6.2: Fraction Addition and Subtraction

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Adding and subtracting fractions is a foundational yet often challenging concept for elementary students. Many struggle to see fractions as single quantities, instead interpreting them as two separate whole numbers—leading to common errors like adding numerators and denominators directly (e.g., thinking 1/3 + 1/2 = 2/5). These misconceptions are compounded by confusion around the need for common denominators, which stems from a lack of understanding that fractions must represent parts of the same whole. Procedural instruction without conceptual grounding further exacerbates the issue, leaving students overwhelmed and prone to mistakes. To address this, future teachers must emphasize conceptual understanding through visual models, real-world contexts, and a clear focus on the meaning behind operations, fostering a more accurate and intuitive grasp of fraction addition and subtraction (NCTM, 2020).

Think About and Discuss

When first learning fraction operations, students view the addition of fractions as adding the numerators and adding the denominators: $\large\frac{3}{4}$ + $\large\frac{1}{2}$ = $\large\frac{4}{6}$. Using this example, think about why such a method for addition is unreasonable. You may want to use fractions strips to help with your understanding. Be prepared to discuss your ideas with your peers.

$A hand-drawn marker illustration showing two rectangles side by side, each representing a fraction. The left rectangle is divided into four equal vertical parts, with three shaded in blue, labeled "3/4" below. The right rectangle is divided into two equal vertical parts, with one shaded in orange, labeled "1/2" below. Both rectangles are the same size and have thick black outlines. The unshaded sections in each rectangle remain white.$

Model Fraction Addition

How many quarters are pictured, if you even know what these are (😆)? One quarter plus $2$ quarters equals $3$ quarters.

$Three U.S. quarters are shown. One is shown on the left, and two are shown on the right.$

Figure $\PageIndex{1}$

Remember, quarters are really fractions of a dollar. Quarters are another way to say fourths. So the picture of the coins shows that

\[\begin{split} \dfrac{1}{4} \qquad \qquad \qquad \dfrac{2}{4} \qquad & \qquad \qquad \dfrac{3}{4} \\ one \; quarter + two \; quarters &= three\; quarters \end{split} \nonumber \]

So again, we see that

\[\dfrac{1}{4} + \dfrac{2}{4} = \dfrac{3}{4} \nonumber \]

Addition of fractions is an extension of whole-number addition and can be motivated using models. To find the sum of $\large\frac{3}{5}$ + $\large\frac{1}{5}$, consider the following models Figure 6.2.1: region, fraction strip, number-line model.

Three purple pie chart-style fraction models show the addition of 3/5 and 1/5 to make 4/5. The left image shows a circle divided into five equal parts with three shaded sectors labeled 1/5 each. The center image shows one separate sector labeled 1/5, indicating the amount to be added. The right image combines both, showing a circle with four shaded sectors labeled 1/5, visually representing the sum 4/5. — Figure $\PageIndex{2}$: Region, Fraction Strip, Number Line

A visual representation of the addition of fractions using purple rectangular fraction tiles. The top row shows three rectangles labeled 1/5 followed by a plus sign and one more 1/5 rectangle, indicating 3/5 + 1/5. The bottom row combines all four 1/5 rectangles into a single row, representing the sum 4/5. — Figure $\PageIndex{2}$: Region, Fraction Strip, Number Line

We see from those models that

\[\dfrac{3}{5} + \dfrac{1}{5} = \dfrac{4}{5} \nonumber \]

Definition: Addition of Fractions Like Denominators

If $a$, $b$, and $c$ are numbers where $c ≠ 0$, then

\[\dfrac{a}{c} + \dfrac{b}{c} = \dfrac{a + b}{c}\]

To add fractions with a common denominator, add the numerators and place the sum over the common denominator.

Example $\PageIndex{1}$

Use a model to find the sum. Show a diagram to illustrate your model. \[\dfrac{1}{8} + \dfrac{4}{8} \nonumber \]

Answer

$\dfrac{5}{8}$

$CNX_BMath_Figure_04_04_004_img.jpg$

Think About and Discuss

How about ⅔ + ¼? Use fractions strips to build a model for this sum. Be prepared to discuss your ideas with your peers.

