6.1: Concept of Fraction, Equivalence, and Order

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The early chapters in our book focused on building a solid understanding of the whole number system, which is essential for long-term success in mathematics. Now, we shift our attention to fractions. Fractions were developed because many real-world problems cannot be accurately described using only whole numbers. As you work through this chapter, you’ll see how a strong grasp of whole numbers provides a helpful foundation for making sense of fraction concepts.

Importance of Fractions

Students often struggle with fractions because they view them as two separate whole numbers rather than a single number representing a part of a whole. This misunderstanding prevents them from seeing the relationship between the numerator and denominator, and from using helpful benchmarks like 0, ½, and 1. Unlike whole numbers, fractions rarely appear in children’s everyday experiences beyond simple examples like ½ or ¼, making them less intuitive. Additionally, students tend to incorrectly apply whole number reasoning to fractions, by believing, for instance, that 1/20 is greater than 1/10 because 20 is larger than 10. (Musser et al., 2014; NCTM, 2020).

Another major challenge is the abstract nature of fractions, which can represent multiple ideas—part-whole, measurement, division, ratio, or operator—depending on the context. Many students only encounter the part-whole model, limiting their understanding. Cognitive overload also plays a role, as tasks with fractions often require juggling several steps, especially when working with unlike denominators (Feike et al., 2018). Moreover, instruction that emphasizes procedural rules—like “invert and multiply”—before students fully grasp the meaning of fractions leads to superficial learning. When students are taught procedures without context, they may memorize steps but lack the conceptual understanding needed for success in more advanced math (NCTM, 2020).

Fractions Children Learn in Elementary School

As we learn the concepts and procedures associated with fractions, it is important that you have context of what students learn about fractions in elementary school. Below is a video explaining how children progress with their understanding of fractions. Also included is a list of some of the standards from the Common Core State Standards (CDE, 2013).

Graham Fletchy - https://youtu.be/9IwVAXmrWK8?si=0DCRvo6Lmqb_1zEq

Common Core Standards (CDE, 2013)

Grade 3: Introduction to Fractions

3.NF.1: Understand a fraction 1/b as one part of a whole divided into b equal parts; a/b as a parts of size 1/b.

3.NF.2: Represent fractions on a number line.
2a: Mark 1/b on a number line between 0 and 1.
2b: Mark a/b by partitioning and counting lengths of 1/b.

3.NF.3: Explain equivalence and compare fractions:
3a: Understand equivalence by same size or location.
3b: Generate simple equivalent fractions.
3c: Express whole numbers as fractions.
3d: Compare fractions with same numerator or denominator using visual models.

Grade 4: Operations and Equivalence

4a–4c: Visualize, apply repeated addition, and solve word problems.

4.NF.1: Explain why a fraction a/b is equivalent to (n × a)/(n × b).

4.NF.2: Compare fractions with different numerators and denominators.

4.NF.3: Build fractions from unit fractions:
3a: Understand addition/subtraction as joining/separating parts.
3b: Decompose fractions into sums.
3c: Add/subtract mixed numbers.
3d: Solve word problems with like denominators.

4.NF.4: Multiply a fraction by a whole number

Grade 5: All Operations with Fractions

5.NF.1-2: Use equivalent fractions to add/subtract fractions with unlike denominators.

5.NF.3: Interpret a fraction as division (e.g., 2 ÷ 3 = 2/3).

5.NF.4: Multiply a fraction or whole number by a fraction.

4a: Visualize multiplication.
4b: Find area of rectangles with fractional side lengths.

5.NF.5: Interpret multiplication outcomes relative to 1.

5.NF.6: Solve real-world problems involving multiplication of fractions and mixed numbers.

5.NF.7: Divide unit fractions by whole numbers and vice versa:

Wonder, Play, Grow

Fraction Talks: Before discussing anything with others, review each picture and determine what fraction of space is occupied by each color. Take some time to think about it and about how you can defend your answer with the class.

