# Mathematics Transition Guide

**B Essential Skills & Competencies For**

General Entry Points of Regent University Mathematics

General Entry Points of Regent University Mathematics

**1. Skills and competencies for Entering Pre-Calculus
(Algebra/Trigonometry)***

**A****) Arithmetic**

Fluency in real number arithmetic-PEMDAS, fraction arithmetic,

rational roots of whole numbers, rules of exponents,

simplification of rational number expressions, including

those with fractional exponents in numerators or denominators.

Basic understanding of complex numbers.

**B) Algebra**

Ability to translate basic word problems involving one or

several unknowns into problems of finding solutions to one

or several equations (or inequalities) each with one or several

variables representing those unknowns. Ability to recognize

reasonableness or un-reasonableness of candidate

solutions. Types of problems may involve proportionality-similarity,

area-length measurements, volumes-surface area

measurements, etc.

**C) Algebra**

Facility with expressions involving variables such as

polynomials, including the ability to understand meaning

of roots and the ability to determine if a given number is a

root of a given polynomial; factorization, ability to describe

and compute all attributes of linear expressions in one variable

(m*x+b), factor quadratics using the coefficients of the

quadratic and the quadratic formula, recognition of complex

roots as roots of quadratics; ability to find factorizations of

higher-order polynomials with additional information (e.g.

factoring out a common power of the variable, or given a root

and polynomial division). Facility with rational expressions

involving one or several variables (e.g. simplifying a rational

expression including one or several variables, including those

with rational exponents in numerators and denominators).

**D) Algebra**

Facility with concepts related to functions-single-valued-ness,

algebraic formulae such as f(x) = 2x^{3} + 7, tabular and

x-y coordinate plane representations of functions and their

graphs, the vertical line test, and meanings of domains and

ranges. The basic function families: polynomial, exponential

and logarithmic, rational and periodic (cf. trigonometry).

Understanding the relationship between a function f(x) and

functions such as f(x)+k, f(x+h), a* f(x), f(a*x), etc. In general,

the elementary analysis of the numerical, functional and

graphical attributes of functions (polynomial, exponential,

logarithmic) is a central topic.

E) Trigonometry

Basic notions of triangle components-angles, side

lengths, right triangle attributes. Similarity and congruence

of triangles. Definitions of the six elementary trigonometric

functions: sine, cosine, etc. via triangle ratios (SOHCAHTOA) and

unit circle coordinate definitions. Basic trig identities such

as the Pythagorean, addition, and subtraction identities,

and the ability to retrieve such information quickly. Use of

such trigonometry in basic geometric and/or story problems

involving proportionality, triangles, etc. Such problems may

include but are not limited to computing measurements of

inaccessible objects such as tall trees and buildings using

measurable objects.

*Not an entry point math course option at ISU

**2. Skills and competencies for Entering Calculus i**

**A****) Algebraic Functions**

Experiences with functions, including analysis of families

of functions (linear, quadratic, general polynomial, exponential,

trigonometric, rational, logarithmic, and piecewise).

Analysis of these functions should include: zeros, maxima

and minima, domain and range, global and local behavior,

intercepts, and asymptotes. Ability to recognize, represent,

transform, compose, and find inverses of functions, and to

represent functions in multiple ways: via algebraic formulae,

graphs, data tables, and descriptions such as “the volume of

a cylinder of fixed height is proportional to the square of its

radius”, and ability to go back and forth between these representations.

Understand and analyze relations that are not

functions such as x^{2} + y^{2} = 5 and x = y^{2}.

**B) Algebraic Equations and Inequalities**

Solve inequalities and equations using algebraic and

graphic methods, including equations involving exponents

and logarithms, and equations of quadratic form such as

e^{2x} - 3e^{x} + 2 = 0. Solve systems of equations, including

inconsistent and dependent systems. Be able to represent

systems of two equations and two unknowns geometrically.

