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Arithmetic and Algebraic Structures

Final Exam Study Guide

The final exam will be on Monday, December 8th 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

• 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)

• 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

• 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