# College Readiness for Mathematic

# College Readiness for Mathematics

Mathematics instructors from the seven campuses of the
University of Maine

System have been working together to reach consensus about the skills and

competencies that entering students need to have mastered in high school in

order to place into and be successful in college level general education math
courses

in the University of Maine System. General education math courses, while
designed

for students who do not plan to major in technical or scientific fields, often
serve as

a gateway to the more advanced math courses that are required in many
professional

undergraduate programs.

Inside we list the mathematical topics and understandings
that constitute college

readiness for general education mathematics through College Algebra. Topics in
italics

that are marked with an asterisk are not required for a general education math
course,

but are required for entry into College Algebra. This document does not address
readiness

for Calculus I, or other mathematics courses specific to majors in science,
engineering and

some professional fields.

We recommend deep and broad understandings, rather than mere exposure. Further,

we want to stress the importance of both procedural and conceptual
understandings.

Students who are ready for college mathematics are able to perform mathematical

operations and manipulations by hand or with a calculator when appropriate; they

understand basic concepts and definitions; and they are able to apply, interpret
and

communicate results.

The topics and understandings for mathematic readiness are organized under five
major

headings: Mathematical Reasoning, Computation, Algebra, Geometry, Data Analysis
and Statistics.

**I. Mathematical Reasoning**

A. Successful students know important definitions, why definitions are necessary
and are able to use

mathematical reasoning to solve problems; they

• Use and understand inductive reasoning and deductive reasoning and understand
the differences

between them;

• Use geometric and visual reasoning;

• Use multiple representations (e.g., analytic, numerical and geometric ) to
solve problems. (This is

extremely important – the concept that there is not just one correct way to
solve a problem);

• Use a variety of strategies to revise solution processes;

• Understand the uses of both proof and counterexample in problem solutions,
know how to use and

generate counterexamples, are able to write simple proofs;

• Are familiar with the process of creating and understanding mathematical
models from word

problems, geometric problems and applications and are able to interpret
solutions in the context of

these source problems;

• Translate simple statements into operations; and

• Understand the role of written symbols in representing mathematical ideas and
the precise use of

special symbols of mathematics.

B. Successful students know how to estimate; they

• Are able to convert between decimal approximations and fractions;

• Know when to use an estimation or approximation in place of an exact answer;

• Recognize the accuracy of an estimation; and

• Know how to make and use estimations.

C. Successful students understand the appropriate use as well as the limitation
of calculators; they

• Recognize when the results produced are unreasonable or represent
misinformation and

• Use calculators for systematic trial-and-error problem solving.

D. Successful students are able to generalize and to go from specific to
abstract and back again. They have

a basic understanding of mathematical modeling.

E. Successful students demonstrate active participation in the process of
learning mathematics; they

• Are willing to experiment with problems that have multiple solutions;

• Demonstrate an understanding of the importance of the mathematical ideas
behind the steps of a

solution, as well as the solution;

• Show an understanding of how to modify patterns and solutions strategies to
obtain different

results; and

• Recognize when a proposed solution does not work, analyze why and use the
analysis to seek a

valid solution

F. Successful students recognize the broad range of applications of mathematical
reasoning. They have an

appreciation of the relevance of mathematical models and know that mathematical
applications are used

in other fields.

**II. Computation**

A. Successful students know basic mathematical operations; they

• Understand, perform and apply arithmetic operations with
real numbers, and percents and

proportions with and without a calculator;

• Understand percent, proportions, and ratios;

• Understand exponents expressed as integer and rational numbers ;

• Understand integer exponents expressed as integer and rational numbers;

• Interpret and write scientific notation and understand order of magnitude;

• Understand the basic structure of the real number line ;

• Understand absolute value;

• Use the correct order of arithmetic operations, particularly demonstrating
facility with the

Distributive Law ; and

• Know terminology for , and relationships among, natural, integer, rational,
irrational and real

numbers.

B. Successful students know and carefully record symbolic manipulations. They
understand the uses of

mathematical symbols as well as the limitations on their appropriate uses (e.g.
equality and inequality

signs, greater than and less than signs, parentheses, superscripts and
subscripts), write standard

notations and convert calculator notations to standard mathematical notation.

**III. Algebra
**

A. Successful students know and apply basic algebraic concepts ; they

• Understand the concept of a variable ;

• Correctly perform addition, subtraction, multiplication and division operations that include

variables, with emphasis on grouping like terms;

• Perform appropriate basic operations on sets (e.g., elements of subsets, union, intersection, and

complements);

• Use the distributive property to multiply a monomial by a polynomial and to multiply two

polynomials ;

• Understand exponents, roots and their properties in expressions involving variables;

• Simplify and perform basic operations on rational expressions, including finding common

denominators: (add, subtract, multiply, divide);

• *Understand the relationship between and properties of common and natural exponentials and

logarithms and

• * Factor polynomials : common terms, difference of two squares, and trinomials.

B. Successful students are able to work with mathematical notation to solve problems and communicate

solutions; they

• Translate simple statements into equations; and

• Understand the role of written symbols in representing mathematical ideas and the precise use of

special symbols of mathematics as used in algebra.

C. Successful students use various appropriate techniques to solve and apply basic equations and

inequalities , using algebraic, numeric and graphic methods —both with and without technology—and are

able to

• Derive, solve, and interpret first degree equations and quadratic equations in one variable;

• Derive, solve, and interpret first degree inequalities in one variable;

• Derive, solve, and interpret systems of linear equations
in two variables ; and

• *Solve quadratic equations using factoring, completing the square and the
quadratic formula.

D. Successful students distinguish between and among expressions, formulas,
equations and functions;

they

• Understand the difference between simplifying, solving, substituting or
evaluating equations and

expressions;

• Understand the concept of a function, including domain and range, and
notation; and

• Understand that functions can be expressed as verbal statements, numbers,
formulas, graphs, and

tables.

E. Successful students understand the relationship between equations and graphs;
they

• Understand basic forms of the equation of a straight line and how to graph the
line without the aid

of a calculator;

• Understand the basic shapes of the graphs of quadratic functions and

know how to find the vertex of a parabola; and

• Know and understand the basic shape of the graphs of exponential and
logarithmic functions.

**IV. Geometry**

A. Successful students understand and use both basic plane and solid geometry;
they

• Understand the concepts of area, perimeter, volume and surface area;

• Know how to calculate area, perimeter, volume and surface area;

• Use similar figures to understand the effects of scale on the figure and to
find unknown angle

measurements and lengths of sides;

• Understand transformations and transformational geometry: rotation,
translation, reflection, and

dilation; and

• Understand and apply the Pythagorean Theorem.

B. Successful students know linear analytic (i.e. coordinate) geometry; they

• Can find the equation of a line given various properties;

• Understand slope as a rate of change;

• *Understand the distance formula and how to find the midpoint of a segment;
and

• *Understand the properties of slopes of parallel and perpendicular lines.

**V. Data Analysis and Statistics**

Successful students apply concepts of statistics and data analysis in the social
and natural sciences. They

know how to represent data in a variety of ways (e.g., scatter plot, line graph
and two-way table) and

select the most appropriate. They

• Use various methods of data representation: tables, graphs, numbers;

• Interpret data from various methods of representation;

• Understand mean, median, mode as measures of central tendency and their proper
applications;

• Understand the definition of probability and why probability events are
expressed as between zero

and one inclusive; and

• Understand data dispersion.

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