Lesson 5: Division Properties of Exponents

1. Look at the exponent laws. Which ones are similar?
2. Redraw the Concept Summary as a spider chart or concept chart in any shape that makes sense to you. Write each exponent law and at least one example of each type.
3. Expand your concept summary by including an example of each exponent law using small real numbers for the base and exponent. How does using real numbers help you to make sense of the exponent laws?

A.APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Lesson Description This lesson covers the various methods used to multiply algebraic expressions of increasing complexity. Students will master the mechanics of: Monomial by Polynomial: Using the distributive property to multiply a single term across multiple terms, applying exponent rules (specifically the Product Rule) as they go. Binomial by Binomial (FOIL): Learning the systematic First, Outer, Inner, Last method to ensure every term in the first binomial is multiplied by every term in the second. Binomial by Trinomial: Extending the distributive property to larger sets, often using the Vertical Method or a Tabular/Box Method to keep track of the six or more resulting terms without losing pieces. Special Products: Recognizing patterns that lead to shortcut results, such as: Difference of Squares: (a-b)(a+b) = a^2 - b^2 Perfect Square Trinomials: (a+b)^2 = a^2 + 2ab + b^2
Purpose
The purpose of Section 13.4 is to develop procedural fluency in algebraic expansion. Multiplication is the inverse operation of factoring (the focus of the next chapter); by learning how to "build" polynomials through multiplication, students gain a deeper understanding of how quadratic and cubic expressions are structured. This skill is vital for solving higher-degree equations and for the "Completing the Square" technique used in Intermediate Algebra (MAT 137).
Depth of Knowledge (DOK) Level
DOK Level 2
Level 2 (Skill/Concept): This section requires the coordination of multiple algebraic rules. Students must simultaneously apply the Distributive Property, the Product Rule for Exponents (x^a times x^b = x^a+b), and the rules for Combining Like Terms. Choosing between different organizational strategies (like FOIL vs. the Box Method) based on the size of the polynomials demonstrates an understanding of the relationships between the terms.

Have students write their own summary of the properties of powers, such as "to multiply expressions with exponents, you add the exponents; to divide, you subtract the exponents."

"you try" on page 310 and practice problem on page 319 #4 all