34 Concept of Fraction

Standards

3.NF.3. Explain equivalence of fractions with denominators [latex]2, 3, 4, 6[/latex] and [latex]8[/latex] in special cases, and compare fractions by reasoning about their size.

1. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
2. Recognize and generate simple equivalent fractions, e.g., [latex]\frac{1}{2}=\frac{2}{4}, \frac{4}{6}=\frac{2}{3}.[/latex] Explain why the fractions are equivalent, e.g., by using a visual fraction model.
3. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express [latex]3[/latex] in the form [latex]3=\frac{3}{1}[/latex]; recognize that [latex]\frac{6}{1}=6[/latex]; locate [latex]\frac{4}{4}[/latex] and [latex]1[/latex] at the same point of a number line diagram.
4. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols [latex]>, =,[/latex] or [latex]<,[/latex] and justify the conclusions, e.g., by using a visual fraction model.

4.NF.A. Extend understanding of fraction equivalence and ordering.

1. Explain why a fraction [latex]\frac{a}{b}[/latex] is equivalent to a fraction [latex]\frac{na}{nb}[/latex] by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. (Denominators are limited to [latex]2, 3, 4, 5, 6, 8, 10, 12,[/latex]  and [latex]100.[/latex])
2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as [latex]\frac{1}{2}.[/latex] Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols [latex]>, =,[/latex] or [latex]<,[/latex] and justify the conclusions, e.g., by using a visual fraction model. (Denominators are limited to [latex]2, 3, 4, 5, 6, 8, 10, 12,[/latex] and [latex]100.[/latex])

 

1. Understanding Fractions with C-Strips

Usually, a fractionis defined as a number that can be written as the quotient (ratio) of an integer, [latex]m,[/latex] and a nonzero integer, [latex]n,[/latex] that is, [latex]\frac{m}{n}.[/latex] In this form, the dividend [latex]m[/latex] is called the numerator and the divisor [latex]n[/latex] is called the denominator. In reality, a fraction is often used as a stand-alone number with its own value, rather than just as a result associated to the operation among two other numbers (that is, the division of [latex]m[/latex] divided by [latex]m[/latex]. This warm-up activity aims to review the concept of fraction using C-strip (see in Figure 9-1). Hereafter we use the capital letters to represent the strips as shown in Figure 9-1.

 

Before we represent fractions using C-strips, let’s recall that the denominator of a fraction indicates how many equal parts make up [latex]1[/latex] whole unit, while the numerator tells you how many of those equal parts are being considered. For example, to represent the fraction [latex]\frac{2}{5}[/latex] the denominator “[latex]5[/latex]” shows that we should divide up the [latex]1[/latex] whole unit into [latex]5[/latex] equal parts, whereas the numerator indicates that [latex]2[/latex] out of the [latex]5[/latex] will be considered. Using C-strips helps students to visualize these terms as shown in the example below.

Example 1

If H (hot pink) represents [latex]1[/latex] unit, then which C-strip represents [latex]\frac{3}{4}[/latex]?

Solution:

We consider the following steps to answer the question.

a. The denominator is [latex]4[/latex]; so it takes [latex]4[/latex] equal parts to make one unit, and each of those [latex]4[/latex] equal parts [latex]= \frac{1}{4}[/latex].

b. Find the C-strip such that a train of [latex]4[/latex] of them is as long as one unit (H). Since a train of [latex]4[/latex] light green C-strips (L) is the length of H (1 unit), then each light green C-strip makes up one part of a whole.

c. And therefore each light green C-strip represents [latex]\frac{1}{4}[/latex].

d. To find which C-strip represents [latex]\frac{3}{4}[/latex], we make a train of [latex]3[/latex] light green C-strips.

e. Find the C-strip having the same length of the train of the 3  light green C-strips. This would be the Blue (B) C-strip.

Therefore, the answer is B, the Blue C-strip, as shown in Figure 9-2.

Example 1 above is solved following the sequence of steps from a. to e.:

a. Look at the denominator of the given fraction, and state how many unknown C-strips (that is, equal parts) make up the given C-strip representing [latex]1[/latex] whole unit.

b. State what is the unknown C-strip that we can use to make up the given C-strip as [latex]1[/latex] whole unit.

c. State the fraction that one C-strip found in b. represents, which is called a unit fraction.

d. Look at the numerator of the given fraction, and state how many C-strips in b. we need to build a “train” to present it.

e. State which C-strip has the same length as the train built in d., and this is the answer!

