# Arithmetic and Algebraic Structures

**Final Exam Study Guide**

The final exam will be on Monday, December 8^{th} from 6-8pm in Lecture Hall F6.

**To prepare for the midterm:**

1. Your notebook will be an important resource during the final exam. Read
through your

notebook, especially the entries related to the content listed below. If there
are ideas or

concepts you have questions about, please be sure to seek clarification from a
classmate or one

of us before the exam.

2. Review all of the homework assignments. If you did not receive full credit for
a problem, make

sure you are now able to solve the problem correctly. You may want to read other
peoples’

assignments to get an idea of different solutions or other ways to explain your
thinking.

• Remember, there is no collaboration on the final. You are allowed to use your
notebooks, but

not the textbook. All of your work must be your own.

**Content that could be covered on the final exam:**

Number Theory:

• List all of the factors of a given number and explain how you know you have
them all

• Identify and define prime numbers, square numbers, multiples of a number,
factors of a

number, even numbers, odd numbers

Fractions and Division:

• Identify the different interpretations of fractions and division

• Determine whether a picture represents a particular fraction

• Write and evaluate word problems that represent each of the two
interpretations of division

(including division of fractions)

Divisibility Rules:

• Explain why the divisibility rules for 2, 4, and 9 works for any numbers

Definitions

• Evaluate definitions, using the criteria of precision and usability (i.e., is
the definition correct

mathematically? Does it rely on terms known to the users?)

• Determine whether specific examples are included or excluded by a particular
definition (e.g.,

given a definition of “even number,” decide whether a particular number is even)

Proofs

• Prove statements about adding, subtracting, or multiplying odd and even
numbers

• Analyze a given proof –– is it convincing? On what is it based–– that is, what
does it use as

its basic definition or assumption, from which the argument grows?

Algorithms and procedures

• Compute using alternative and standard algorithms for basic operations (+, -,
x, ÷)

• Explain why alternative and standard algorithms for basic operations work

• Explain why “invert and multiply” rule works

• Explain why the procedure for converting mixed numbers to improper fractions
works

Basic operations

• Write story problems for different interpretations of basic operations (+, -,
x, ÷) involving

both whole numbers and fractions

• Analyze common student errors in relation to basic operations

Percents

• Calculate a percentage of a given total

• Determine total amount given a certain percentage and a certain amount

Algebraic expressions, graphs, and patterns

• Determine whether two expressions are equivalent

• Evaluate given expressions, equations

• Determine pattern and find corresponding equation/formula

Mathematical practices and sensibilities

• Giving mathematical explanations: Evaluate the adequacy of an explanation, or
give an

explanation, for a solution to a problem.

• Careful attention to language: Examine a mathematical statement to determine
whether it is

ambiguous or precise