Using The Distributive Property - Area Model

🔑 Key Concepts

  • The area of a rectangle is found by multiplying its length and width.
  • The two smaller rectangles share a height of \(2\).
  • The left rectangle has area \(2x\), and the right rectangle has area \(16\).
  • Adding the partial areas gives the total area \(2x+16\).
  • The full rectangle has dimensions \(2\) and \(x+8\), so its area is \(2(x+8)\).
  • The expressions \(2x+16\) and \(2(x+8)\) are equivalent.

✏️ Worked Example — Write the Total Area Expression

🧠 Math Vocabulary

  • Area: The amount of space inside a two-dimensional figure, measured in square units.
  • Area model: A rectangular visual model used to represent multiplication and the distributive property.
  • Distributive property: A property that multiplies a factor by every term in a sum or difference.
  • Equivalent expressions: Expressions that have the same value for every possible value of their variables.
  • Factor: A number, variable, or expression multiplied by another quantity.
  • Term: A number, variable, or product separated from other terms by addition or subtraction.

💡 Main Idea

An area model can represent the same quantity in more than one way. Adding the two partial areas gives \(2x+16\). Treating the figure as one large rectangle gives \(2(x+8)\). The distributive property shows that these expressions are equivalent because \(2(x+8)=2x+16\).

📚 What You Should Already Know

Students should know how to find the area of a rectangle, multiply whole numbers and variables, add algebraic expressions, and interpret a variable as an unknown quantity. Students should also understand that adjacent rectangles can be combined to form one larger rectangle.

🚀 What Comes Next

Students can extend this reasoning to expressions with more terms, negative coefficients, fractional coefficients, and multiple variables. Area models also prepare students to factor expressions, combine like terms, and solve equations involving the distributive property.

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