Learning Outcomes
- Model fraction addition
- Add fractions with a common denominator
- Add fractions with a common denominator that contain variables
Model Fraction Addition
How many quarters are pictured? One quarter plus [latex]2[/latex] quarters equals [latex]3[/latex] quarters.
Remember, quarters are really fractions of a dollar. Quarters are another way to say fourths. So the picture of the coins shows that
[latex]{\Large\frac{1}{4}}+{\Large\frac{2}{4}}=\Large{\frac{3}{4}}[/latex]
[latex]\text{one quarter }+\text{ two quarters }=\text{ three quarters} [/latex]
Let’s use fraction circles to model the same example, [latex]\Large\frac{1}{4}\normalsize+\Large\frac{2}{4}[/latex].
| Start with one [latex]\Large\frac{1}{4}[/latex] piece. |  |
[latex]\Large\frac{1}{4}[/latex] |
| Add two more [latex]\Large\frac{1}{4}[/latex] pieces. |  |
[latex]+\Large\frac{2}{4}[/latex] |
| The result is [latex]\Large\frac{3}{4}[/latex] . |  |
[latex]\Large\frac{3}{4}[/latex] |
So again, we see that
[latex]\Large\frac{1}{4}\normalsize+\Large\frac{2}{4}\normalsize=\Large\frac{3}{4}[/latex]
Doing the Manipulative Mathematics activity “Model Fraction Addition” will help you develop a better understanding of adding fractions
example
Use a model to find the sum [latex]\Large\frac{3}{8}\normalsize+\Large\frac{2}{8}[/latex].
Solution:
| Start with three [latex]\Large\frac{1}{8}[/latex] pieces. |  |
[latex]\Large\frac{3}{8}[/latex] |
| Add two [latex]\Large\frac{1}{8}[/latex] pieces. |  |
[latex]+\Large\frac{2}{8}[/latex] |
| How many [latex]\Large\frac{1}{8}[/latex] pieces are there? |  |
[latex]\Large\frac{5}{8}[/latex] |
There are five [latex]\Large\frac{1}{8}[/latex] pieces, or five-eighths. The model shows that [latex]\Large\frac{3}{8}\normalsize+\Large\frac{2}{8}\normalsize=\Large\frac{5}{8}[/latex].
try it
Use a model to find each sum. Show a diagram to illustrate your model.
[latex]\Large\frac{1}{8}\normalsize+\Large\frac{4}{8}[/latex]
Use a model to find each sum. Show a diagram to illustrate your model.
[latex]\Large\frac{1}{6}\normalsize+\Large\frac{4}{6}[/latex]
The following video shows more examples of how to use models to add fractions with like denominators.
Add Fractions with a Common Denominator
The example above shows that to add the same-size pieces—meaning that the fractions have the same denominator—we just add the number of pieces.
Fraction Addition
If [latex]a,b,\text{ and }c[/latex] are numbers where [latex]c\ne 0[/latex], then
[latex]\Large\frac{a}{c}\normalsize+\Large\frac{b}{c}\normalsize=\Large\frac{a+b}{c}[/latex]
To add fractions with a common denominators, add the numerators and place the sum over the common denominator.
Example
Find the sum: [latex]\Large\frac{3}{5}\normalsize+\Large\frac{1}{5}[/latex]
Try It
Example
Find the sum: [latex]\Large\frac{x}{3}\normalsize+\Large\frac{2}{3}[/latex]
Note that we cannot simplify this fraction any more. Since [latex]x\text{ and }2[/latex] are not like terms, we cannot combine them.
Try It
Example
Find the sum: [latex]-\Large\frac{9}{d}\normalsize+\Large\frac{3}{d}[/latex]
Tip: A negative sign on a fraction can be written in the following locations: by the numerator, by the denominator or out in front of the fraction bar. It is your choice!
Try It
Example
Find the sum: [latex]\Large\frac{2n}{11}\normalsize+\Large\frac{5n}{11}[/latex]
Try It
Example
Find the sum: [latex]\Large-\frac{3}{12}+\left(-\frac{5}{12}\right)[/latex]
Try It
in the next video we show three more examples of adding fractions with common denominators. Pay close attention to the second example, it addresses a common mistake made by students when simplifying fractions with variables.