 

After completing the five steps, we found that, if H (the Hot Pint C-strip) represents [latex]1[/latex] unit, then B (the Blue C-strip) represents [latex]\frac{3}{4}[/latex]. The five-step process helps students unpack the meaning of denominator and numerator step by step. We may shorten the solution of Example 1 as

a.  [latex]4[/latex]             b.  L             c.  [latex]\frac{1}{4}[/latex], or “one fourth”             d.  [latex]3[/latex]             e.  B

 

Exercises 1-6 provide more practice similar to Example 1.

Exercises 1-6

For exercises 1 – 6, explain how to find the solution. Do each step from a. to e. using your C-strips and record your answer for each step in the blank accordingly.  

a. State how many C-strips (each an equal part of the whole) make up one unit.

b. State which C-strip makes up one part of the whole.

c. State the fraction that the C-strip in part b represents.

d. State how many of the C-strips in part b you need to make into a train.

e. State which C-strip is the length of the train you made in part c, this is the answer!!!

1. If H represents 1 unit, then which C-strip represents [latex]\frac{5}{6}[/latex]?

2. If H represents 1 unit, then which C-strip represents [latex]\frac{5}{12}[/latex]?

3. If O represents 1 unit, then which C-strip represents [latex]\frac{2}{5}[/latex]?

4. If B represents 1 unit, then which C-strip represents [latex]\frac{2}{3}[/latex]?

5. If D represents 1 unit, then which C-strip represents [latex]\frac{5}{3}[/latex]?

6. If N represents 1 unit, then which C-strip represents [latex]\frac{3}{4}[/latex]?

 

In Example 2, the given C-strip represents a fractional part, rather than [latex]1[/latex] whole unit. and our task is to find the unknown C-strip that represents the [latex]1[/latex] unit. To complete this task, we need to go through a reverse process of Example 1, as shown below. 

Example 2

If N (brown) represents [latex]\frac{4}{5}[/latex], then which C-strip represents 1 unit?

Solution

We consider the following steps to answer the question.

a. The denominator is 5, so it takes 5 equal parts to make up 1 whole unit, where each equal part is [latex]\frac{1}{5}[/latex]. Since N is only [latex]\frac{4}{5}[/latex], then a train of only 4 of the 5 equal parts will be the length of N.

b. A train of 4 red C-strips is the same length as N. So a red C-strip is one of the 5 equal parts that make up a whole.

c. Since it takes 5 equal parts (5 reds) to make one unit, each red strip represents [latex]\frac{1}{5}[/latex].

d. And we form a train of 5 reds and see which C-strip has this length.

e. A train of 5 reds has the same length as the orange C-strip (O).

Therefore, the answer is O.

 So, the solution includes (a.) looking at numerator, which indicates how many equal parts are in the given C-strip; (b.) figuring out which C-strip represents [latex]1[/latex] equal part of the whole; (c) recognizing the unit fraction that the C-strip in (b) represents; (d.) recognizing how many C-strip we found in b. can make a whole according to the denominator; and, lastly, (e.) finding the length of the whole and the C-strip that represents it.  Similarly as we did for Example 1, the solution of Example 2 may be shorten as

a.  4             b.  R             c.  [latex]\frac{1}{5}[/latex] , or “one fifth”            d.  5             e.  O

 

Exercises 7-11 provide more practice for Example 2.

Exercises 7-11

For exercises 7 – 11, explain how to find the solution. Do each step using your C-strips and record your answer for each step in the blank accordingly.

a. State how many C-strips will make up the named C-strip stated in the problem. Look at the numerator.

b. Which C-strip makes up one equal part?

c. State the fraction that the C-strip in part b represents. Look at the denominator.

d. State how many of the C-strips in part b will make up one unit.

e. Form the unit by making a train from the equal parts (C-strip in part b) and state which C-strip has the same length as that train.

7. If D represents [latex]\frac{1}{2}[/latex], then which C-strip is 1 unit?

8. If L represents [latex]\frac{1}{3}[/latex], then which C-strip is 1 unit?

9. If P represents [latex]\frac{2}{3}[/latex], then which C-strip is 1 unit?

10. If N represents [latex]\frac{4}{5}[/latex], then which C-strip is 1 unit?

11. If R represents [latex]\frac{2}{5}[/latex], then which C-strip is 1 unit?

 

The type of problems in Example 3 are a little more challenging. They take more steps. From the first piece of information, figure out which C-strip is the whole unit – just like you did in problems 7 – 11. Then, start over using that unit C-rod, and figure out the second part of the question – just like you did in problems 1 – 6.

