2

Front 15

Truncation Error

Back 15

You won't get the exact right answer, or the answer will be changed by truncating a number, or chopping it off.

number being used is 3.1728934673142983

by changing it to 3.1, you have cut off the other digits and it will change your answer when put into a problem, as a opposed to the answer you will get from using all the exact digits.

Front 16

Triangle Inequality Theorem

Back 16

The triangle inequality theorem states that any side of a triangle is always shorter than the sum of the other two sides.

Front 17

Medians of a Triangle

Back 17

A median of a triangle is a line joining a vertex to the midpoint of the opposite side.
A triangle therefor has three medians.

Front 18

Altitude of a triangle

Back 18

The altitude of a triangle is a line segment from one vertex of a triangle to the opposite side so that the line segment is PERPENDICULAR to the side. Look at the pictures. In other words,

(1) it starts at a vertex and
(2) is perpendicular to the side.

Note how the altitude of an obtuse triangle can fall outside of the triangle.

Front 19

Diagonals of a Polygon

Back 19

Definition: The diagonal of a polygon is a line segment linking two non-adjacent vertices.

Front 20

Vertices of a Polygon

Back 20

The vertices of a polygon are the points where its sides intersect.

Front 21

Inscribed Angles of a Circle

Back 21

An inscribed angle is an angle with its vertex "on" the circle, formed by two intersecting chords.

Inscribed Angle = 1/2 Intercepted Arc

Front 22

Secant Line

Back 22

A line which passes through at least two points of a curve. Note: If the two points are close together, the secant line is nearly the same as a tangent line.

Front 23

Tangent Line

Back 23

A line that touches a curve at a point without crossing over. Formally, it is a line which intersects a differentiable curve at a point where the slope of the curve equals the slope of the line.

Note: A line tangent to a circle is perpendicular to the radius to the point of tangency.

Front 24

Region on a plane

Back 24

A plane is a 2-dimensional space. A plane region is, well, a region on a plane, as opposed to, for example, a region in a
3-dimensional space. We'll calculate the area A of a plane region bounded by the curve that's the graph of a function f
continuous on [a, b] where a < b, the x-axis, and the vertical lines x = a and x = b. See Figs. 1.1 and 1.2. For the sake
of simplicity we'll take the freedom to refer to such an area as “area between f and [a, b]”. The area of the region
bounded by a curve that's the graph of a function f and the x-axis, without the specification of the vertical lines or from
what x-value to what x-value, is the area of the region bounded by the curve, the x-axis, and the vertical lines at the
smallest and largest x-intercepts of the curve. For the sake of simplicity we'll take the freedom to refer to such an area as
“area between f and the x-axis”.

Front 25

Step Function

Back 25

Step Function

A function that has a graph resembling a staircase. Examples are the floor function (below) and the ceiling function.

Front 26

Absolute Value Functions

Back 26

Absolute value functions are explored, using an applet, by comparing the graphs of f(x) and h(x) = |f(x)|.



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Function f(x) used is a quadratic function of the form


f(x) = ax2 + bx + c

The exploration is carried out by changing the parameters a, b and c included in f(x) above.

Front 27

Double Bar Charts

Back 27

A double bar chart is similar to a regular bar chart, except that it provides two piece of information for each category rather than just one. Often, the charts are color-coded with a different colored bar representing each piece of information.

Front 28

Double line graphs

Back 28

Double line graphs let people compare two sets of data over time. Often they use two colors and a code to show the sets of data.

Front 29

Box Plot

Back 29

One of the more effective graphical summaries of a data set, the box plot generally shows mean, median, 25th and 75th percentiles, and outliers. A standard box plot is composed of the median, upper hinge, lower hinge, higher adjacent value, lower adjacent value, outside values, and far out values. An example is shown below. Parallel box plots are very useful for comparing distributions. See also: step, H-spread.

Front 30

Weighted Mean

Back 30

A method of computing a kind of arithmetic mean of a set of numbers in which some elements of the set carry more importance (weight) than others.



Example: Grades are often computed using a weighted average. Suppose that homework counts 10%, quizzes 20%, and tests 70%.

If Pat has a homework grade of 92, a quiz grade of 68, and a test grade of 81, then

Pat's overall grade = (0.10)(92) + (0.20)(68) + (0.70)(81)
= 79.5

Front 31

Sample Mean

Back 31

The mean of a random sample is an unbiased estimate of the mean of the population from which it was drawn. Another way to say this is to assert that regardless of the size of the population and regardless of the size of the random sample, it can be shown (through The Central Limit Theorem) that if we repeatedly took random samples of the same size from the same population, the sample means would cluster around the exact value of the population mean.

Front 32

Measures of Dispersion

Back 32

Another important characteristic of a data set is how it is distributed, or how far each element is from some measure of central tendancy (average). There are several ways to measure the variability of the data. Although the most common and most important is the standard deviation, which provides an average distance for each element from the mean,

range
spread of data
outliers

Front 33

Outlier

Back 33

A data point that is distinctly separate from the rest of the data. One definition of outlier is any data point more than 1.5 interquartile ranges (IQRs) below the first quartile or above the third quartile.

