MATH 210, Study Guide for the Final Exam

Date and time: Tuesday, December 18, 11:30--1:30pm in CR 5114.

Exam coverage

1. Place Value and Models for Arithmetic
   1. Counting. Whole Numbers. Decimal, Egyptian and Roman Numerals (1.1)
   2. Place Value. Place Value Systems with Bases Other than Ten (1.2)
   3. Addition (1.3)
   4. Subtracton (1.4)
   5. Multiplication (1.5)
   6. Division (1.6)
2. Mental Math and Word Problems
   1. Mental Math (2.1)
   2. Word Problems (2.2, 2.3)
3. Algorithms
   1. Addition (2.1)
   2. Subtraction (3.2)
   3. Multiplication (3.3)
   4. Long Division by One-Digit Numbers (3.4)
   5. Estimation (3.5)
   6. Long Division: general case (3.6)
4. Prealgebra
   1. Letters, algebraic symbols, and expressions (4.1)
   2. Identities, properies, rules (4.2)
   3. Exponents (4.3)
5. Number theory
   1. On logic and proofs (5.1)
   2. Divisibility tests (5.2)
   3. Primes and the Fundamental Theorem of Arithmetic (5.3)
   4. Test of primeness. Factorials and their properties (5.4)
   5. Greatest common factor (divisor) and the least common multiple (5.5)
6. Fractions
   1. Fraction basics: fractional unit, interpretations and models (6.1)
   2. Mixed numbers and improper fractions, addition and subtraction (6.2)
   3. Multiplication of fractions (6.3)
   4. Division of fractions (6.4)
   5. Division word problems (6.5)
   6. Fractions in algebra (6.6)
7. Ratios, Percentages and proportions
   1. Ratios and proportions (7.1)
   2. Changing ratios and percentages (7.2)
   3. Solving percent problems by the unitary method (7.3)
   4. Rates, speed and dimensional analysis (7.4)
9. Decimals, Rational and Real Numbers
   1. Decimals (9.1)
   2. Rational numbers and decimals (9.2)
   3. Irrational numbers. The set of real numbers (9.3)
8. Negative Numbers and Integers (lecture notes and/or Sections 8.1, 8.2 and 8.3)
   1. Interpretations of negative numbers. The set of integers
   2. Rules of arithmetic relating to subtraction: associative and distributive properties
   3. Working with integers -- integer arithmetic. Models of integers: vectors, chips, money...
   4. Powers and exponents with negative numbers. Negative exponents

Key types of questions

• Writing out numbers in words; using different types of numerals
• Part-whole (missing addend), take-away and comparison interpretations of subtraction: come up with a word problem for each type
• Definition of division as missing factor. Answer why division by zero does not make sense.
• Partitive and measurement interpretations of division
• Use various models (bundle, chip, or measurement) to illustrate arithmetic operations
• Mental math calculations; present your your way of thinking as teacher's solution
• Apply and explain the use of arithmetic algorithms
• Apply and interpret the Quotient-Remainder Theorem
• Identify and give visual proofs of arithmetic properties (commutative, associative, distributive ... )
• Write an algebraic expression, evaluate an algebraic expression for given values of symbols
• Derive an algebraic identity (using pictures,or algebra)
• Teacher's solutions of word problem, using bar diagrams, or algebra -- you need to know how to do both!
• Test a number for divisibility by 2, 3, 4, 5, 6, 8, 9, 10, 11, 13... or for being prime
• Find a prime factorization of a whole number
• Find GCD and LCM using prime factorization (Euclid's algorithm is excluded)
• Word problems on GCD and LCM
• Relatively prime numbers; properties of GCD and LCM
• Word problems on fractions and percentages
• Come up with a word problem on division of fractions
• Calculations with fractions; double fractions
• Rectangular arrays and bar diagrams for arithmetic operations on fractions
• Word problems on ratios; solving proportions
• Word problem on rates and speed
• Conversion from common fractions to decimals and percentages and back
• Multiplication and division algorithms for decimals
• Scientific notation: the leading number has to be between 1 and 10. Thus, 1200 should be expressed as 1.2 10³, not as 12 10²
• Estimation and rounding of decimals
• Rational and irrational numbers
• Negative numbers: illustrating arithmetic with models: vector, color chips, money...
• Negative numbers: algebra and the rules for signs
• Explain why (-1)X(-1) = 1 using arithmetic rules, and similar questions

List of sample problems, with solutions is available at the Math Department's MATH 210 Page. This list is not all inclusive; make sure to check the homework assignment/quiz problems for other types of questions! Solutions to homework questions are also available, see the main page.