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. REPRESENTATIONS OF 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.

Note It is crucial to define your unit before beginning any exercise!!

Now we introduce the fundamental law of fractions, which is followed by several examples that we can solve using this fundamental law.

According to the fundamental law of fractions, if the denominator and the numerator of a fraction get changed to their multiples of the same factor, then the resulted fraction is equivalent to the original one. The skill of writing a fraction into its equivalent form, which is with a different denominator but has the same value of the fraction, is a critical skill when we do 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.

There are two ways to rewrite a fraction in its simplest form, that is, to simplify a fraction. One way to simplify fractions is to prime factor the numerator and denominator, and then “cancel out” any common factors. Another way is using the greatest common factor (GCF) of the numerator and denominator and following the steps (i) GCF of the numerator and denominator, (ii) rewrite both the numerator and denominator as the product of the GCF(a,b) and another factor, and then (iii) “cancel out” any common factors. When you cancel out everything from either the numerator and/or denominator, there is always a factor of 1 that is still there.

Example 6 Write [latex]\frac{315}{350}[/latex] in simplest form by using each of the two methods just described.

Solution

Method 1: Prime factorization:

[latex]\frac{315}{350}=\frac{3\times3\times5\times7}{2\times5\times5\times7}=\frac{3\times3}{2\times5}=\frac{9}{10}[/latex]

Method 2: GCF(315, 350) = 35:

[latex]\frac{315}{350}=\frac{9\times35}{10\times35}=\frac{9}{10}[/latex]

Exercise

[Note – More problems may be put here and they  may be designed as H5P interactive problems, instead of showing them all as the text of the section.]

22. Write each fraction in simplest form using each of the two methods: (1) prime factorization and (2) finding GCF as shown in the previous example. Show the actual factorization for each method, and then reduce to the simplest form.

1. [latex]\frac{378}{675}[/latex]                                 b. [latex]\frac{247}{323}[/latex]

Theorem II of Equivalent Fractions

Two fractions, [latex]\frac{a}{b}[/latex] and [latex]\frac{c}{d}[/latex], are equivalent if the two fractions are equal when they are both written in simplest form.

Example 7 Determine if the following statements are true or false.  State whether the pair of fractions are equivalent or not.

1. [latex]\frac{6}{8}=\frac{9}{12}[/latex]

Solution

In simplest form, [latex]\frac{6}{8}=\frac{3}{4}[/latex]  and [latex]\frac{9}{12}=\frac{3}{4}[/latex].  Therefore, [latex]\frac{6}{8}[/latex] and [latex]\frac{9}{12}[/latex] are equivalent.

1. [latex]\frac{15}{24}=\frac{10}{18}[/latex]

Solution

In simplest form, [latex]\frac{15}{24}=\frac{5}{8}[/latex], and [latex]\frac{10}{18}=\frac{5}{9}[/latex].  Therefore, [latex]\frac{15}{24}[/latex] and [latex]\frac{10}{18}[/latex] are not equivalent.

Exercise

[Note – More problems may be put here and they  may be designed as H5P interactive problems, instead of showing them all as the text of the section.]

23. Determine if the following statements are true or false by writing each fraction in simplest form.  State whether the pair of fractions are equivalent or not.  Show your work and reasoning.

1. [latex]\frac{14}{25}=\frac{28}{50}[/latex]

1. [latex]\frac{25}{35}=\frac{10}{14}[/latex]

1. [latex]\frac{12}{16}=\frac{25}{40}[/latex]

Theorem III of Equivalent Fractions

Assume a and c are whole numbers, and b and d are counting numbers. Two fractions, ab and cd, are equivalent if and only if ad=bc.  In other words,

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

Note The symbol means “if and only if”, so if ad=bc, then the two fractions ab and cd are equivalent, and vice versa.  Using this method to determine if two fractions are equivalent is called comparing cross products.

Example 8 Determine if the following statements are true or false by comparing cross products.  State whether the pair of fractions are equivalent or not.

1. [latex]\frac{6}{8}=\frac{9}{12}[/latex]

Solution

Since 6 · 12 = 8 · 9, the statement is true. Therefore, [latex]\frac{6}{8}[/latex] and [latex]\frac{9}{12}[/latex] are equivalent fractions.

1. [latex]\frac{15}{24}=\frac{10}{18}[/latex]

Solution

Since 15 · 18 ≠ 24 · 10, the statement is false. Therefore, [latex]\frac{15}{24}[/latex] and [latex]\frac{10}{18}[/latex] are not equivalent fractions.

Exercise

[Note – More problems may be put here and they  may be designed as H5P interactive problems, instead of showing them all as the text of the section.]

24. Determine if the following statements are true or false by comparing cross products.  State whether the pair of fractions are equivalent or not. Show your work and reasoning.

1. [latex]\frac{14}{25}=\frac{28}{50}[/latex]

1. [latex]\frac{25}{35}=\frac{10}{14}[/latex]

1. [latex]\frac{12}{16}=\frac{25}{40}[/latex]

4. Compare Rational Numbers

Each rational number, that is, fraction, has its own value, as other numbers (such as whole numbers and integers) do, so we can compare two fractions according to their values. However, the comparison of fractions is not as obvious as what we do to compare whole numbers or integers. This is because the values of fractions are determined by both numerators and denominators. Below is the summary of comparing positive fractions.

Comparing fractions with like denominators

Given two positive fractions with like denominators, the one with the greater numerator is the greater fraction. That is,

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].

