MATH
100:12
Mathematical
Concepts
Fall/Winter
2008-2009
Key
Ideas and Sample
Exercises |
This page contains sample exercises
from the textbook and other examples along with the key ideas. These are good practice questions to help you understand
the main ideas and they are a good indication of what you might see on
quizzes, the midterm or the final. However, keep in mind that there may
be other types of questions!
Winter:
Chapter 4: Numeration Systems
- Section 4.1 The
Evolution of Number Systems. Key Ideas: different types of
number systems (simple grouping, multiplicative), historical examples
(Egyptian, Roman, Chinese), be able to
convert between different systems, Egyptian multiplication method. Questions: 5, 6, odd 9-73, 77, 79,
83, 85
- Section 4.2
Place Value Systems. Key Ideas: Place value systems (ex.
Babylonian, know symbols), expanded form of numbers. Questions: 5,
9, 13, 17, 19, 29, 31, 33
- Section 4.3
Calculating in Other Bases. Key Ideas: different
bases (converting, doing addition and subtraction). Questions: odd
9-63, 73, 75, 79, 81, 87
- Of Further Interest Section
on Modular Systems. Key ideas: Understand what a
modulo m system is, what it means for two numbers to be congruent for a
given modulus. Questions (pages 239-241): 13-47 odd, 53, 55, 57
- NOTES on mathematical
systems and the symmetries of the square. Key Ideas:
different properties of operations (closed, commutative, associative, identity
and inverse), the 8 symmetries of the square.
Chapter 5: Number Theory and
the Real Number System
- Section 5.1 Number
Theory. Key Ideas: definition of prime/composite, how many
primes are there, Mersenne primes (not in the book), Sieve of
Eratosthenes, divisibility tests, factoring, prime factors, GCD, LCM,
using GCD and LCM to solve problems. Questions: odd 9- 63, 73,
75, 77, 79
- Section 5.2 Integers.
Key Ideas: rules for adding, subtracting, multiplying and dividing
integers. Questions: odd 7-93
- Section 5.3 The
Rational Numbers. Key Ideas: determining when two
fractions are equal, arithmetic of fractions, word problems with fractions,
converting fractions to decimals and decimals to fractions. Questions: odd 9-115
- Section 5.4
The Real Number System. Key Ideas: what irrational numbers
look like (infinite non-repeating decimals), working with radical
expressions, properties of number systems (closed, commutative, associative,
identity, inverse, distributive). Questions: odd 9-89
Chapter 6: Algebraic Models
- Section 6.1 Linear
Equations. Key Ideas: know what it means for two variables to
be linearly related, two forms of linear equations (standard and
slope-intercept), what the sign and magnitude of slope tells
you, solving or rearranging linear equations, intercepts, converting word
descriptions to linear models. Questions: odd 9-61, 67-87
- Section 6.2 Modeling
with Linear Equations. Key ideas: Build a linear model
given a point and a slope or given two points, figuring out what the variables
are, making predictions. Questions: odd 7-21, 23, 25, 27, 31,
33
- Section 6.4
Exponential Equations and Growth. Key Ideas: understand
difference between linear and exponential functions, use exponential
models for growth and decay, compound interest, populations growth, using
log functions to solve equations, logistic models for populations.
Questions: odd 9-57
Chapter 7: Systems of Linear
Equations
- Section 7.1 Systems of
Linear Equations. Key Ideas: 3 cases for systems with
2 linear equations, converting word problems to linear systems and
solving, supply and demand questions. Questions: odd 9-49
Chapter 8: Geometry
- Section 8.1 Lines,
Angles and Circles. Key Ideas: Euclid's five axioms,
terminology and notation for lines and angles, types of angles (right,
straight, acute, obtuse), parallel lines, figuring out angles in a diagram,
definition of a circle, relation between angle and arc length.
Questions: odd 9-61
- Section 8.2 Polygons.
Key Ideas: Closed, simple, convex. Polygons: types of triangles,
quadrilaterals, etc. Angles in a triangle, general polygon.
Similar polygons. Questions: odd 9-47
- Section 8.3 Perimeter and
Area. Key Ideas: be able to find perimeter and area for
basic shapes (triangles, quads, circles, etc) and use this for more
complicated shapes. Two ways to find area of a triangle.
Using Pythagoras' Theorem. Questions: odd 5-77.
- Section 8.6 Geometric Symmetry and
Tessellations. Key Ideas: know the four basic rigid
motions: reflections, translations, glide reflections and rotations.
Know what a symmetry of a geometric object is. Questions: odd 4-25
Old quizzes that are good for practicing:
Fall:
Chapter 1: Problem Solving and Set
Theory
- Section 1.1 Problem solving.