Example 3

If N represents [latex]\frac{2}{3}[/latex], then which C-strip represents [latex]\frac{3}{4}[/latex]?

Solution

We consider the following steps to answer the question.

a. Begin these the same way the previous problems were done by first figuring out what the unit C-strip is. After doing the same steps you did for exercises 1-6, you will conclude that H is the unit C-strip.

b. Now, on to part 2: Start over with H as the unit C-strip, and find [latex]\frac{3}{4}[/latex] the same way you did it for the first 13 exercises. The key is to start over by looking only at the stated unit C- strip (H in this case), and not getting that confused with the first part of the problem. In other words, now that you have determined that the unit is H, determine what is 3/4 (in relation to the unit – H). You will find that the answer is B

 

Exercises 12-14 provide more practice for Example 3.

Exercises

12. If P represents [latex]\frac{2}{3}[/latex], then which C-strip represents [latex]\frac{3}{2}[/latex]?

13. If O represents [latex]\frac{5}{6}[/latex], then which C-strip represents [latex]\frac{3}{4}[/latex]?

14. If D represents [latex]\frac{2}{3}[/latex], then which C-strip represents [latex]\frac{1}{3}[/latex]?

 

2. Representing Fractions 

Commonly used representations to visualize fractions are set model, bar model, area model, number line model.

 

Figure 9-5 represents a class of 14 students as a set with two subsets, one of which includes 8 girls and the other 6 boys.  It shows [latex]\frac{8}{14}[/latex] of the class are girls and [latex]\frac{6}{14}[/latex] of the class are boys.

The bar model below (Figure 9-6) represents [latex]\frac{4}{6}[/latex], in which the whole bar is divided into 6 equal parts with 4 of them shaded.

 

 

The circle below (Figure 9-7) is evenly divided into 8 parts and five of them are shaded to represent [latex]\frac{5}{8}[/latex] by an area model. An area model may be in a square or rectangle as well.

A fraction as a number has its unique value so that it can be represented on a number line according to its value. This is the so-called number line model (as shown in Figure 9-8).

 

 

There are also other manipulatives for hands-on activities that are helpful fostering students’ understanding of fractions, such as fraction tiles, fractions bars, colored rods, pattern blocks, and tangram, etc.

 

Exercises – Write the Fractions Represented in the Models

For Exercises 15-17, fill in the blanks using appropriate fractions represented by the models.

15. The unshaded portion is _______ of the whole figure below.

 

 

16. The shaded portion is _____ of the whole square. The unshaded portion is ______ of the whole square.

 

 

17.  The shaded portion is _____ of the whole circle. The unshaded portion is ______ of the whole circle. The unshaded portion is _______ of the half circle.

 

3. Equivalent Fractions

Notice there are 2 dots for the answer in part a, where each dot represents [latex]\frac{1}{5}[/latex].  So, the 2 dots represent the number [latex]\frac{2}{5}[/latex].  For part b, the answer has 4 dots.  Since each dot represents [latex]\frac{1}{10}[/latex], this also represents[latex]\frac{4}{10}[/latex]  Therefore, [latex]\frac{2}{5}[/latex] and [latex]\frac{4}{10}[/latex] represent the same number. In other words, [latex]\frac{2}{5}[/latex] and [latex]\frac{4}{10}[/latex] are equivalent fractions.

By using dot models, students may notice that the same fraction can represent different quantities of dots, depending on how many dots make up the whole unit. Therefore, it is crucial to define the unit before beginning any exercise involving dot models.

Now we introduce the fundamental law of fractions, which is followed by two theorems on equivalent fractions and examples respectively.

According to the fundamental law of fractions, if both the numerator and the denominator of a fraction are multiplied by the same factor, the resulting fraction is equivalent to the original one. The ability to write a fraction in an equivalent form—with a different denominator but the same value—is a critical skill when performing addition and subtraction of fractions.

Below are more exercises to practice the skill in Example 5.

The fundamental law also helps us to simplify a fraction into simplest form. Firstly let’s define what is a simplest form.

There are two common ways to rewrite a fraction in its simplest form—that is, to simplify a fraction. One method is to use prime factorization: factor both the numerator and the denominator into primes, then cancel out any common factors. Another method involves using the greatest common factor (GCF) of the numerator and denominator. This process includes three steps:
(i) find the GCF of the numerator and denominator,
(ii) rewrite both numbers as the product of the GCF and another factor, and
(iii) cancel the common factor.

