Ray's Arithmetic · Algebra

Ray's New Elementary Algebra

Joseph Ray's first algebra (Primary Elements of Algebra), for students about 12 to 15: signs and terms, the four rules, theorems and factoring, fractions, equations, powers and roots, quadratics and progressions. Each guide is one 30 to 45 minute session, with the right page of the book beside it.

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How to use it

  • One session, one guide. Each guide covers a few of the book's numbered articles. Three or four sessions a week finishes a chapter in two or three weeks.
  • Read the rule aloud. Ray states every rule in words. Have your student read it, say it back in their own words, then use it.
  • Pencil first. Work every example by hand on paper. The site's calculator (/homeschool/calculator/) is for checking a finished answer, never for producing it.
  • Check with numbers. Ray checks algebra by putting numbers in for the letters. Make it a habit: if both sides give the same number, the work is probably right.
  • Answers are in the book. Ray prints the answer after almost every exercise. Cover it with a card, work the problem, then uncover and compare.
  • Go back when stuck. Most trouble later on comes from signs. If a lesson goes badly, spend a session on the rule for signs before moving on.

Unit 1 Chapter I: Fundamental rules

Definitions and signs, then addition, subtraction, multiplication and division of algebraic quantities (Arts. 1 to 79). Lessons 1–10.

  1. 1What algebra isArts. 1–17✓
  2. 2Signs, coefficients, powers and rootsArts. 18–38✓
  3. 3Terms, polynomials and numerical valuesArts. 39–52✓
  4. 4AdditionArts. 53–55✓
  5. 5SubtractionArts. 56–59✓
  6. 6What plus and minus really meanArts. 60–64✓
  7. 7Multiplying monomials and polynomialsArts. 65–71✓
  8. 8Signs in multiplicationArt. 72✓
  9. 9Dividing monomials, and a polynomial by a monomialArts. 73–78✓
  10. 10Dividing one polynomial by anotherArt. 79✓

Unit 2 Chapter II: Theorems, factoring, etc.

The algebraic theorems, factoring, greatest common divisor and least common multiple (Arts. 80 to 112). Lessons 11–16.

  1. 11The first algebraic theoremsArts. 80–86✓
  2. 12Divisors, prime factors and common monomial factorsArts. 87–93✓
  3. 13Factoring with the theorems; quadratic trinomialsArts. 94–96✓
  4. 14Greatest common divisor of monomials and numbersArts. 97–102✓
  5. 15Greatest common divisor of polynomialsArts. 103–106✓
  6. 16Least common multipleArts. 107–112✓

Unit 3 Chapter III: Algebraic fractions

What fractions are, then reducing, common denominators, and the four operations with fractions (Arts. 113 to 144). Lessons 17–23.

  1. 17What an algebraic fraction isArts. 113–127✓
  2. 18Reducing fractions to lowest termsArts. 128–129✓
  3. 19Mixed quantities and the signs of fractionsArts. 130–132✓
  4. 20Common and least common denominatorsArts. 133–136✓
  5. 21Adding and subtracting fractionsArts. 137–138✓
  6. 22Multiplying fractionsArts. 139–140✓
  7. 23Dividing fractions; complex fractions; seriesArts. 141–144✓

Unit 4 Chapter IV: Simple equations

Equations of the first degree with one, two and three unknowns, and the word problems that lead to them (Arts. 145 to 163). Lessons 24–32.

  1. 24What an equation is; the axiomsArts. 145–152✓
  2. 25Transposition and clearing of fractionsArts. 153–154✓
  3. 26Solving simple equationsArt. 155✓
  4. 27From words to equationsArt. 156 (problems 1–8)✓
  5. 28Word problems: work, mixtures and sharesArt. 156 (problems 9–40)✓
  6. 29Word problems: travel, ages and moreArt. 156 (problems 41–70)✓
  7. 30Two unknowns: three ways to eliminateArts. 157–160✓
  8. 31Problems with two unknownsArt. 161✓
  9. 32Three or more unknownsArts. 162–163✓

Unit 5 Chapter V: Supplement to simple equations

Formulas and general rules, negative answers, the Problem of the Couriers, and indeterminate and impossible problems (Arts. 164 to 177). Lessons 33–36.

  1. 33Formulas and general rulesArts. 164–168✓
  2. 34General problemsArts. 169–171✓
  3. 35Negative answers and the discussion of a problemArts. 172–173✓
  4. 36The couriers; indeterminate and impossible problemsArts. 173–177✓

Unit 6 Chapter VI: Powers, roots and radicals

Powers and the binomial theorem, square roots of numbers and expressions, and working with radicals (Arts. 178 to 205). Lessons 37–44.

  1. 37Powers of monomials, polynomials and fractionsArts. 178–182✓
  2. 38The binomial theoremArts. 183–186✓
  3. 39Square roots of numbersArts. 187–190✓
  4. 40Roots of fractions and decimals; approximate rootsArts. 191–194✓
  5. 41Square roots of monomials and polynomialsArts. 195–197✓
  6. 42Radicals and reducing themArts. 198–199✓
  7. 43Adding, subtracting, multiplying and dividing radicalsArts. 200–203✓
  8. 44Rational denominators and equations with radicalsArts. 204–205✓

Unit 7 Chapter VII: Quadratic equations

Pure quadratics, completing the square, the Hindoo method, problems, properties of roots, and pairs of equations (Arts. 206 to 219). Lessons 45–50.

  1. 45Pure quadratic equationsArts. 206–211✓
  2. 46Completing the squareArt. 212✓
  3. 47The Hindoo methodArt. 213✓
  4. 48Problems producing quadratic equationsArt. 214✓
  5. 49Properties of the rootsArts. 215–218✓
  6. 50Quadratic equations with two unknownsArt. 219✓

Unit 8 Chapter VIII: Progressions and proportion

Arithmetical and geometrical progressions, ratio, and the propositions of proportion (Arts. 220 to 255). Lessons 51–54.

  1. 51Arithmetical progressionArts. 220–225✓
  2. 52Geometrical progressionArts. 226–230✓
  3. 53Ratio and proportionArts. 231–243✓
  4. 54The propositions of proportion; final reviewArts. 244–255✓

The book itself

Ray's New Elementary Algebra

1866 · First algebra, about ages 12 to 15

Algebra from the beginning: signs and letters, the four operations, factoring, fractions, equations, powers and roots, and quadratics, with many exercises and their answers.

🧑‍🏫 Teaching guides ⬇ Download PDF (5.2 MB) Internet Archive ↗

Scanned from a Cornell University copy. The title page reads “Primary Elements of Algebra”.

The answers

Each guide covers a few of the book's numbered articles, shown beside the title; guide numbers simply count the sessions in order. The guides are written for this site; the book is in the public domain.

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