Picture 1:

$A rectangular image divided into three equal vertical sections. Each section contains two right triangles of different colors forming a square. From left to right: The first section has an orange triangle on the bottom left and a blue triangle on the top right. The second section has a blue triangle on the bottom left and a red triangle on the top right. The third section has two red triangles, with one in the top left and the other in the bottom right, forming a square split diagonally.$

Picture 2:

$A rectangle divided into three colored regions. The left half is filled with a large red rectangle. The right half is split vertically into two parts: a green region on the top and left side, and a smaller blue rectangle in the bottom right corner, partially overlapping the green region. The image is outlined with a bold black border.$

Picture 3:

$A photo of wooden pattern blocks on a wooden surface. On the left, a hexagon shape is formed by two green triangles on top and two blue trapezoids on the bottom. On the right, there is a single yellow hexagon block. The image compares two ways of forming a hexagon—using multiple shapes versus a single shape.$

Concept of Fraction

In elementary math, fractions are used in two distinct ways:

🧠Let's Listen to Learn❤️

Part-to-Whole Model:
This is the most familiar introduction to fractions. A fraction like 3/4 represents 3 parts out of 4 equal parts of a whole. The numerator (3) tells how many parts are being considered, and the denominator (4) tells what kind of parts (fourths, fifths, etc.). This model emphasizes fractions as a way to name parts of a whole, whether it's pieces of a pie (¾), segments of a stick (3/6), or cups of liquid (4/6).

$An image showing three different visual representations of fractions using red shading: On the left, a circle is divided into four equal parts, with three parts shaded red, representing 3/4. In the middle, a rectangle is divided into eight equal sections, with five shaded red, representing 5/8. On the right, a cylindrical container is partially filled with red liquid, with fill marks showing the liquid reaches approximately three-fourths of the way up.$

Fractions as Numbers:
Fractions are also abstract numbers that represent a relative amount, independent of any particular object. For instance, 5/8 means the same proportion regardless of the size or shape of the whole—it expresses a point on the number line, just like whole numbers do. This view shifts the meaning of fractions from a specific object-based context to a general numerical concept. Each of the figures below shows the relative amount of 5/8.

$An image showing three different visual models representing fractions: On the left, a circle is divided into 8 equal parts with 5 of them shaded blue, representing the fraction 5/8. In the middle, a square is divided into 8 triangular sections, with 4 of them shaded orange, representing 4/8 or 1/2. On the right, there are 8 small circles arranged in two rows; 5 are shaded dark green and 3 are light green, representing the fraction 5/8.$

Definition: Fractions

A fraction is a number that can be represented by an ordered pair of whole numbers ($\frac{a}{b})$ (or (a/b)), where $b\neq 0$. In set notation, the set of fractions is \[ F = \left\{ \frac{a}{b} \,\middle|\, a \text{ and } b \text{ are whole numbers, } b \ne 0 \right\}. \]

Language Note: The word fraction comes from the Latin word fractius which means ”to break”. When an object is divided into an equal number of parts then each part is called a fraction. The number a is called the numerator, derived from the Latin word numeros, meaning ”number”, and represents the number of parts in consideration. The number b is called the denominator, derived from the Latin word denominare, meaning ”namer”, and represents how many equal parts in the unit. Keep in mind that this number can never be zero since division by zero is undefined.

Visual Representation of a Fraction

Several physical and pictorial representations are useful in the elementary school classroom to illustrate fraction concepts.

SET MODEL - The figure below shows a set of 7 apples that contains a subset of 3 that are wormy. Therefore, we would say that $\frac{3}{7}$ of the apples are wormy.

$An image showing 7 red apples arranged in rows. Three of the apples each have a cartoon white worm with a smiling face coming out of them, while the other four apples are plain. This visual represents the fraction 3/7, indicating that 3 out of the 7 apples are wormy.$

NUMBER LINE (Measurement or Length Model) - Fractions can be shown on a number line by dividing the segment from 0 to 1 into equal parts. The fraction $\frac{1}{b}$ is the point at the end of the first segment when that interval is divided into b equal parts.

