Ray's New Elementary Algebra · Arts. 39–52 · Unit 1: Chapter I: Fundamental rules
3. Terms, polynomials and numerical values
Goal: Your student can name monomials, binomials and trinomials, spot similar terms, find the value of an expression for given numbers, and write a word phrase in algebra.
You'll need
- The book, open to the page
- Pencil and plenty of paper (or a whiteboard)
- A card to cover printed answers
What's in the book
Algebraic expressions, monomials and polynomials, degree, parentheses and the vinculum, similar terms and reciprocals; then two sets of examples: finding numerical values when a = 4, b = 3, c = 5, d = 10, x = 2, y = 6, and writing phrases in symbols. Answers are printed.
The lesson, step by step
Warm-up
Ask: What is the coefficient of x in 7x? What does a² mean? (Quick review of the last session.)
Names for expressions
Read Arts. 40 to 43.
A Monomial, or Term, is an algebraic expression, not united to any other by the sign + or —.
A Binomial is a polynomial composed of two terms.
Write 3ab, a + b, x² − 2x + 1. Your student names each.
Similar terms
Read Art. 50.
Similar or Like quantities are those composed of the same letters, affected with the same exponents.
Ask: Are 3a²b and 5a²b similar? 3a²b and 3ab²? (Yes; no: the exponents are on different letters.) This idea is the key to addition next session.
Numerical values, worked together
Read Art. 44 and the line before the Examples:
Let the numerical values be found, on the supposition that a=4, b=3, c=5, d=10, x=2, and y=6.
Work example 9 together:
(a+b) (a—b) Ans. 7.
Saya + b is 7, a − b is 1, and 7 times 1 is 7. Always do what is in the parentheses first.
Guided practice
Cover the answers. Your student does these two while you watch:
4a— x. . . . Ans. 14.
a(a+b). . . . Ans. 28.
Then example 4 (6a²x, Ans. 192). If it comes out 2304, the 6 was squared too: the exponent belongs only to a.
Writing phrases in symbols
Read the second set of examples. Do the first two together:
Three times a, plus 6, minus four times c.
a minus b, into three times c.
The answers are 3a + 6 − 4c and (a − b)3c. Point out that "into" means "times," so a minus b needs parentheses.
Independent practice
Your student works the rest of the numerical-value examples (1 to 13) and the twelve phrases, then uncovers the printed answers and checks. Redo any that disagree.
Wrap-up
Ask your student to explain the difference between the phrases in examples 5 and 6 (a plus b, all divided by 3c; and a, plus b divided by 3c).
Tip
A few of these examples use fraction bars that are hard to read in the scan. If one is unclear, skip it; the others teach the same thing.
Answers to the book's problems
The book prints the answer after each example. With a = 4, b = 3, c = 5: example 12, 3(a+c)(a−c) + 3a² − 3c², works out to −27 + 48 − 75 = −54, as printed.
Worked out for this site from the scan. If a number in your copy differs, trust the book. Every answer is in the Key ↗.