NOTE:
This ‘original cat book’ (below) was
filled out to become
an entire
First Course in Algebra.
If you go through that entire Algebra I course,
then you're
guaranteed to get everything from
the book below—plus lots more!
However, if you're trying to use my higher-level materials (e.g.,
Geometry or
Algebra II)
without having gone through the
Algebra I course,
then I highly recommend that you read this original cat book.
TABLE OF CONTENTS
Introduction
Brief Table of Ideas, Symbols, and Notation
English versus Mathematics
This diagram summarizes the language ideas discussed in the first section,
The Language of Mathematics.
Table of Contents
This is a printable version of the Table of Contents, with page numbers.
- THE LANGUAGE OF MATHEMATICS (pages 1–10)
expressions versus sentences; sentences have verbs and state a complete
thought; conventions; definitions; simplifying an expression; connectives
Solutions to Exercises: Chapter 1
- THE REAL NUMBERS (pages 13–22)
the real numbers and where they live; living to the right or left of
zero; size (distance from zero) versus order (left/right positioning);
opposites; integers; averaging
Solutions to Exercises: Chapter 2
- MATHEMATICIANS ARE FOND OF COLLECTIONS (pages 26–37)
introduction to sets; membership in a set; finite versus infinite sets;
notation used in connection with sets; the idea of "infinity"; interval notation
Solutions to Exercises: Chapter 3
- HOLDING THIS, HOLDING THAT (pages 40–55)
letters used to represent other objects—variables; variables "hold"
objects; variables are used to state a general principle, represent a sequence
of operations, represent an "unknown" quantity; solving a sentence
Solutions to Exercises: Chapter 4
- I LIVE TWO BLOCKS WEST OF YOU (pages 58–71)
order properties of the real numbers—position of one number relative
to another; mathematical sentences used to describe order relationships; the
phrases "greatest," "least," "at least," "at most"; describing a whole class
of sentences in one fell swoop; equations versus inequalities; solution of
a sentence in one variable; graph of a sentence; reading a sentence backwards
Solutions to Exercises: Chapter 5
- NUMBERS HAVE LOTS OF DIFFERENT NAMES! (pages 75–91)
different names can reveal various properties that a number has; theorems
are to a mathematician as tools are to a carpenter; the power of adding zero
and multiplying by one; simplifying an expression versus solving a sentence;
unit conversion
Solutions to Exercises: Chapter 6
- THESE SENTENCES CERTAINLY LOOK DIFFERENT (pages 94–109)
comparing expressions versus comparing sentences; equality of expressions
versus equivalence of sentences; connectives; truth tables; synonyms for "is
equivalent to"; the mathematical words "and" and "or"
Solutions to Exercises: Chapter 7
- TRANSFORMING TOOL #1 (the Addition Property of Equality) (pages 112–125)
the Addition Property of Equality; facts versus commands; "for all"
sentences; translating a theorem; preferred formats for solving a sentence;
why the Addition Property of Equality is true; preferred format for
checking solutions
Solutions to Exercises: Chapter 8
- TRANSFORMING TOOL #2 (the Multiplication Property of Equality) (pages 128–138)
the Multiplication Property of Equality; what goes wrong with multiplying
by zero; undoing the last operation first; why the Multiplication Property
of Equality is true
Solutions to Exercises: Chapter 9
INDEX
SAMPLE QUIZZES:
Students at Miss Hall's School were given the following quizzes, to
test their understanding of each chapter.
