2.4: Multiplication of Rational Numbers
Learning Objectives
At the end of this lesson, students will be able to:
- Multiply by \begin{align*}-1\end{align*}.
- Multiply rational numbers.
- Identify and apply properties of multiplication.
- Solve real-world problems using multiplication.
Vocabulary
Terms introduced in this lesson:
- argument
- multiplicative properties
- multiplicative identity property
- distributive property
Teaching Strategies and Tips
Changing the sign of a number is equivalent to multiplying it by \begin{align*}-1\end{align*}. In Example 1, students find the opposite of several numbers and expressions being careful to use parentheses when appropriate and simplifying the result.
Use Examples 2 and 3 to justify why, when multiplying two fractions, the numerators multiply together and the denominators multiply together.
Use the product of three or more fractions as in Examples 4c and 4d as an extension of the multiplication rule.
Additional examples:
- Multiply the following rational numbers.
\begin{align*}\frac{3} {11} \cdot \frac{5} {7}\end{align*}
Hint: The product of two rational numbers is the product of their numerators divided by the product of their denominators.
- Multiply the following rational numbers.
\begin{align*}\frac{11} {5} \cdot \frac{7} {4} \cdot \frac{3} {10}\end{align*}
Hint: Multiply all the numerators and all the denominators. Do not covert the improper fraction to mixed form.
- Multiply the following rational numbers.
\begin{align*}\frac{5} {7} \cdot 12\end{align*}
Hint: Rewrite the \begin{align*}12\end{align*} as \begin{align*}12/1\end{align*}, using the “invisible \begin{align*}1\end{align*}”.
Students first learn about the convenience of canceling before multiplying in Examples 4d and 5.
Additional example:
- Multiply the following rational numbers.
\begin{align*}\frac{24} {33} \cdot \frac{8} {27} \cdot \frac{9} {64}\end{align*}
Solution:
\begin{align*}\frac{24} {33} \cdot \frac{8} {27} \cdot \frac{9} {64} = \frac{3 \cdot 8} {3 \cdot 11} \cdot \frac{8} {3 \cdot 9} \cdot \frac{9} {8 \cdot 8} = \frac{\cancel{3} \cdot \cancel{8}} {\cancel{3} \cdot 11} \cdot \frac{\cancel{8}} {3 \cdot \cancel{9}} \cdot \frac{\cancel{9}} {\cancel{8} \cdot \cancel{8}} = \frac{1} {33}\end{align*}
Use Examples 6-8 to introduce the four properties of real numbers which involve multiplication: the commutative, associative, multiplicative identity, and distributive properties.
- A geometric interpretation of the commutative property is to consider finding the area of a rectangle. \begin{align*}L \times W\end{align*} is the same number no matter how you draw the rectangle or what you call \begin{align*}L\end{align*} and \begin{align*}W\end{align*}; therefore, \begin{align*}L \times W = W \times L\end{align*}. Similarly, the commutative property says that the order for multiplying any two real numbers does not matter. See Example 6.
- The associative property of multiplication concerns three or more numbers. Just as for addition, the sum is the same regardless of how they are grouped and in which pair the multiplication takes place first.
- State the rule being used in each example you do in the classroom.
Error Troubleshooting
Example 1b: The opposite of \begin{align*}\pi\end{align*} is simply \begin{align*}-1 \cdot (\pi) = -\pi\end{align*}. There is no need to use the decimal expansion.
Example 1c: Multiply both terms of the expression by \begin{align*}-1\end{align*}. This will make more sense to students after covering the distributive law in the next lesson.
Additional example:
- Find the opposite of the expression
\begin{align*} x - 4y + 1\end{align*}
Hint: multiply each of the three terms by \begin{align*}-1\end{align*}.
The difference between absolute value and other grouping symbols is that multiplying absolute value by \begin{align*}-1\end{align*} will not affect the argument; that is, a negative will not distribute into the absolute value. See Example 1d.
General Tip: It is helpful to note that
- \begin{align*}|x|\end{align*} and \begin{align*}|-x|\end{align*} are always positive
- \begin{align*}-|x|\end{align*} is always negative.
General Tip: A common mistake is to forget to cancel like factors before multiplying the fractions, as the numbers will only get larger and thus harder to factor. Have students factor numerators and denominators first to remove any repetitions by canceling. Then carry out the remaining easier multiplication.