MODEL

$A visual model showing the addition of fractions 2 / 3 + 1 / 4 =$

🧠Let's Listen to Learn❤️

Theorem $\PageIndex{1}$

Addition of Fractions with Unlike Denominators

Let $\large\frac{a}{b}$ and $\large\frac{c}{d}$ be any fractions. Then $\large\frac{a}{b}$ + $\large\frac{c}{d}$ = $\large\frac{ad+bc}{bd}$

PROOF: $\large\frac{a}{b}$ + $\large\frac{c}{d}$ = $\large\frac{ad}{bd} $ + $\large\frac{bc}{bd}$ = $\large\frac{ad + bc}{bd}$

In words, to add fractions with unlike denominators, find equivalent fractions with common denominators. Then the sum will be represented by the sum of the numerators over the common denominator

Example $\PageIndex{2}$

Find the following sums and simplify.

a) $\large\frac{5}{9}\text{+}\large\frac{3}{4}$

b) $\large\frac{17}{15}\text{+}\large\frac{5}{12}$

c) $\large\frac{3}{5}\text{ + (}\large\frac{4}{7}\text{ + }\large\frac{2}{5}\text{)}$

Answer

a) 20/36 + 27/36 = 47/36

b) Do not cross multiply, it may be easier to first find a common denominator. 68/60 + 25/60 = 93/60. 93/60 simplifies to 31/20.

c) (3/5 + 2/5) + 4/7 = 1 + 4/7 = 1$\large\frac{4}{7}$ or 1+4/7 = 7/7 + 4/7 = 11/7

Use the Desmos Scientific Calclator to check your work.

Improper Fractions and Mixed Numbers

As we saw in the previous example, the number 1 + 4/7 can be expressed as the mixed number and as an improper fraction. For example, 2$\large\frac{1}{2}$, can be seen as 2 + 1/2, which can be represented as 4/2 + 1/2 = 5/2. Any mixed number can be changed to an improper fraction, and vice versa.

Example $\PageIndex{3}$

a) Express 3$\large\frac{2}{5}$ as an improper fraction.

b) Express 36/7 as a mixed number.

Answer

a) 3 + 2/5 = 15/5 + 2/5 = 17/5. Do you know a shortcut?

b) 36/7 = 35/7 + 1/7 = 5 + 1/7 = 5$\large\frac{1}{7}$. Do you know another way to do this?

Divide the numerator by the denominator.

The quotient is the whole number part of the mixed number.
The remainder becomes the new numerator.

Keep the same denominator from the original fraction.

Think About and Discuss

When converting a mixed number to an improper fraction, why do we multiply the whole number times the denominator of the fractional part, add this product to the numerator of the fractional part, and write the result over the denominator of the fractional part?

1.Multiply the whole number by the denominator.

2.Add the numerator to this product.

3. Write this new sum over the original denominator.

IMPORTANT: Children may know the rule for converting a mixed number to an improper fraction, but they often do not understand why they do this. Make sure you do before moving on.

Fraction Subtraction

Subtraction of fractions can be viewed in two ways as we did with whole-number subtraction—either as (1) take-away or (2) using the missing-addend approach. The same rules follow as fraction addition.

Definition: Fraction Subtraction

If $a$, $b$, and $c$ are numbers where $c ≠ 0$, then

\[\dfrac{a}{c} - \dfrac{b}{c} = \dfrac{a-b}{c}\]

To subtract fractions with a common denominator, we subtract the numerators and place the difference over the common denominator.

$\dfrac{4}{5} − \dfrac{1}{5}$ = $\dfrac{4-1}{5}$ = $\dfrac{3}{5}$.

$The bottom reads 4 fifths minus 1 fifth equals 3 fifths. Above 4 fifths, there is a circle divided into 5 equal pieces, with 4 pieces shaded in orange. Above 1 fifth, the same circle is shown, but 1 of the 4 shaded pieces is shaded in grey. Above 3 fifths, the 1 grey piece is no longer shaded, so there is a circle divided into 5 pieces with 3 of the pieces shaded in orange.$

Theorem $\PageIndex{2}$

Subtraction of Fractions with Unlike Denominators

Let $\large\frac{a}{b}$ and $\large\frac{c}{d}$ be any fractions. Then $\large\frac{a}{b}$ - $\large\frac{c}{d}$ = $\large\frac{ad-bc}{bd}$

NOTE: In this case we have $\large\frac{a}{b}$ ≥ $\large\frac{c}{d}$.