Note: The IQR definition given here is widely used but is not the last word in determining whether a given number is an outlier.

Front 34

Finite

Back 34

Describes a set which does not have an infinite number of elements. That is, a set which can have its elements counted using natural numbers.

Formally, a set is finite if its cardinality is a natural number.

Front 35

Geometric probability

Back 35

dealing with the areas of
regions instead of the "number" of outcomes.


area of favorable region
probability = ------------------------------
area of total region

Front 36

Logical connective

Back 36

In logic, two sentences (either in a formal language or a natural language) may be joined by means of a logical connective to form a compound sentence. The truth-value of the compound is uniqely determined by the truth-values of the simpler sentences. The logical connective therefore represents a function, and since the value of the compound sentence is a truth-value, it is called a truth-function and the logical connective is called a "truth-functional connective".

Front 37

Logical quantifiers

Back 37

Mathematics
We will begin by discussing quantification in informal mathematical discourse. Consider the following statement

1·2 = 1 + 1, and 2·2 = 2 + 2, and 3 · 2 = 3 + 3, ...., and n · 2 = n + n, etc.
This has the appearance of an infinite conjunction of propositions. From the point of view of formal languages this is immediately a problem, since we expect syntax rules to generate finite objects. Putting aside this objection, also note that in this example we were lucky in that there is a procedure to generate all the conjuncts. However, if we wanted to assert something about every irrational number, we would have no way enumerating all the conjuncts since irrationals cannot be enumerated. A succinct formulation which avoids these problems uses universal quantification:

For any natural number n, n·2 = n + n.
A similar analysis applies to the disjunction,

1 is prime, or 2 is prime, or 3 is prime, etc.
which can be rephrased using existential quantification:

For some natural number n, n is prime.

Front 38

Negation

Back 38

In logic, a negation of a statement is formed by placing the
word "not" into the original statement. The negation will always have the opposite truth value of the original statement.
Under negation, what was TRUE, will become FALSE -
or - what was FALSE, will become TRUE.
The negation of statement p is "not p." The negation of p is symbolized by "~p." The truth value of ~p is the opposite of the truth value of p.

Front 39

Converse

Back 39

The converse of a conditional statement is formed by interchanging the hypothesis and conclusion of the original statement.
In other words, the parts of the sentence change places.
The words "if" and "then" do not move.
HINT: Try to associate the logical CONVERSE with
Converse™ sneakers -- think of the two parts of the sentence "putting on their sneakers" and "running" to their new positions.
Conditional: "If the space shuttle was launched, then a cloud of smoke was seen."

Converse: "If a cloud of smoke was seen, then the space shuttle was launched."

Front 40

Inverse

Back 40

The inverse of a conditional statement is formed by negating the hypothesis and negating the conclusion of the original statement.
In other words, the word "not" is added to both parts of the sentence.
HINT: Remember that to create an INverse, you will need
to INsert the word NOT into both portions of the sentence.
Since you are actually negating each part of the sentence,
you may also use other words (in addition to NOT)
to create the negation.
Conditional: "If you grew up in Alaska, then you have seen snow."

Inverse: "If you did not grow up in Alaska, then you have not seen snow."

Front 41

Contrapositive

Back 41

The contrapositive of a conditional statement is formed by negating both the hypothesis and the conclusion, and then interchanging the resulting negations.
In other words, the contrapositive negates and switches the parts of the sentence. It does BOTH the jobs of the INVERSE and the CONVERSE.
HINT: Remember that the contrapositive (a big long word) is really the combining together of the strategies of two other words: converse and inverse.
Conditional:
"If 9 is an odd number, then 9 is divisible by 2."
(true) (false) This statement is logically FALSE.



Contrapositive:
"If 9 is not divisible by 2, then 9 is not an odd number."
(true) (false)

Front 42

Union of Sets

Back 42

Combining the elements of two or more sets. Union is indicated by the U (cup) symbol.

{a,b,c} U {a,c,e} = {a,b,c,e}

Front 43

Set

Back 43

A group of numbers, variables, geometric figures, or just about anything. Sets are written using set braces {}. For example, {1,2,3} is the set containing the elements 1, 2, and 3.

Note: Order does not matter in a set. The sets {a,b,c} and {c,a,b} are the same set. Repetition does not matter either, so {a,b} and {a,a,b,b,b} are the same set.

Front 44

Intersection of Sets

Back 44

The elements two or more sets have in common. Intersection is indicated by the ∩ (cap) symbol.

{a,b,c}n {a,c,e} = {a,c}

Front 45

Subset

Back 45

Set A is a subset of set B if all of the elements (if any) of set A are contained in set B. This is written A C B.

Note: The empty set is a subset of every set.

{a,b,c} C {a,b,c,d}

Front 46

Disjoint Sets

Back 46

Two or more sets which have no elements in common. For example, the sets A = {a,b,c} and B = {d,e,f} are disjoint.