$A number line diagram showing the fraction 1/b. The segment from 0 to 1 is divided into b equal parts, with tick marks representing each division. The top label indicates one of these segments as "1/b". Below the number line, handwritten red text labels the segment from 0 to 1 as "b equal parts", with the numbers 0 and 1 marked at each end.$

To locate $\frac{a}{b}$, start at 0 and count a segments of length $\frac{1}{b}$. The end of the ath segment marks the location of a/b.

$A number line diagram showing how to locate the fraction a/b. The interval from 0 to 1 is divided into b equal parts, with one tick mark labeled as 1/b. Another point further along is labeled a/b, indicating it is located by counting a copies of 1/b from 0. Red handwritten annotations label the segment from 0 to 1 as “b equal parts” and the distance from 0 to a/b as “a copies.”$

REGION/AREA MODEL - A shape is chosen to represent the unit and is then subdivided into equal parts. A fraction is visualized by coloring some of the parts as shown in the figure below.

$A rectangle divided into 6 equal sections arranged in 2 rows and 3 columns. Five of the sections are shaded medium blue, and one section in the bottom right corner is shaded a lighter blue. This image visually represents the fraction 5/6.$

FRACTION STRIPS (Bars) -

Here the unit is defined by a rectangular strip. A fraction $\frac{a}{b}$ is modeled by shading a parts of the b equally sized subrectangles. Sample fraction strips are shown in figure below.

$An image of fraction strips used to visually compare fractional amounts. The top strip (orange) is labeled 1/1, representing one whole. Below it, five equal purple parts each labeled 1/5 add up to make a strip slightly shorter than the whole, showing 4/5 with a blue label and arrow pointing to it. Beneath the purple strip are two small yellow pieces each labeled 1/12, with a bracket and label indicating the total as 2/12.$

Example $\PageIndex{1}$

Explain how to complete each diagram below to represent $\frac{3}{10}$.

$Two mathematical representations of the number 1. On the left, a number line shows a segment from 0 to 1 with an arrow pointing right. On the right, a rectangular region model is divided into 10 equal boxes (2 rows and 5 columns), illustrating the whole as a set of 10 equal parts.$

Answer: $wo visual representations of the fraction 3/10. On the left, a number line from 0 to 1 is divided into 10 equal segments, with the point at 3/10 marked in red above the third tick mark. On the right, a rectangular region model divided into 10 equal parts (2 rows and 5 columns) has 3 of the sections shaded light blue, representing 3 out of 10 parts.$

There are 3 different types of fractions:

• Proper Fractions - Proper fractions have the numerator part smaller than the denominator part, for example, $\frac{2}{5}$.

• Improper Fractions - Improper fractions have the numerator part greater or equal to the denominator for example, $\frac{7}{3}$.

• Mixed Fractions - Mixed fractions have a whole number plus a fraction, for example, 3$\frac{1}{2}$.

Equivalent Fractions

Two fractions that represent the same relative amount are said to be equivalent fractions. As suggested by the fraction strips below, the fractions all represent the same relative amount as $\frac{1}{3}$. And make a note that whole number 1 can be represented by a fraction $\frac{1}{1}$.

$A visual model showing equivalent fractions. At the top, an orange bar labeled "1/1" represents a whole. Below it, a red section labeled "1/3" fills one-third of the whole. Below that, two blue sections labeled "1/6" each together also cover one-third. At the bottom, four yellow sections labeled "1/12" each together again fill the same one-third space. To the right, a cloud-like shape contains the equation: "1/3 = 2/6 = 4/12" in handwritten black text, illustrating equivalence.$

Equivalent fractions are also located at the same point on a number line. The equivalence can be seen by their common location on the number line as seen below.

$A number line from 0 to 1 is divided into eighths, with tick marks labeled in blue as 1/8, 2/8, 3/8, 4/8, 5/8, and 6/8. Above the line, red lines mark equivalent fractions: 1/4 at 2/8, 2/4 at 4/8, and 3/4 at 6/8. Black dots highlight the positions of these equivalent values. Below the number line, light blue callout bubbles show the equivalences: "1/4 = 2/8", "2/4 = 4/8", and "3/4 = 6/8".$

It is useful to have a simple test to determine whether fractions such as $\frac{3}{4}$ and $\frac{6}{8}$ represent the same relative amount without having to draw a picture of each representation.