**C) Algebraic Expressions**

Ability to meaningfully manipulate algebraic expressions

to get equivalent forms by simplifying, factoring, expanding,

composing, decomposing, using order of operations, using

laws of exponents and logarithms and applying properties of

real numbers. Understand the difference between equations

and expressions.

**D) Rate of Change**

Familiarity with such examples as the speed of a car, the

number of people per year by which a population increases,

and slope of a line. Ability to analyze average and instantaneous

rate of change in multiple ways including: numeric, algebraic,

and graphic representations. Students should distinguish

between a constant rate of change and a non-constant rate

of change. They should be thoroughly comfortable with using

appropriate units for both functions and their rate of change.

**E) Geometry and Trigonometry–Coordinates**

Use (x,y) coordinates to describe points, lines, circles and

rectangles. Use formulas for distance, midpoint and slope,

and the Pythagorean Theorem, and have some degree of

understanding where they come from. Find and analyze

equations that represent lines, circles and parabolas. Familiarity

with the other conic sections–ellipses and hyperbolas.

**F) Geometry**

Find perimeter, area, volume and surface area of various

objects. Estimate these quantities as a check on accuracy. Recognize

and use appropriate units, and use dimensional analysis.

**G) Trigonometry**

Study right-triangle trigonometry and unit circle trigonometry.

Understand the trigonometric ratios, the Pythagorean

Theorem and its converse, and the special-case triangles:

30-60-90 and 45-45-90. Understand the trigonometric

functions including their domains, ranges and periodicity.

Understand the effect of various transformations on their

amplitudes and periodicity. Know basic identities such as

the Pythagorean, addition, and subtraction identities, and be

able to use the Law of Sines and Law of Cosines. Use trigonometry

to solve applied problems. Be able to solve simple

trigonometric equations with multiple solutions, such as

and sin(x) = 0.6.

**H) Theorems and Proofs**

Ability to read and understand a simple proof or mathematical

argument, at the level given in a collegiate textbook.

Understand that a theorem does not imply its converse –

e.g. “Differentiability implies continuity” does not mean that

continuity implies differentiability. Ability to formulate and

test conjectures, and understand the difference between

noticing a pattern and proving that the pattern always holds.

**3. Skills and competencies for Entering Calculus II**

In addition to the material below, it is assumed that an
incoming Calculus II student has mastered

all the material recommended for success in Calculus I, as listed above.

**A) Limits and Continuity**

Understand and be able to compute limits in graphic,

numeric and algebraic contexts. Understand continuity in

these contexts as well. Be familiar with the Intermediate

Value Theorem.

**B) Derivatives**

**Definitions and Theorems:** Understand the limit definition

of the derivative, the connection between differentiability

and continuity, the Mean Value Theorem, and Rolle’s

Theorem. Understand the derivative in terms of slope of a

tangent line, and an instantaneous rate of change

**Computation: **Be able to write the formula for f’ quickly

and accurately given a formula for a function f, either defined

by a formula or defined implicitly (this is the equivalent to

knowing one’s multiplication tables).

**Geometry:** Given a graph of a function f, be able to sketch

a graph of f’ that, while not necessarily perfect, should be

positive when f is increasing, negative when f is decreasing,

and flat when f has a local extremum. The intervals where f’

is increasing and decreasing should correspond to where f is

concave up or concave down.

**Applications:** Given a description of a function f, such

as “The height H of a column of mercury in a thermometer

is given by H=f(t), where t is the temperature in degrees

Fahrenheit,” be able to describe the meaning of f’, including

understanding its units. Ability to set up and solve global

optimization problems. Find exact values of local maxima,

local minima, and points of inflection of a function. Other

applications, such as Newton’s Method and L’Hopitals Rule

are desirable as well.

**C) Integrals**

Most Calculus II courses start with integrals, but many of

them assume a familiarity with antiderivatives from Calculus I.

Ideally, know how to find antiderivatives of basic functions,

and be familiar (if not fluent) with u-substitution. Understand

the meaning of the constant of integration. Understand the

definite integral as the limit of Riemann sums. Be able to

approximate the definite integral of a function over a given

interval with a calculator. Understand and be able to apply

the Fundamental Theorem of Calculus.

