animated gif by Robin Senseney, MHS Class of 2006 |

DR. CAROL JV FISHER BURNS

Ideas for anyone, grade 7 and up,

who wants to understand mathematics

You're about to learn that

The hope is that the next time you find yourself faced with

you'll think:

and laugh a bit...

making mathematics a little less stressful,

and a little more fun.

who wants me to say everything

first

copyright Dr. Carol JVF Burns

This is not a true “web book”. The book contains lots of mathematics,
and at the time I put this online, mathematical typesetting was not well supported
on the web. The original book is typeset using TeX (the technical
typesetting language used by the mathematical community), and I created
“snapshots” of each page. To view and print these “snapshots”, you'll need
the ability to work with *portable document format* (pdf) file types:
Adobe Acrobat Reader is free and available for download at

```
https://get.adobe.com/reader/otherversions/
```

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.

This ‘original cat book’ (below) was

If you go through that entire Algebra I course,

then you're

However, if you're trying to use my higher-level materials (e.g., Geometry or Algebra II)

then I highly recommend that you read this original cat book.

Table of Contents

This is a printable version of the Table of Contents, with page numbers.

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*.

- 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.**