2.1: Integers and Rational Numbers
Learning Objectives
At the end of this lesson, students will be able to:
- Graph and compare integers.
- Classify and order rational numbers.
- Find opposites of numbers.
- Find absolute values.
- Compare fractions to determine which is bigger.
Vocabulary
Terms introduced in this lesson:
- integers
- whole numbers
- greatest
- least
- even/odd numbers
- rational numbers
- ratio
- numerator
- denominator
- proper fractions
- improper fractions
- equivalent fractions
- reducing
- simplest form
- lowest common denominator
- common denominator
- opposites
- absolute value
- entire expression
Teaching Strategies and Tips
Through the use of the number line in Example 1, students learn that “greater than” and “farther to the right” are equivalent, and that “less than” and “farther to the left” are equivalent.
Use Example 2 to show that odd numbers are two apart. The frog will never land on an even number because:
- the frog begins at an odd number
- the frog only makes a jump of \begin{align*}2\end{align*}
Students get a visual reinforcement that odd numbers are always two hops apart, even though the number \begin{align*}2\end{align*} is even. This will help them understand later why two consecutive, odd numbers can be written as \begin{align*}x\end{align*} and \begin{align*}x + 2\end{align*}.
In Example 4, ask students to divide the pie diagrams into equal parts according to the denominators of each fraction and to shade according to their numerators. Common denominators have not been introduced at this point.
Use Example 5 to contrast Example 4 and motivate the process of finding common denominators. The problem of determining the larger of two, nearly equal fractions is a harder one. This task becomes straightforward once students learn to rewrite fractions as equivalent ones. Note the switch from “pies” in Example 4 to “pie rectangles” in Example 5.
Determining the larger of two rational numbers may be expressed symbolically: Given \begin{align*}\frac{a} {b}\end{align*} and \begin{align*}\frac{c} {d}\end{align*}, each is equivalent to \begin{align*}\frac{ad} {bd}\end{align*} and \begin{align*}\frac{cb} {bd}\end{align*} respectively, where the choice of common denominator will never need to be larger than \begin{align*}bd\end{align*}. Therefore, if \begin{align*}ad > cb\end{align*}, then the fraction \begin{align*}\frac{a} {b}\end{align*} is larger. If \begin{align*}ad < cb\end{align*}, then the fraction \begin{align*}\frac{c} {d}\end{align*} is larger.
Additional examples:
- Which is greater \begin{align*}\frac{9} {13}\end{align*} or \begin{align*}\frac{2} {3}\end{align*} ?
Solution: Since \begin{align*}9 \cdot 3 = 27 > 26 = 13 \cdot 2\end{align*}, then \begin{align*}\frac{9} {13} > \frac{2} {3}\end{align*}.
- Which is greater \begin{align*}\frac{31} {7}\end{align*} or \begin{align*}\frac{40} {9}\end{align*} ?
Hint: Convert to mixed-numbers and see Example 5.
- Challenge: Without a calculator, arrange from least to greatest. \begin{align*}\frac{7} {19}, \frac{5} {17}, \frac{1} {3}.\end{align*}
Hint: Compare any two fractions. Compare the larger of these two with the third.
A mirror placed at the origin, perpendicular to the number line, reflects each whole number the same distance into the mirror as the distance it measures from the mirror. This demonstration shows that every real number has an opposite, with the exception of zero. One of the more common questions teachers get from their students is whether zero is positive or negative; it is clear from the demonstration above that zero has no reflection and therefore zero cannot be either. Positives and negatives are reflections of each other. See Example 7.
Distance is briefly explained in terms of absolute value. They are covered in the chapter Graphing Linear Inequalities.
Use Example 8 to show that absolute value expressions are grouping symbols and the order of operations applies when evaluating them – students must treat them like a parenthesis in that what’s inside must be simplified first.
Error Troubleshooting
General Tip: Often, when students cancel all the factors in a numerator, they write an answer of , since nothing is left. For example:
\begin{align*}\frac{6} {30} = \frac{\cancel{2} \cdot \cancel{3}} {\cancel{2} \cdot \cancel{3} \cdot 5} \neq \frac{0} {5}\end{align*}
When canceling repeated factors from the numerator and denominator of a fraction, remind students that a \begin{align*}1\end{align*} remains.
In Example 7e, urge students to apply the \begin{align*}-1\end{align*} to the whole expression since “opposite of” means multiplying an entire expression by \begin{align*}-1\end{align*}.
Example:
- The opposite of \begin{align*}-2x+1\end{align*} is \begin{align*}2x-1\end{align*} and not \begin{align*}2x+1\end{align*}.
General Tip: Parentheses are an organizational tool. Students are encouraged to put them around an expression if it helps them prevent the above mistake.
Example:
- Find the opposite of \begin{align*}-2x+1\end{align*}.
Solution:
Start by putting parentheses around the expression: \begin{align*}-(-2x+1)\end{align*}
Use Example 8d to help students see that \begin{align*}- |-15| \neq 15\end{align*} and in general, \begin{align*}-|x| \neq |x|\end{align*}. Overly confident students reciting that the absolute value is always positive can incorrectly simplify \begin{align*}-|x|\end{align*} as \begin{align*}|x|\end{align*} instead of \begin{align*}|-x|\end{align*} as \begin{align*}|x|\end{align*}.
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