**Questions Eliciting Thinking**What does it mean for expressions to be equivalent?
Is *x* + 2 equivalent to 2*x*? If you evaluated each expression for *x* = 5, would you get the same result?
Can you explain the Distributive Property? What does the word distribute mean?
What do you think the parentheses mean? |

**Instructional Implications**Explain what it means for expressions to be equivalent (i.e., the value of each expression is the same when evaluated for the same values of the variables). Demonstrate that two expressions [e.g., 2(*a* + 3) and 2*a* + 6] are equivalent by asking the student to evaluate each expression for various values of *a*. Also, provide an example of two expressions such as 2(*a* + 3) and 2*a* + 3 and demonstrate that they are not equivalent by evaluating each expression for a particular value of *a*. Be sure the student understands that a demonstration that two expressions are equivalent for a variety of values does not constitute a proof that they are equivalent. To prove two expressions are equivalent, properties and theorems must be used.
Provide instruction on the Distributive Property and be very clear in describing what the property means. Explain that the Distributive Property of Multiplication over Addition states that *a*(*b* + *c*) = *ab* + *ac* for all values of *a*, *b*, and *c*. Initially demonstrate the property with a numerical example, such as 4(8 + 3), by rewriting it as (4 8) + (4 3). Then evaluate each expression using the order of operations rules to show that the expressions are equivalent. Next, demonstrate the property with algebraic expressions such as 5(2*x* + 9). Initially, rewrite 5(2*x* + 9) as (5 2*x*) + (5 9) and then as 10*x* + 45. Describe this use of the Distributive Property as “expanding” the expression. Be sure the student understands that the Distributive Property applies to subtractions as well since any subtraction can be rewritten as an addition. For example, an expression such as 6(4*y* – 5) = 6[4*y* + (-5)] = (6 4*y*) + [6 (-5)] = 24*y* + (-30) or 24*y* – 30.
Make clear that since the Distributive Property says that *a*(*b* + *c*) = *ab* + *ac* for all values of *a*, *b*, and *c* then it is also the case that *ab* + *ac* = *a*(*b* + *c*) for all values of* a*, *b*, and *c*. Guide the student to use the Distributive Property to rewrite expressions such as 7*m* + (7 3) as 7(*m* + 3). Describe this process as factoring since the expression is being rewritten as a product of the factors 7 and (*m* + 3). Explain the usefulness of factoring by guiding the student through problem #2. Show the student that by rewriting the expression for the area of the rectangle, 8*x* + 16, as 4(2*x* + 4), the length of the rectangle, (2*x* + 4), is revealed. Challenge the student to factor expressions such as 5*n* + 25 and 12*d* – 18 in as many ways as possible using the Distributive Property.
Provide additional opportunities to both expand and factor expressions using the Distributive Property. |

**Instructional Implications**Make clear that since the Distributive Property says that *a*(*b* + *c*) = *ab* + *ac* for all values of* a*, *b*, and *c*, then it is also the case that *ab* + *ac* = *a*(*b* + *c*) for all values of *a*,* b*, and* c*. Guide the student to use the Distributive Property to rewrite expressions such as 7*m* + (7 x 3) as 7(*m* + 3). Describe this process as factoring since the expression is being rewritten as a product of the factors 7 and (*m* + 3). Explain the usefulness of factoring by guiding the student through problem #2. Show the student that by rewriting the expression for the area of the rectangle, 8*x* + 16, as 4(2*x* + 4), the length of the rectangle, (2*x* + 4), is revealed.
Provide additional opportunities to both expand and factor expressions using the Distributive Property. |