Interactive H5P Quick-check Problems

[The problems in this category are for students to have a quick check on if they understand the concepts and/or skills.]

Comparing fractions with like numerators

Given two positive fractions with like numerators, the one with the greater denominator is the smaller fraction. That is,

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].

Interactive H5P Quick-check Problems

[The problems in this category are for students to have a quick check on if they understand the concepts and/or skills.]

Comparing fractions with unlike denominator and unlike numerators

Below is a way to compare two fractions that are unequal by using cross products.

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

As you write the fractions, [latex]\frac{a}{b}<\frac{c}{d}[/latex], [latex]a[/latex] is the first number written and [latex]d[/latex] is the last number written. [latex]a[/latex] and [latex]d[/latex] are called the [latex]extremes[/latex] (outside numbers when written as a/b[latex]<[/latex]c/d), and their product is written on the left of the inequality sign when you take the cross products.  The other two numbers, [latex]b[/latex] and [latex]c[/latex], are called the [latex]means[/latex] (when written as  a/b[latex]<[/latex]c/d, these are the inside numbers), and their product is written on the right of the inequality sign when you take the cross products. There are only three cases possible when comparing two fractions, [latex]\frac{a}{b}[/latex] and [latex]\frac{c}{d}[/latex].  Either the first fraction ([latex]\frac{a}{b}[/latex]) is equal to the second ([latex]\frac{c}{d}[/latex]), in which case [latex]ad=bc[/latex]; or the first ([latex]\frac{a}{b}[/latex]) is less than the second ([latex]\frac{c}{d}[/latex]), in which case ad[latex]<[/latex]bc; or the first ([latex]\frac{a}{b}[/latex]) is more than the second ([latex]\frac{c}{d}[/latex]), in which case ad[latex]>[/latex]bc.

Example 9 Compare [latex]\frac{2}{5}[/latex] and [latex]\frac{3}{7}[/latex] using cross products.

Solution

Multiply the extremes (2 and 7) and put on the left. Multiply the means (5 and 3) and put on the right. Then compare the products:

Since [latex]2·7[/latex] [latex]<[/latex] [latex]5 · 3[/latex], then [latex]\frac{2}{5}[/latex] [latex]<[/latex] [latex]\frac{3}{7}[/latex].   Example 10 Compare [latex]\frac{6}{7}[/latex] and [latex]\frac{5}{6}[/latex] using cross products.   Solution Multiply the outside numbers (6 and 6) and put on the left.  Multiply the inside numbers (7 and 5) and put on the right. Then compare the products.   [latex]Since 6\cdot6 >7\cdot, then \frac{6}{7}>\frac{5}{6}[/latex]

Exercise

[Note – More problems may be put here and they  may be designed as H5P interactive problems, instead of showing them all as the text of the section.]^[1]

Use cross products to compare each of the following fractions.  Use < or >.

1. [latex]\frac{4}{5} and \frac{5}{8}[/latex]
2. [latex]\frac{12}{35} and \frac{11}{18}[/latex]
3. [latex]\frac{13}{15} and \frac{14}{17}[/latex]

Use Dot Models to Compare Fractions

We are now going to work again with models, using dots, to compare two fractions. 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.

Example 11 Use models to compare [latex]\frac{2}{5}[/latex] and [latex]\frac{3}{7}[/latex].

Solution

Step 1: Let 1 unit = 5 rows of dots by 7 columns of dots, for a total of 35 dots, as shown below. Since there are 35 dots to a unit, each dot = [latex]\frac{1}{35}[/latex] of a unit.

Step 2: Since there are 5 rows, the unit can be broken up into 5 equal parts by circling the rows, as shown below:

Therefore, each row is [latex]\frac{1}{5}[/latex] of a unit. Notice there are 7 dots in [latex]\frac{1}{5}[/latex] of a unit:

[latex]\frac{1}{5}=[/latex]

Step 3: Similarly, since there are 7 equal columns, each column is [latex]\frac{1}{7}[/latex] of a unit:

[latex]\frac{1}{7}[/latex] =

Step 4: Now that we have properly defined the unit, we are ready to show what [latex]\frac{2}{5}[/latex] looks like. Since [latex]\frac{1}{5}[/latex] is 1 row of dots, then [latex]\frac{2}{5}[/latex]must be 2 rows of dots.  Therefore, [latex]\frac{2}{5}[/latex] is shown below:

Notice that [latex]\frac{2}{5}[/latex] contains 14 dots, which is equivalent to [latex]\frac{14}{35}[/latex].

Step 5: Similarly, we can show what [latex]\frac{3}{5}[/latex] looks like.  Since [latex]\frac{1}{7}[/latex] is 1 column of dots, then [latex]\frac{3}{7}[/latex]must be 3 columns of dots.  Therefore, [latex]\frac{3}{7}[/latex] is shown below:

Notice that [latex]\frac{3}{7}[/latex] contains 15 dots, which is equivalent to [latex]\frac{15}{35}[/latex].

Since [latex]\frac{2}{5}[/latex] contains less dots than [latex]\frac{3}{7}[/latex],  must be less than [latex]\frac{3}{7}[/latex].  That is, [latex]\frac{2}{5}[/latex] < [latex]\frac{3}{7}[/latex].

Exercise

[Note – More problems may be put here and they  may be designed as H5P interactive problems, instead of showing them all as the text of the section.]

29. Compare [latex]\frac{3}{4}[/latex] and [latex]\frac{4}{5}[/latex] using models.  Show all of the steps, and explain the procedure as shown in the previous example.

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.