Definition: Fraction Equality

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

NOTE: Two fractions are equal fractions if and only if their cross-products, that is, products ad and bc obtained by cross-multiplication, are equal.

Theorem $\PageIndex{1}$

Let $\large\frac{a}{b}$ be any fraction and n a nonzero whole number. Then $\large\frac{a}{b}$=$\large\frac{an}{bn}$=$\large\frac{na}{nb}$.

Example $\PageIndex{2}$

Verify the following equations using the definition of fraction equality or the preceding theorem.

a) $\large\frac{6}{14}$ =$\large\frac{9}{21}$

b) $\large\frac{6}{21}$ =$\large\frac{60}{210}$

Answer

a) (6 x 21) = (9 x 14) = 126, equivalent fractions.

b) $\large\frac{60}{210}$ = $\large\frac{30x2}{30x7}$=$\large\frac{2}{7}$ and $\large\frac{6}{21}$ = $\large\frac{3x2}{3x7}$=$\large\frac{2}{7}$, equivalent fractions.

Occasionally, the term reducing is used to describe the process in 6.2 example b). However, the term reducing can be misleading, since fractions are not reduced in size (the relative amount they represent) but only in complexity (the numerators and denominators are smaller). We use the term “simplify” or say 60/210 is in “simplified form” when it is written as 2/7.

Example $\PageIndex{3}$

Find the simplest form for each fraction.

a) $\large\frac{72}{240}$

b) $\large\frac{9}{32}$

c) $\large\frac{399}{483}$

Answer: a) $\large\frac{72}{240}$=$\large\frac{3x24}{10x24}$ = 3/10 b) 9/32 c) $\large\frac{399}{483}$=$\large\frac{19x21}{23x21}$=19/23

Ordering Fractions

🧠Let's Listen to Learn❤️

Think About and Discuss

Before discussing these ideas with your peers, make sure to independently respond to the prompts.

Task 1: Consider each fraction strip (1/2, 1/3, 1/4, 1/5, 1/6, 1/8, 1/9, 1/10, and 1/12). One fraction is bigger than another if it has more length than the other fraction. Compare them and put them in order from smallest to largest, and write them in order below, using the less than symbol ( < ). Where would 1/7 go?

Task 2: Use fraction strips to compare 5/6, 5/8, 5/9, 5/10, and 5/12. Put them in order from smallest to largest using the less than symbol ( < ). Where would 5/7 go?

Task 3: Use fraction strips to compare these fractions: 1/2, 2/3, 3/4, 4/5, 5/6, 7/8, 8/9, 9/10, and 11/12. Put them in order from smallest to largest using the less than symbol ( < ).

Task 4: Discuss with your peers and explain any patterns you learned.

Online Fraction Strips

https://polypad.amplify.com/p#fractions

$A colorful fraction wall chart displaying unit fractions from 1/1 to 1/12. Each row is a different color and represents a fraction bar broken into equal parts. The top row is orange and shows one whole (1/1). Subsequent rows divide the same length into equal parts labeled as 1/2, 1/3, 1/4, and so on, down to 1/12 at the bottom. Each unit fraction is shown repeated across the row to fill the same total length as the whole. This visual illustrates how unit fractions become smaller as the denominator increases and shows how different fractions can be compared based on size.$

The concepts of less than and greater than in fractions are extensions of the respective whole-number concepts. As it was in the case of whole numbers and what you learned in the Think About and Discuss activity, the smaller of two fractions is to the left of the larger fraction on the fraction number line. Also, what we learned suggest the following definition, where fractions having common denominators can be compared simply by comparing their numerators (which are whole numbers).

Definition: Less Than for Fractions

Let $\large\frac{a}{c}$ and $\large\frac{b}{c}$ be any fractions. Then $\large\frac{a}{c}$ < $\large\frac{b}{c}$ if and only if a < b.

NOTE: Although the definition is stated for “less than,” a corresponding statement holds for “greater than.” Similar statements hold for “less than or equal to” and “greater than or equal to.”