Example $\PageIndex{4}$: difference

Find the following differences.

a) $\dfrac{4}{7} − \dfrac{3}{8}$.

b) $\dfrac{25}{12} − \dfrac{7}{18}$.

Answer

a) $\dfrac{4}{7} − \dfrac{3}{8}$ = $\dfrac{4•8}{7•8} − \dfrac{3•7}{8•7}$ = $\dfrac{32-21}{56}$ = 11/56

b) NOTE: LCM(12, 18)=36

$\dfrac{25}{12} − \dfrac{7}{18}$=$\dfrac{25•3}{12•3} − \dfrac{7•2}{18•2}$ = $\dfrac{75-14}{36}$ = 61/36

Mental Math and Estimation

Mental Math and Estimation for Addition and Subtraction Mental math and estimation techniques similar to those used with whole numbers can be used with fractions.

Example $\PageIndex{5}$

Estimate using the indicated techniques or your own techniques that come to mind.

a) 3$\large\frac{3}{7}$ + 6$\large\frac{2}{5}$ using range estimation.

b) 15$\large\frac{1}{4}$ - 7$\large\frac{3}{5}$ using front-end with adjustment.

c) 3$\large\frac{5}{6}$ + 5$\large\frac{1}{8}$ + 8$\large\frac{3}{8}$ using rounding to the nearest 1/2 or whole.

Answer

a) 3$\large\frac{3}{7}$ + 6$\large\frac{2}{5}$ is between 3 + 6 = 9 and 4 + 7 = 11.

b) 15$\large\frac{1}{4}$ - 7$\large\frac{3}{5}$ ≈ 7$\large\frac{1}{2}$ since 15-7 = 8 and $\large\frac{3}{5}$ ≈ $\large\frac{1}{2}$.

c) 3$\large\frac{5}{6}$ + 5$\large\frac{1}{8}$ + 8$\large\frac{3}{8}$ ≈ 4 + 5 + 8$\large\frac{1}{2}$ = 17$\large\frac{1}{2}$

Think About and Discuss

Which is larger? $\large\frac{3}{8}$ or $\large\frac{2}{3}$ ?

Think of a way to figure that out without a calculator and then be prepared to discuss this with your peers.

Using Benchmark Fractions

Expected of fifth grade students is the ability to use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers (CDE, 2013). One way to estimate the size of a fraction is to compare it to common benchmarks like 0, $\frac{1}{2}$, and 1. For example, to compare $\frac{5}{6}$ and $\frac{5}{9}$, you might reason that $\frac{5}{6}$ is very close to 1, while $\frac{5}{9}$ is a little more than $\frac{1}{2}$ (since $\frac{4.5}{9}$)= $\frac{1}{2}$. Therefore, you can estimate that $\frac{5}{6}$ is larger than $\frac{5}{9}$. You can also focus on the size of the unit fraction (the fraction with a numerator of 1). For example, when comparing $\frac{1}{8}$ and $\frac{1}{3}$, you can reason that thirds are larger parts than eighths (if the whole is the same), so $\frac{1}{3}$ is larger.

Wonder, Play, Grow

Work collaboratively to share your thinking about the following questions. Keep in mind everything you have learned and that there will multiple ways to respond.

i) Which is larger, 5/6 or 5/9?

ii) Which is smaller, 2/11 or 1/9?

It's important to note that estimation skills develop slowly and require a strong conceptual understanding of fractions. Encouraging children to think about the meaning of fractions and use visual models can support the development of these estimation abilities (Feike et al., 2018).

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Definition: Addition of Fractions Like Denominators

Example \(\PageIndex{1}\)

Think About and Discuss

Theorem \(\PageIndex{1}\)

Example \(\PageIndex{2}\)

Example \(\PageIndex{3}\)

Think About and Discuss

Definition: Fraction Subtraction

Theorem \(\PageIndex{2}\)

Example \(\PageIndex{4}\): difference

Example \(\PageIndex{5}\)

Think About and Discuss