Even when all common factors are canceled from the numerator or denominator, a factor of 1 remains—it’s just often not shown explicitly.

The process of reducing fractions to their simplest forms can help us compare two or more fractions. Unlike whole numbers or integers,  comparing fractions is not always straightforward. As we’ve seen in the Fundamental Law of Fractions and the examples above, two fractions with different denominators and different numerators can still represent the same value, which are called  equivalent fractions.

In this theorem, the phrase if and only if, denoted by the symbol “[latex]\Longleftrightarrow[/latex]“, means that the statement works in both directions: if [latex]ad=bc[/latex], then the two fractions, [latex]\frac{a}{b}[/latex] and [latex]\frac{c}{d}[/latex], are equivalent, and if the two fractions are equivalent, then [latex]ad=bc[/latex].  Using this method to determine whether two fractions are equivalent is called comparing cross products.

4. Compare Rational Numbers

Each rational number—that is, each fraction—has its own value, just like other numbers such as whole numbers and integers. Therefore, we can compare two fractions based on their values. However, comparing fractions is not as straightforward as comparing whole numbers or integers, since a fraction’s value depends on both its numerator and its denominator. Pedagogically, it is helpful to introduce fraction comparison in stages: fractions with like denominators, fractions with like numerators, and fractions with unlike numerators and unlike denominators. These stages provide opportunities for students to develop a solid conceptual understanding on denominator and numerator, as well as the idea of equivalent fractions.

Comparing Fractions with Like Denominators

When two positive fractions have the same denominator, that common denominator indicates that both fractions are made up of the same kind of parts–each coming from a whole of the same size, divided into the same number of equal parts. The numerator of each fraction tells us how many of those equal parts the fraction includes. Therefore, the one with more parts (a larger numerator) must represent a greater quantity.

For example, in [latex]\frac{3}{8}[/latex] and [latex]\frac{5}{8}[/latex], both fractions describe quantities made up of eighths. Since [latex]5[/latex] eighths is more than [latex]3[/latex] eighths, [latex]\frac{5}{8} > \frac{3}{8}[/latex].

In conclusion, among positive fractions with like denominators, the one with the larger numerator is greater. Mathematically, we express this as:

If [latex]a[/latex] is a non-zero whole number, [latex]b[/latex] and [latex]c[/latex] are whole numbers, then [latex]\frac{b}{a}>\frac{c}{a}[/latex] if, and only if, [latex]b>c[/latex].

Comparing Fractions with Like Numerators

In contrast, when two positive fractions have the same numerator, they represent the same number of parts—but the size of each part depends on the denominator. The denominator tells us how many equal parts the whole is divided into. Therefore, a smaller denominator means fewer, larger parts, while a larger denominator means more, smaller parts.

Since both fractions have the same number of parts (same numerator), the one with the larger parts (i.e., the smaller denominator) represents a greater quantity. For example, in [latex]\frac{3}{5}[/latex] and [latex]\frac{3}{8}[/latex], both fractions include [latex]3[/latex] parts. But fifths are larger than eighths, so [latex]\frac{3}{5} > \frac{3}{8}[/latex].

In conclusion, among positive fractions with like numerators, the one with the larger denominator is smaller. Mathematically, we express this as:

If [latex]a[/latex] is a non-zero whole number, [latex]b[/latex] and [latex]c[/latex] are whole numbers, then [latex]\frac{b}{a}<\frac{c}{a}[/latex] if, and only if, [latex]b>c[/latex].

[H5Ps for the two cases above]

Comparing Fractions with Unlike Denominators and Unlike  Numerators

When comparing two positive fractions that have different numerators and different denominators, we cannot determine which is greater by looking at the numerators or denominators alone. This is because the fractions are made up of different numbers of different-sized parts, so a direct comparison is not possible. There are two main strategies that can be used to compare them: using a common denominator and comparing cross products. At the end of this section, we use dot models to visualize the comparison.

Using a Common Denominator

By rewriting both fractions as equivalent fractions with the same denominator, we make their parts the same size. Then, as in the first case, we can compare the numerators directly.

The practice of using a common denominator to compare fractions helps students master the skill of writing equivalent fractions and deepens their understanding of the meaning of denominators and numerators. It also gives students opportunities to  practice clear and precise mathematical writing.