Theorem $\PageIndex{2}$

Cross-Multiplication of Fraction Inequality

Let $\large\frac{a}{b}$ and $\large\frac{c}{d}$ be any fractions. Then $\large\frac{a}{b}$<$\large\frac{c}{d}$ if and only if ad<bc.

Example $\PageIndex{4}$

Arrange in order.

a) $\large\frac{7}{8}$ and $\large\frac{9}{11}$

b) $\large\frac{2}{9}$ and $\large\frac{7}{36}$

b) $\large\frac{17}{32}$ and $\large\frac{19}{40}$

Answer: a) 7/8 > 9/11 b) 7/36 < 2/9 c) 19/40 < 17/32

Another COOL way to compare fractions - Dr. Howie Hua does it again.

https://youtu.be/7pbiKIYsWF0?si=NFS_08iO8gCRYqRG

Wonder, Play, Grow

Let's see what else Dr. Hua is up to? Work with your peers to find a fraction that is between $\large\frac{3}{4}$ and $\large\frac{4}{5}$. For an extra challenge, can you find more than one?

$A tweet by Howie Hua reads: "Today's class conversation starter: What is a fraction that is between 3/4 and 4/5? How would you do this?" The image attached is a humorous meme featuring three cats lying close together. The cat on the left is labeled "3/4", the cat on the right is labeled "4/5", and the uncomfortable-looking white cat squished between them is labeled with a question mark (?). To the left of the cats is a whiteboard-style prompt that reads: "What is a fraction that is between 3/4 and 4/5?" with space for "Responses:" below.$

$A fraction bar model showing three levels of unit fractions stacked above a whole. At the bottom, a long orange bar labeled 1/1 represents one whole. Above it, five purple bars labeled 1/5 are placed side by side to equal the whole. On the top row, three pink bars labeled 1/4 are shown, covering most—but not all—of the whole. This visual demonstrates how different unit fractions compare in size and how more parts are needed to make a whole as the denominator increases.$

Using common denominators is great to find a fraction between two others. For example, 3/4 = 30/40 and 4/5 = 32/40. Therefore we know that 3/4 < 31/40 < 4/5. However there is simple technique of adding the numerators and denominators to determine a fraction between two other fractions.

Theorem $\PageIndex{3}$

Let $\large\frac{a}{b}$ and $\large\frac{c}{d}$ be any fractions where $\large\frac{a}{b}$ < $\large\frac{c}{d}$. Then

$\large\frac{a}{b}$ < $\large\frac{a+c}{b+d}$ <$\large\frac{c}{d}$.

Example $\PageIndex{5}$

Find a fraction between these pairs of fractions.

a) $\large\frac{7}{11}$ and $\large\frac{8}{11}$

b) $\large\frac{9}{13}$ and $\large\frac{12}{17}$

Answer

a) $
\frac{7}{11} = \frac{14}{22} \quad \text{and} \quad \frac{8}{11} = \frac{16}{22}. \quad \text{Hence} \quad \frac{15}{22} \text{ is between } \frac{7}{11} \text{ and } \frac{8}{11}.
$

b) $
\frac{9}{13} = \frac{9 \cdot 17}{13 \cdot 17} = \frac{153}{13 \cdot 17} \quad \text{and} \quad \frac{12}{17} = \frac{12 \cdot 13}{17 \cdot 13} = \frac{156}{17 \cdot 13}. \quad \text{Hence both } \frac{154}{13 \cdot 17} \text{ and } \frac{155}{13 \cdot 17} \text{ are between } \frac{9}{13} \text{ and } \frac{12}{17}.
$

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Definition: Fractions

Example \(\PageIndex{1}\)

Definition: Fraction Equality

Theorem \(\PageIndex{1}\)

Example \(\PageIndex{2}\)

Example \(\PageIndex{3}\)

Think About and Discuss

Definition: Less Than for Fractions

Theorem \(\PageIndex{2}\)

Example \(\PageIndex{4}\)

Theorem \(\PageIndex{3}\)

Example \(\PageIndex{5}\)

Fratction Finder