Look for and make use of structure. Introduction 10 minutes I will begin with the essential question:

Vignette In the Classroom In this vignette, students use associative, commutative, and distributive properties to generate equivalent algebraic expressions for the area of a rectangle. Today's task is to generate as many expressions as you can to represent the area of this rectangle.

To help us think about this, let's start by using algebra tiles to represent this rectangle. I already drew a picture to show what that might look like.

What algebraic expression would you use to represent the area? Is there another way to write that expression that may be a little simpler? You could add all the x s together and add all the 1 s together.

So what you did was grouped the "like" terms. We often use parentheses in mathematics to show groups, so you could write your work like this: I know an easier way to do the problem. What would that be? Well, I remember that the fastest way to find the area of a rectangle is to multiply the length by the width.

I agree that the area of a rectangle can be found by multiplying the length by the width, but I think there's something wrong with your expression.

The algebra tiles show that there are 3 x s and 15 units. Somehow you lost two x s, but I don't know how. According to the order of operations, multiplication does come before addition so your observation is correct.

How did the two x s get lost? It's because you only multiplied the 5 by 3, and didn't multiply the x by 3. Oh, you're saying that 3 only got distributed over the 5 but not the x. How can we rewrite the expression to make it accurate and show that the 3 needs to get distributed, or multiplied by both 5 and the x?

Show me what you mean. That means that both the x and the 5 get multiplied by 3. I'm having a little trouble making sense of what you wrote, because you've used an x to represent the variable and an x to indicate multiplication. Is there another way to write the expression that may be a little less confusing.

I see what you mean.

Why do you have to write any symbol at all? What are you suggesting? That we write it like this: It is true that in algebra, when we write quantities right next to each other without any symbol in between, multiplication is implied.

For example, 6y means 6 times whatever value the variable y has.The Distributive Property is an algebra property which is used to multiply a single term and two or more terms inside a set of parentheses. Take a look at the problem below. Take a look at the problem below.

Applying the Distributive Property Math Journal 2, p. Students use the distributive property Simplifying and Solving Equations Objective To simplify and solve equations.

Common Core State are equivalent is to substitute a value for the variable. Write pairs of expressions. The distributive property is the ability of one operation to "distribute" over another operation contained inside a set of parenthesis. Most commonly, this refers to the property of multiplication distributing over addition or subtraction, such that \(x(a+b) = xa + xb\).

USING THE PROPERTIES TO SIMPLIFY EXPRESSIONS Using the distributive property on an expression involving the sum of like terms allows us to combine the like terms as shown in the next example.

Since dividing by 5 is equivalent to multiplying by 1 5, we have (10x) 10 x (2)x 2x. We can instead write the expression as $$2\left (3p+5 \right)+\left (-1 \right)\left (p+2 \right)$$ By using the distributive property we can rewrite the expression as.

Use the distributive property to express a sum of two whole numbers with a common factor as a multiple of a sum of two whole numbers with no common factor.

"I Can Find the Greatest Common Factor and Least Common Multiple of a Set of Numbers.".

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4 Ways to Use Distributive Property to Solve an Equation - wikiHow