2.2: Addition of Rational Numbers
Learning Objectives
At the end of this lesson, students will be able to:
- Add using a number line.
- Add rational numbers.
- Identify and apply properties of addition.
- Solve real-world problems using addition of fractions.
Vocabulary
Terms introduced in this lesson:
- adding a negative number
- LCD/LCM
- least common multiple
- equivalent fractions
- commutative property
- associative property
- additive identity
- additive properties
Teaching Strategies and Tips
Use Examples 1-3 to provide a visual frame of reference for adding real numbers. Eventually, students will need to add numbers without using the number line.
General Tip: Move to the left on the number line when faced with a negative number or subtraction.
Additional example:
- Represent the sum \begin{align*}-7-5\end{align*} on the number line.
Solution:
Starting at \begin{align*}-7\end{align*}, move \begin{align*}5\;\mathrm{units}\end{align*} to the left: \begin{align*}-7-5=-12\end{align*}
Use Examples 4 and 5 to introduce LCD (lowest common denominator) and LCM (lowest common multiple). Although they have different meanings, the difference is subtle.
Additional examples:
- Find the LCM of \begin{align*}24\end{align*} and \begin{align*}30\end{align*}.
Solution:
The lowest number that both \begin{align*}24\end{align*} and \begin{align*}30\end{align*} divide into without remainder is \begin{align*}120\end{align*}.
- Simplify \begin{align*}\frac{5} {24} + \frac{1} {30}\end{align*}.
Solution:
In order to combine these fractions we need to rewrite them over a common denominator. We are looking for the lowest common denominator (LCD). We need to first identify the lowest common multiple (LCM) of \begin{align*}24\end{align*} and \begin{align*}30\end{align*}.
\begin{align*}\frac{5} {24} + \frac{1} {30} = \frac{25} {120} + \frac{4} {120} = \frac{29} {120}\end{align*}
Teachers are encouraged to reinforce the notions of LCD and LCM using visual representations as illustrated in Example 4.
To help students learn the difference between the associative and commutative properties state the rule that is used when doing each example in the classroom and then have them note the differences.
Error Troubleshooting
General Tip: Students who have difficulty with LCD and LCM at this point, might benefit from factor trees, which are discussed in the next lesson.
For instance, in Example 6, students are asked to combine two denominators having a common factor. The factor trees for \begin{align*}12\end{align*} and \begin{align*}9\end{align*} are:
- \begin{align*}12 = 2 \times 2 \times 3\end{align*}
- \begin{align*}9 = 3 \times 3\end{align*}
To find the LCM, start by taking all of the prime numbers \begin{align*}\left \{2, 2, 3 \right \}\end{align*} of \begin{align*}12\end{align*}. Then ask what prime numbers of \begin{align*}9\end{align*} are missing from this list? The second \begin{align*}3\end{align*} is missing. Therefore, the LCM of \begin{align*}12\end{align*} and \begin{align*}9\end{align*} is the product of the numbers in the list \begin{align*}\left \{2, 2, 3, 3\right \}\end{align*}
\begin{align*}2 \times 2 \times 3 \times 3 = 36\end{align*}
General Tip: Students need constant reminding that adding fractions cannot take place without common denominators. Although memorizing rules such as,
- \begin{align*}\frac{a} {c} + \frac{b} {c} = \frac{a + b} {c}\end{align*}
- \begin{align*}\frac{a} {b} + \frac{c} {d} = \frac{ad + bc} {bd}\end{align*}
may prove helpful, teachers are encouraged to explain why denominators must be the same. Use an approach similar to the following:
- Measurements involving feet and inches cannot be added until a common unit is chosen. For example, the sum of \begin{align*}7\;\mathrm{inches}\end{align*} and \begin{align*}3\;\mathrm{feet}\end{align*} is not equal to \begin{align*}10\;\mathrm{inches}\end{align*}, \begin{align*}10\;\mathrm{feet}\end{align*}, or \begin{align*}10\;\mathrm{feet-inches}\end{align*}. Converting \begin{align*}3\;\mathrm{feet}\end{align*} to \begin{align*}36\;\mathrm{inches}\end{align*} allows us to add the \begin{align*}7\;\mathrm{inches}\end{align*} to it: \begin{align*}7\;\mathrm{in} + 36\;\mathrm{in} = 43\;\mathrm{in}\end{align*}.
- The same is true with fractions. “Halves” and “fourths” must be first converted to the common unit “fourths.” Similarly, fractions representing “halves” and “thirds,” will need to be converted to a third unit, “sixths,” the LCM of \begin{align*}2\end{align*} and \begin{align*}3\end{align*}.
- Example 4 uses a visual argument to justify the need for common denominators. In one direction, the “cake” is sliced according to the first fraction, \begin{align*}3/5\end{align*}; in the other direction it is sliced according to the second fraction \begin{align*}1/6\end{align*}. The sum of \begin{align*}3/5\end{align*} and \begin{align*}1/6\end{align*} can now be found – but only in terms of the little squares since the total number of squares represent the common unit.