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# 2.5: The Distributive Property

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

At the end of this lesson, students will be able to:

• Apply the distributive property.
• Identify parts of an expression.
• Solve real-world problems using the distributive property.

## Vocabulary

Terms introduced in this lesson:

distributive property

## Teaching Strategies and Tips

In the lesson introduction:

• Students see that distributing gift bags among the children is equivalent to distributing numbers.
• Care must be taken in using the abbreviations p,f,\begin{align*}p, f,\end{align*} and c\begin{align*}c\end{align*} for photo, favor, and candy, respectively. These are not variables but units of measurement. Students can misconstrue the notation and intention – these quantities are not unknown and do not need to be solved for.

Use Examples 1 and 2 to show that some expressions can be simplified in more than one way. Use the distributive property and apply the order of operations to obtain the same result.

• Use the distributive property to simplify:

4(57)\begin{align*}-4 (5 - 7)\end{align*}

Solution: 4(57)=45+(4)(7)=20+28=8\begin{align*}-4(5 - 7) = -4 \cdot 5 + (-4) \cdot (-7) = -20 + 28 = 8\end{align*}

Note: Use care with the negatives.

• Use order of operations to simplify:

4(57)\begin{align*}-4(5 - 7)\end{align*}

Solution: 4(57)=4(2)=8\begin{align*}-4(5 - 7) = -4(-2) = 8\end{align*}

In Examples 2 and 3b-3d, have students deal with the minus sign by committing to (1) using additive inverses, or (2) subtraction, but not both. For instance, Example 3b can be solved in two ways:

1. 7(3x5)=7(3x+(5))=7(3x)+7(5)=21x35\begin{align*}7(3x - 5) = 7(3x + (-5)) = 7(3x) + 7(-5) = 21x - 35\end{align*}
2. 7(3x5)=7(3x)75=21x35\begin{align*}7(3x - 5) = 7(3x) - 7 \cdot 5 = 21x - 35\end{align*}

Use Example 4 to demonstrate the hidden distributive property. Fractions with two or more terms in their numerators can be simplified using the distributive property. The hidden distributive property is based on the rule for multiplication of rational numbers:

ab=1ba\begin{align*}\frac{a} {b} = \frac{1} {b} \cdot a\end{align*}

Point out in Example 5 that the distributive property equally applies on the right. Steel required =(4+5)8\begin{align*}= (4 + 5)8\end{align*} is also a correct answer.

• Simplify using the distributive property.

(3x1)2\begin{align*}(3x - 1)2\end{align*}

Solution: (3x1)2=(3x)212=6x2\begin{align*}(3x - 1)2 = (3x)2 - 1 \cdot 2 = 6x - 2\end{align*}

In Example 6,

• Some students will want to round up rather than down. Have students complete the problem by rounding up to see that they would not have enough money.
• Point out that the distributive property equally applies to factors with three or more terms.

## Error Troubleshooting

In Example 3d, the xs\begin{align*}x'\mathrm{s}\end{align*} will eventually cancel. Students can forget to reduce in the end.

The 3 in Example 4b does not cancel. This is a common mistake made by students.

• Simplify the following expression.

ax2aax2a=x2\begin{align*}\frac{ax - 2} {a} \neq \frac{\cancel{a}x - 2} {\cancel{a}} = x - 2\end{align*}

Solution: ax2a\begin{align*}\frac{ax - 2} {a}\end{align*} can be rewritten as 1a(ax2)\begin{align*}\frac{1} {a}(ax - 2)\end{align*} Distribute the 1a\begin{align*}\frac{1} {a}\end{align*}:

1a(ax2)=axa2a=x2a\begin{align*}\frac{1} {a} (ax - 2) = \frac{ax} {a} - \frac{2} {a} = x - \frac{2} {a}\end{align*}

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