Key ideas: knowing that there are
different ways to solve a problem, the 4 steps of problem solving
(understand the question, devise a plan, carry out plan, verify answer),
looking for patterns, predicting next number in a sequence (sometimes more
than one right answer- just make sure to explain your reasoning), Gauss's
method of finding sums (1 + 2 + ... + n = n(n+1)/2 etc), Pascal's triangle-
know how to fill out, know a few facts about Pascal, know how to use the
triangle to find the number of ways to choose objects (like the pizza
question). There are other problem-solving strategies in the book that
we didn't go over in class. Make sure to read these examples to see
other approaches. Questions from the book: 13, 15, 19, 21, 25, 27,
33, 35, 61, 63, 69
- Section 1.3 The
Language of Sets. Key Ideas: ways to specify a set (listing and set-builder notation), what it means for a set to be
well-defined, set notation (empty set, element of, not an element
of), the cardinal number of a set. Questions from the book: 7,
9, 11, 13, 15, 17, 23, 25, 27, 29, 31, 33, 35, 37, 43, 49, 51, 55, 59, 61, 65,
67, 73, 75, 79
- Section 1.4 Comparing sets.
Key Ideas: what does it mean for a set to be a subset of another
set, when are two sets equal, when two sets are equivalent, number
of subsets of a set. Questions: 11, 13, 15, 17, 21, 23, 25, 27,
31, 33, 35, 37, 39, 59, 61, 63
- Section 1.5 Set
Operations. Key Ideas: know how to do union, intersection,
complement and difference, use Venn diagrams. Questions: 1, 3, 7,
11, 13, 15, 17, 19, 21, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53,
55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75
- Section 1.6 Survey Problems.
Key Ideas: know how to convert a word problem to a Venn diagram and
get info from the diagram. Questions: 7, 9, 11, 13, 15, 17, 19,
21, 23, 25, 27, 29, 31, 33, 35
Chapter 10: Voting
- Section 10.1 Voting
Methods. Key Ideas: Understand the 4 voting methods (Plurality,
Borda-Count, Plurality with Elimination and Pair-wise Comparison) and how
to use them with examples, reasons for using each of the methods. Questions:
9-24, 25
- Section 10.2 Defects
in Voting Methods. Key Ideas: Understand the 4 fairness
criteria (Majority, Condorcet, Independence of Irrelevant Alternatives,
Monotonicity) , be able to give an example of a voting method that
fails each criterion. Questions: 7-18, 19, 21, 37
- Section 10.3 Weighted
Voting Systems. Key Ideas: Understand what a weighted
voting system is and how to apply it, what it means to be a critical
voter, the Banzhaf power index to measure voting power.
Questions: odd 7-21, 27, 29, 31, 33, 35, 37
Chapter 2: Logic
- Section 2.1 Inductive
and Deductive Reasoning. Key ideas: know what the difference
between inductive and deductive reasoning is, be able to give
examples using each type of reasoning. Questions: 7, 9, 11, 17,
19, 25, 31, 39, 45, 51
- Section 2.2
Statements, Connectives and Quantifiers. Key ideas: know what
a statement is, know how to tell if a statement is simple or
compound, different connectives (not, and, or, if...then, if and only
if), symbols for connectives, universal quantifier (all, every, always),
existential quantifier (some, sometimes, there is at least one), negation of
quantified statements. Questions: 1, 3, 5, 7, 11, 13, 15, 17, 21,
23, 25, 27, 29, 37, 39, 41, 43, 45, 47, 49, 63, 65, 67, 69
- Section 2.3 Truth
Tables. Key ideas: know how to do truth tables for the
five types of connective statements, know how many rows there should be
based on the number of simple statements in the compound statement, be able
to fill out table for more complicated statements, DeMorgan's laws for logic.
Questions: 1, 5, 7, 9, 11, 13, 15, 21-28, 29, 36, 37, 39, 41, 43, 49,
51, 53, 55, 57, 61, 63, 65
- Section 2.4 The
Conditional and Biconditional. Key Ideas: know what conditional
and biconditional statements look like, and their corresponding
truth tables. You also need to know what the converse, inverse
and contrapositive of a conditional statement are, and which
ones are logically equivalent. Questions: 5, all odd 9-43
- Section 2.5 Verifying Arguments.
Key Ideas: be able to write arguments symbolically, know how to
determine if an argument is valid or invalid, standard arguments (don't
need to memorize name, but know that they are valid: law of detachment, law
of contraposition, law of syllogism, disjunctive syllogism), standard invalid
arguments (fallacy of converse, fallacy of inverse). Questions: odd
11-47
Chapter 12: Counting
- Section 12.1
Introduction to Counting Methods. Key Ideas: use different
approaches like lists or tree diagrams to count.
Generalize ideas for counting. Questions: odd 5-53
- Section 12.2
Fundamental Counting Principle. Key ideas: fundamental counting
principle, dealing with conditions. Questions: odd 5-39
- Section 12.3
Combinations and permutations. Key ideas: combinations,
permutations, counting, connection with Pascal's triangle, solving counting
problems. Questions: 7-69
Chapter 13: Probability
- Section 13.1 Basics of
Probability. Key Ideas: what probability means, finding it from
experimental data or from counting theory, what the value tells
you, sample space, event. Questions: 9-33, 41-47
- Section 13.2 Complements. Key
Ideas: the complement of an event and how to find its probability.
Questions: 5, 7, 9, 11
Old quizzes:
Last updated April 1, 2009.