(the original ‘cat’ book)

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(algebra book)

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The purchased pdf file does not have any red notes in the headers.

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Want a fresh start on math? View it in a totally new way? Then read this little book.

If you're using my higher-level materials (Geometry, Algebra II, Precalculus, Calculus) without having gone through my Algebra I course, then I highly recommend that you read the cat-on-swing book. It will quickly bring you up-to-speed with all the math language ideas you need to fully benefit from my lessons.

You'll learn that numbers have lots of different names, and math is the renaming tool. You'll learn that ‘$\,x\,$’ is to math as ‘cat’ is to English. It's an easy read. If you can add, subtract, multiply and divide with numbers like 1, 2, and 3, then you can read this book.

Supplement any math class, 7th grade through college. Use for self-learners, home-schoolers, math lovers, math haters. Reviewers and users write:

An entire website grew from this book. Free randomly-generated exercises. Free unlimited worksheets. Algebra Pinball. All the same style. All the same voice. All building on the ideas in this little book.

Bound, printed copies are great for students: they can highlight, write margin notes, and do exercises right in the book.

Here's a review of the cat-on-swing book.
Here are some of my testimonials from over the years.

Teachers who use the book to supplement a math class can particularly benefit:
Teachers who purchase the \$5.00 pdf file can make as many copies as needed for students. (Just be sure to preserve the copyright notices.)

If you have any questions, don't hesitate to contact me (Dr. Carol JVF Burns, author and website creator):

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Table of Contents


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.

    expressions versus sentences; sentences have verbs and state a complete thought; conventions; definitions; simplifying an expression; connectives
    Solutions to Exercises: Chapter 1
  2. THE REAL NUMBERS (pages 13–25)
    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
    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
  4. HOLDING THIS, HOLDING THAT (pages 40–57)
    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
  5. I LIVE TWO BLOCKS WEST OF YOU (pages 58–74)
    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
    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
    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
  8. TRANSFORMING TOOL #1 (the Addition Property of Equality) (pages 112–127)
    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
  9. TRANSFORMING TOOL #2 (the Multiplication Property of Equality) (pages 128–142)
    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


Students at Miss Hall's School were given the following quizzes, to test their understanding of each chapter.