**4. General skills & competencies for Entering
Statistics**

**A) Algebra**

Use and apply algebraic concepts that include ratio, proportion,

percentage, slope and intercept. Be able to transfer

algebra competencies to statistics applications.

**B) Calculators and software**

Have experience using calculators to solve algebraic equations.

Be familiar with (or ability to independently learn) software

such as jmp.

**C) Modeling**

Be able to do least squares regression and inference.

Engineering statisticsShould be preceded by an introduction to calculus. Knowledge of functional notation, summation notation, facility with linear functions, and the ability to solve algebraic equations (e.g., quadratic equations) as part of a good working knowledge of high school algebra. Be familiar with exponential and logarithmic functions. Know basic techniques of integration and differentiation. Have exposure to exponential, Poisson, and normal distributions. |

**5. Skills & competencies for Entering Liberal Arts Math**

Liberal Arts Mathematics covers a range of liberal arts
mathematics courses. At the Regent universities, these courses include:

Iowa State University: Math 105 Introduction to Mathematical Ideas

University of Northern Iowa: 800:023 Math for Decision Making

**A) Arithmetic**

Fluency in real number arithmetic and fraction arithmetic,

rules of exponents, square roots. Knowing when it is appropriate

to multiply, divide, add, subtract or exponentiate.

**B) Modeling**

Ability to translate a verbal situation into a mathematical

problem, and to check the solution for reasonableness.

**C) Algebra**

Ability to solve linear equations and inequalities in one variable,

facility with linear equations and inequalities (slope of

a line, two-point form, point-slope form, slope-intercept form,

etc.), finding the intersection of two lines/the solution of two

linear equations in two unknowns.

D) Symbolic Manipulation

Facility with effectively working with symbolism. Be able

to distinguish between expressions like (P+Q’) and (P+Q)’ in

both reading and their writing.

**E****) Representation**

The ability to use graphs, Venn-diagrams, pie charts, data-tables,

and other pictures that are used to represent mathematical

situations.

**F) Generalization**

The ability to generalize from examples, and then test

the generalization by applying it to additional situations.

Mathematics involves finding answers to specific questions

according to prescribed rules, but it also involves generalizing

from specific problems to general principles.

G) Independence

The ability to go beyond being told what to do at every step

and take the responsibility to understand the context of the

topics being discussed and to independently proceed using

appropriate methods to solve problems.

**6. Skills & competencies for Entering Elementary
Education Math**

**A****) Overall**

Positive disposition toward the study and learning of

mathematics. Belief that all children can learn mathematics.

Experience in the use of technology for learning

mathematics.

B) Number and Operations

Work flexibly with and understand the meaning and

effects of arithmetic operations with rational numbers,

their position in the real number system, and can use that

understanding to solve problems. Use properties of addition

and multiplication to simplify computations.

**C) Algebra**

Able to represent, analyze and generalize a variety of

patterns with tables, graphs, words and symbolic rules.

Possess a conceptual understanding of different uses of

variables, graphs and the nature of changes in quantities.

D) Geometry

Able to describe, classify and understand relationships

among two- and three-dimensional objects using their

defining properties. Can create and critique inductive and

deductive arguments concerning geometric ideas and relationships,

such as congruence, similarity and the Pythagorean

relationship.

**E) Measurement**

Understand relationships among the angles, side

lengths, perimeters, areas and volumes of similar objects.

Can select and use units of appropriate size and type

when working with such measures. Select and apply techniques

and tools for measuring.

**F) Data Analysis and Probability**

Can interpret measures of center and spread. Understand

the appropriate use and how to analyze histograms,

box plots, and scatter plots. Able to use observations

about differences between two or more samples to make

conjectures about the populations from which the samples

were taken. Command a basic understanding of probability

to make and test conjectures about the results of experiments

and the concept of randomness.