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Section 1.6 Properties of Real Numbers and Simplifying Expressions
Introductory Algebra, Miller/O’Neill/Hyde
Andrea Hendricks
I. Overview – This section will illustrate the following:
A. Commutative Properties of Real Numbers
B. Associative Properties of Real Numbers
C. Identity and Inverse Properties of Real Numbers
D. Distributive Property of Multiplication over Addition
E. Simplifying Algebraic Expressions
F. Clearing Parentheses and Combining Like Terms
II. Commutative Properties of Real Numbers
A. Order does not affect outcome (putting on shoes)
B. Commutative Property of Addition
C. Commutative Property of Multiplication
D. Subtraction and Division – not Commutative
III. Associate Properties of Real Numbers
A. Grouping of numbers under addition and multiplication does not affect outcome.
B. Associative Property of Addition
Illustrate that (5 + 2) + 7 is the same as 5 + (2 + 7)
C. Associate Property of Multiplication
Illustrate that (5 * 2)7 = 5(2*7)
D. Emphasize that the grouping is changed but the order of the numbers is the same.
Teaching Tip: Commutative reminds me of commuting to work or school – driving
from home to school is the same as driving from school to work; only the order is
changed. Associative reminds me of a group of associates or friends. It doesn’t
matter how we are grouped since we are all friends.
IV. Identity and Inverse Properties of Real Numbers
A. Identity – number that doesn’t change the result of addition or multiplication
B. Zero - Identity Element of Addition (Additive Identity)
C. One – Identity Element of Multiplication (Multiplicative Identity)
D. Identity Properties
1) a + 0 = 0 + a = a
2) a 1  1 a  a
E. Inverse – number that takes you back to the identity element in addition and
multiplication
F. Additive Inverse (-a is the additive inverse of a)
G. Multiplicative Inverse (1/b is the multiplicative inverse of b)
H. Inverse Properties
1) a  (a)  a  a  0
1 1
2) b    b  1
b b
V. Distributive Property of Multiplication over Addition
A. Compare the two expressions 6(2 + 3) and 6(2) + 6(3)
B. Distributive Property: a(b + c) = ab + ac and (b + c)a = ab + ac
C. Illustrate the opposite of an expression is equivalent to distributing by -1
VI. Simplifying Algebraic Expressions
A. Algebraic Expression
B. Variable Terms
C. Constant Term
D. Coefficient
E. Like Terms
F. Unlike Terms
VII. Combining Like Terms
A. To add like terms, combine the coefficients of the like terms and leave the
variable factors unchanged.
B. Some things to remember
1) x = 1x
2) –x = -1x
3) Clear parentheses using the distributive property before combining
VIII. Summary
You should be able to … (repeat objectives).