The purpose of this lesson is to extend the work with the distributive property in the previous lesson to situations where one of the quantities is represented by a variable, as in \(2(x+3) = 2x + 2\boldcdot3\). Students use the same rectangle diagrams as before to represent these situations, reinforcing the idea that the work they do with expressions is simply an extension of the work they previously did with numbers. They see that the distributive property can arise out of writing areas of rectangles in two different ways, which emphasizes the idea of equivalent expressions as being two different ways of writing the same quantity.
- Generate algebraic expressions that represent the area of a rectangle with an unknown length.
- Justify (orally and using other representations) that algebraic expressions that are related by the distributive property are equivalent.
Let's use rectangles to understand the distributive property with variables.
- I can use a diagram of a split rectangle to write different expressions with variables representing its area.
Equivalent expressions are always equal to each other. If the expressions have variables, they are equal whenever the same value is used for the variable in each expression.
For example, \(3x+4x\) is equivalent to \(5x+2x\). No matter what value we use for \(x\), these expressions are always equal. When \(x\) is 3, both expressions equal 21. When \(x\) is 10, both expressions equal 70.
A term is a part of an expression. It can be a single number, a variable, or a number and a variable that are multiplied together. For example, the expression \(5x + 18\) has two terms. The first term is \(5x\) and the second term is 18.
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