Download Academic Algebra I Nonlinear Expressions Unit Plan

Academic Algebra I Nonlinear Expressions Unit Plan
Introduction: Basic Concepts
Recall that an expression is called linear if all its variables have an exponent of 1. In this section we will
learn rules for simplifying non-linear expressions, that contain terms such as 𝑥 2 .
Recall that a term is any mathematical symbol or set of mathematical symbols linked by multiply and/or
divide. Terms are separated from each other by add and subtract.
For example, consider the expression 3𝑥 2 + 5𝑥 + 7
How many terms does it have? [Answer: 3]
The 3𝑥 2 counts as a single term, even though the “x” and the “2” are linked by the power operation,
because power is just repeated multiplication.
A monomial is an expression with only one term. A polynomial is an expression with more than one term.
A term can consist of any number of elements called factors.
Factor – Any mathematical symbol or set of mathematical symbols linked by power and/or root.
Factors are separated from each other by add, subtract, multiply or divide.
Two factors are considered like factors if and only if:
1.) They are both numerical factors with an exponent of 1. (That is, “regular old numbers.”) OR
2.) They both have the exact same base.
[Recall that in a power, the base is the large number, or the number before the “^”, while the exponent is
the little number on the top right, or the number after the “^”]
[Show “relation, expression, term, factor” slideshow.]
Topic I: Properties of Powers
Like factors can be combined into a single factor using the Properties of Powers rules:
1.) Product of Powers Property:
Recall that “product” means the result of a multiply operation.
[Show examples.]
Key Idea: When we combine factors with like bases by multiplying, the base stays the same and we add
their exponents.
2.) Distribution of Powers Property: Just like multiply and divide distribute over add and subtract (regular
distribution and “the butterfly,” the power operations distributes over multiply and divide.
Key Idea: The exponent on the outside multiplied by the exponent of every factor on the inside (including
those factors “without” exponents.
[Recall that a factor “without” an exponent actually has a “hidden” exponent of 1”]
HW: p. 453 #1-54; Product of Powers Topic Practice, 8.1 Practice problems
3.) Quotient of Powers Property:
Recall that “quotient” means the result of a divide operation.
[Show some examples.]
Key Idea: When we combine factors with like bases by dividing, the base stays the same and we subtract
their exponents.
HW: p. 466 all
4.) Zero and Negative Exponents
[Show example of “counting down” from 5^3 to 5^0 and beyond.]
Key Ideas:
-
Any number to the power of 1 is that number.
Any number to the power of 0 (except 0) is 1.
0^0 is undefined
Any number to the power of a negative exponent is 1/(that number to the opposite of the
original exponent). That is:
𝑥 −𝑎 =
1
𝑥𝑎
What do we mean when we say that an expression that has powers has been simplified?
- All like bases have been combined
- No negative powers remain
- All numerical powers have been evaluated
HW: p. 459 #14-45;
Quotient of Powers Topic Practice;
6.1 Practice B #1-20 (first section no calculators)
2.1 Practice B;
Summary HW: p. 469 Quiz 1 #1-23
Properties of Powers No Calculators Quiz
Topic II: GCF and LCM of Monomials
Recall: Definitions of GCF and LCM of numbers, and how to find them. (Practice: p. 778 all)
Key Idea: Just like numbers, monomials have GCF’s and LCM’s.
To find the GCF of a monomial, find the GCF of the coefficients, and then take the lowest exponent for
each variable.
[Show example.]
To check your answer, divide each monomial by the GCF. You should get coefficients which are relatively
prime (have no common factors except 1) and no negative exponents.
To find the LCM of a monomial, find the LCM of the coefficients, and then take the highest exponent for
each variable.
[Show example.]
To check your answer, divide each monomial into the LCM. You should get coefficients which are
relatively prime and no negative exponents.
HW: GCF and LCM of monomials topic practice; GCF and LCM extra practice (if necessary)
Topic III: Multiplying Polynomials
Binomial – An expression with exactly two terms.
Strategies for multiplying polynomials: Vertical method, Geometric Method, Double Distribution, FOIL
(only works for binomials.)
HW: p. 587 all; p. 593 #6-38; Multiplying Polynomials Topic Practice
Topic IV: Dividing Polynomials
If the denominator is a monomial, we can divide polynomials using the “butterfly” method and the
division property of powers:
[Show example.]
HW: p. 687 #15-22; p. 697 #1,2; p. 702 #23; p. 703 #25; p. 807 #18; Dividing Polynomials Topic Practice
Nonlinear Expressions Worksheets
Nonlinear Expressions Test