Front 47

Permutation

Back 47

A selection of objects in which the order of the objects matters.
Example: The permutations of the letters in the set {a, b, c} are:

abc acb
bac bca
cab cba

Front 48

Combination

Back 48

A selection of objects from a collection. Order is irrelevant.



Example: A poker hand is a combination of 5 cards from a 52 card deck. This is a combination since the order of the 5 cards does not matter.

Front 49

Base 10

Back 49

When we write a number in base ten, for example 376; the number is expressed in terms of powers of ten like this 376 = 3*ten^2 + 7*ten + 6. The positions of the digits tells the power of ten multiplied by that digit. In other systems the base is no longer ten, but the idea is the same.

Front 50

Base 5

Back 50

In base five, for example, we have only the digits 0, 1, 2, 3, and 4. Six has to be expressed as 10, seven as 11, and eight as 12; since six = 1*five + 1, 7 = 1*five + 2, and eight = 1*five + 3.


Still in base five we have the number 1433. What is this number in base ten? Well, 1433(base five) = 1*five^3 + 4*five^2 + 3*five + 3. You can work this out as 125 + 4*25 + 3*5 + 3 = 243.

Front 51

Base 8

Back 51

Octal numbers (base eight) are written using eight distinct symbols: 0, 1, 2, 3, 4, 5, 6, and 7. Once we have counted up to 7 we run out of digits, so we must place a 1 in the "eights' column." Counting from one to twenty in base eight goes like this: 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21, 22, 23, 24. Find 71(base ten) expressed in base eight. The answer is 107, since 71 = 1*eight^2 + 0*eight + 7.

Front 52

Base 2

Back 52

Binary numbers (base two) use only 0 and 1. Counting from 1 to 10 in binary we have 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010. Here we run out of digits very soon -- after counting to 1 infact. Notice that the powers of two -- 2, 4, 8, 16, etc. -- are represented in base two by 10, 100, 1000, 10000, and so on -- just as the powers of ten are in base ten notation.

Front 53

Base 16

Back 53

Hexadecimal numbers (base 16). Hex digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. Here A = ten, B = eleven, ... , F = sixteen. So a number like A5 = A*sixteen + 5 = 10*16 + 5 = 165. And 3BF = 3*sixteen^2 + B*sixteen + F = 3*256 + 11*16 + 15 = 768 + 176 + 15 = 959(base 10).

Front 54

Triangular number

Back 54

The image shows how the triangular numbers can be formed by adding consecutive integers.
By pictuing each integer as a row of dots the image shows why this set of numbers has the name "triangular".
The first triangular number is 1.
The second triangular number is 1+2=3.
The third triangular number is 1+2+3=6.
The fourth triangular number is 1+2+3+4=10.
The nth triangular number is 1+2+3+...+n.
A number which can be represented by a triangular array of dots. 1, 3, 6, 10, are all triangular numbers.
The nth triangular number is n(n+1)/2.
Every integer is the sum of at most three triangular numbers.
If t is a triangular number, 8t+1 is a square.
The square of the nth triangular number is equal to the sum of the first n cubes.
Certain triangular numbers are also squares, but no triangular number can be a third, fourth or fifth power.
If t is a triangular number, 8t+1 is a square number.
The square of the nth triangular number is equal
to the sum of the first n cubes.
Certain triangular numbers are also squares, but no triangular number can be a third, fourth or fifth power.

Front 55

flowchart

Back 55

A flowchart is a schematic representation of an algorithm or a process.

A flowchart is one of the seven basic tools of quality control, which also includes the histogram, Pareto chart, check sheet, control chart, cause-and-effect diagram, and scatter diagram. They are commonly used in business/economic presentations to help the audience visualize the content better, or to find flaws in the process. Alternatively, one can use Nassi-Shneiderman diagrams.

A flowchart is described as "cross-functional" when the page is divided into different "lanes" describing the control of different organizational units. A symbol appearing in a particular "lane" is within the control of that organizational unit. This technique allows the analyst to locate the responsibility for performing an action or making a decision correctly, allowing the relationship between different organizational units with responsibility over a single process.

Front 56

algorithm

Back 56

A set of instructions used to solve a problem or obtain a desired result. For example, the "shampoo algorithm" explains how to wash one's hair: wet hair, lather, rinse, repeat. Gaussian elimination is an algorithm for solving linear systems of equations.
An algorithm is a step-by-step procedure designed to achieve a certain objective in a finite time,
often with several steps that repeat or “loop” as many times as necessary. The most familiar
algorithms are the elementary school procedures for adding, subtracting, multiplying, and
dividing, but there are many other algorithms in mathematics.

Front 57

Conjecture

Back 57

An educated guess.

Front 58

Subscript

Back 58

A quantity displayed below the normal line of text (and generally in a smaller point size), as the "" in , is called a subscript. Subscripts are commonly used to indicate indices ( is the entry in the th row and th column of a matrix ), partial differentiation ( is an abbreviation for ), and a host of other operations and notations in mathematics.

Front 59

Network

Back 59

A graph or directed graph together with a function which assigns a positive real number to each edge