Comparing Cross Products

This method is also known as cross-multiplication. It avoids finding a common denominator explicitly and instead relies on comparing the cross products of the two fractions. Mathematically, we express this strategy as:

[latex]\frac{a}{b}\lt\frac{c}{d}\Longleftrightarrow ad\lt bc[/latex]

[latex]\frac{a}{b}>\frac{c}{d}\Longleftrightarrow ad>bc[/latex].

In a comparison like [latex]\frac{a}{b}<\frac{c}{d}[/latex], the first number ([latex]a[/latex]) and the last number ([latex]d[/latex]) are called the extremes. These are the outside numbers when the fractions are written side by side. The other two numbers, [latex]b[/latex] and [latex]c[/latex], are called the means, or the inside numbers. To compare the fractions, we multiply the extremes and the means  and compare the resulting products.

The two methods–using a common denominator and comparing cross products–may seem unrelated at first glance, but they are mathematically equivalent–both rely on expressing the fractions in a comparable form to determine which is greater.

Using Dot Models to Visualize Fraction Comparison

We are now going to work again with models, using dots, to compare two fractions. When you use dot models for each problem, it is CRUCIAL that you begin each problem by explicitly stating the following:

1. Be specific about what you are using for the unit.  It will be easiest if you use an array of dots, where the denominator of one fraction is the number of rows in the array, and the denominator of the other fraction is the number of columns in the array.
2. State the value of each dot, each column and each row.

Note how to choose the dot array according to the common denominator of the two fractions.

Use Manipulatives to Compare Fractions

[Here may design e-manipulatives for students working online.]

Manipulatives may be used to help students understand the comparison of fractions more solidly. Below is a set of activities that have the knowledge of comparing fractions embedded in.

Materials: Fraction Circles, Fraction Arrays, Multiple Strips, C-strips

Directions:

Use your marked fraction circles to do these first few activities.

1. Take one wedge from each fraction circle ([latex]\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6},\frac{1}{7},\frac{1}{8},\frac{1}{9},\frac{1}{10},[/latex], and [latex]\frac{1}{12},[/latex]).  One fraction is bigger than another if it covers more space 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 ( < ).

2. From what you discovered in exercise 1, circle the larger fraction in each case.

1.   [latex]\frac{1}{90} or \frac{1}{95}[/latex]                       b.   [latex]\frac{1}{32} or \frac{1}{33}[/latex]

If you take 5 of the fraction pieces each of which represents [latex]\frac{1}{8}[/latex], then together you have [latex]\frac{5}{8}[/latex].  Use this fact to do the next exercise.

3. Use the fraction circles to compare [latex]\frac{5}{8},\frac{5}{9},\frac{5}{10}, and \frac{5}{12}[/latex].  Put them in order from smallest to largest using the less than symbol ( < ).

4. From what you discovered in exercise 3, circle the larger fraction in each case.

1.   [latex]\frac{15}{37} or \frac{15}{40}[/latex]                         b.   [latex]\frac{89}{100} or \frac{89}{200}[/latex]

5. Use the fraction array to order these fractions: [latex]\frac{4}{5},\frac{4}{6},\frac{4}{7},\frac{4}{8},\frac{4}{9},\frac{4}{10},\frac{4}{11},and \frac{1}{12}[/latex].  Put them in order from smallest to largest using the less than symbol ( < ).

6. Explain the pattern you learned from doing exercises 1-5.

7. Use fraction circles to compare these fractions:[latex]\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},\frac{5}{6}\,\frac{7}{8},\frac{8}{9},\frac{9}{10},and \frac{11}{12}[/latex].  Put them in order from smallest to largest using the less than symbol ( < ).

8. Use the fraction array to order these fractions: [latex]\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},\frac{5}{6},\frac{7}{8},\frac{8}{9},\frac{9}{10}, and \frac{11}{12}[/latex]. Put them in order from smallest to largest using the less than symbol ( < ).

9. From what you discovered in exercises 7 and 8, circle the larger fraction in each case.

1. [latex]\frac{89}{90} or \frac{94}{95}[/latex]                        b.   [latex]\frac{56}{57} or \frac{31}{33}[/latex]

10. Order these fractions: 4546, 7172, 3435, 99100, 2526, 1314, 5152.  Put them in order from smallest to largest using the less than symbol ( < ).

11. Explain the pattern you learned from doing exercises 7 – 10.

12. Draw a model of your own that would help to convince someone why 45 was bigger than 23.