4.1 Visualize Fractions

Learning Objectives

By the end of this section, you will be able to:

  • Understand the meaning of fractions
  • Model improper fractions and mixed numbers
  • Convert between improper fractions and mixed numbers
  • Model equivalent fractions
  • Find equivalent fractions
  • Locate fractions and mixed numbers on the number line
  • Order fractions and mixed numbers

Understand the Meaning of Fractions

Andy and Bobby love pizza. On Monday night, they share a pizza equally. How much of the pizza does each one get? Are you thinking that each boy gets half of the pizza? That’s right. There is one whole pizza, evenly divided into two parts, so each boy gets one of the two equal parts.

In math, we write 1/2 to mean one out of two parts.

An image of a round pizza sliced vertically down the center, creating two equal pieces. Each piece is labeled as one half.

On Tuesday, Andy and Bobby share a pizza with their parents, Fred and Christy, with each person getting an equal amount of the whole pizza. How much of the pizza does each person get? There is one whole pizza, divided evenly into four equal parts. Each person has one of the four equal parts, so each has 1/4 of the pizza.

An image of a round pizza sliced vertically and horizontally, creating four equal pieces. Each piece is labeled as one fourth.

On Wednesday, the family invites some friends over for a pizza dinner. There are a total of 12 people. If they share the pizza equally, each person would get 1/12of the pizza.

An image of a round pizza sliced into twelve equal wedges. Each piece is labeled as one twelfth.

A fraction is a way to represent parts of a whole. The denominator b represents the number of equal parts the whole has been divided into, and the numerator a represents how many parts are included. The denominator, b, cannot equal zero because division by zero is undefined.

In Figure 4.2, the circle has been divided into three parts of equal size. Each part represents 13 of the circle. This type of model is called a fraction circle. Other shapes, such as rectangles, can also be used to model fractions.

A circle is divided into three equal wedges. Each piece is labeled as one third.

Figure 4.2

What does the fraction 23 represent? The fraction 23 means two of three equal parts.

A circle is divided into three equal wedges. Two of the wedges are shaded.

In Example 4.1 and Example 4.2, we used circles and rectangles to model fractions. Fractions can also be modeled as manipulatives called fraction tiles, as shown in Figure 4.3. Here, the whole is modeled as one long, undivided rectangular tile. Beneath it are tiles of equal length divided into different numbers of equally sized parts.

One long, undivided rectangular tile is shown, labeled “1”. Below it is a rectangular tile of the same size and shape that has been divided vertically into two equal pieces, each labeled as one half. Below that is another rectangular tile that has been divided into three equal pieces, each labeled as one third. Below that is another rectangular tile that has been divided into four equal pieces, each labeled as one fourth. Below that is another rectangular tile that has been divided into six pieces, each labeled as one sixth.

Figure 4.3

We’ll be using fraction tiles to discover some basic facts about fractions. Refer to Figure 4.3 to answer the following questions:

How many 1/2 tiles does it take to make one whole tile?It takes two halves to make a whole, so two out of two is
2/2
=1.
How many 1/3 tiles does it take to make one whole tile?It takes three thirds, so three out of three is 3/3=1.
How many 1/4 tiles does it take to make one whole tile?It takes four fourths, so four out of four is 44=1.
How many 1/6 tiles does it take to make one whole tile?It takes six sixths, so six out of six is 6/6=1.
What if the whole were divided into 2/4 equal parts? (We have not shown fraction tiles to represent this, but try to visualize it in your mind.) How many 1/24 tiles does it take to make one whole tile?It takes 24 twenty-fourths, so 24/24=1.

What if we have more fraction pieces than we need for 1 whole? We’ll look at this in the next example.

Model Improper Fractions and Mixed Numbers

In Example 4.4 (b), you had eight equal fifth pieces. You used five of them to make one whole, and you had three fifths left over. Let us use fraction notation to show what happened. You had eight pieces, each of them one fifth, 1/5, so altogether you had eight fifths, which we can write as 8/5. The fraction 8/5 is one whole, 1, plus three fifths, 3/5, or 1 3/5, which is read as one and three-fifths.

The number 1/35 is called a mixed number. A mixed number consists of a whole number and a fraction.


Convert between Improper Fractions and Mixed Numbers

Model Equivalent Fractions

Find Equivalent Fractions

Two pizzas are shown. The pizza on the left is divided into 2 equal pieces. 1 piece is shaded. The pizza on the right is divided into 8 equal pieces. 4 pieces are shaded.

Figure 4.4

This is another way to show that 1212 is equivalent to 48.48.

How can we use mathematics to change 12 into 48? How could you take a pizza that is cut into two pieces and cut it into eight pieces? You could cut each of the two larger pieces into four smaller pieces! The whole pizza would then be cut into eight pieces instead of just two. Mathematically, what we’ve described could be written as:

1 times 4 over 2 times 4 is written with the 4s in red. This is set equal to 4 over 8.

These models lead to the Equivalent Fractions Property, which states that if we multiply the numerator and denominator of a fraction by the same number, the value of the fraction does not change.

When working with fractions, it is often necessary to express the same fraction in different forms. To find equivalent forms of a fraction, we can use the Equivalent Fractions Property. For example, consider the fraction one-half.

The top line says that 1 times 3 over 2 times 3 equals 3 over 6, so one half equals 3 sixths. The next line says that 1 times 2 over 2 times 2 equals 2 over 4, so one half equals 2 fourths. The last line says that 1 times 10 over 2 times 10 equals 10 over 20, so one half equals 10 twentieths.

Locate Fractions and Mixed Numbers on the Number Line

Now we are ready to plot fractions on a number line. This will help us visualize fractions and understand their values.

In Introduction to Integers, we defined the opposite of a number. It is the number that is the same distance from zero on the number line but on the opposite side of zero. We saw, for example, that the opposite of 77 is −7  and the opposite of −7 7.

A number line is shown. It shows the numbers negative 7, 0 and 7. There are red dots at negative 7 and 7. The space between negative 7 and 0 is labeled as 7 units. The space between 0 and 7 is labeled as 7 units.

Fractions have opposites, too. The opposite of 34 is −34 . It is the same distance from 00 on the number line, but on the opposite side of 0.

A number line is shown. It shows the numbers negative 1, negative 3 fourths, 0, 3 fourths, and 1. There are red dots at negative 3 fourths and 3 fourths. The space between negative 3 fourths and 0 is labeled as 3 fourths of a unit. The space between 0 and 3 fourths is labeled as 3 fourths of a unit.

Thinking of negative fractions as the opposite of positive fractions will help us locate them on the number line. To locate −158 on the number line, first think of where 158 is located. It is an improper fraction, so we first convert it to the mixed number 1 7/8 and see that it will be between 1 and 2 on the number line. So its opposite, −15/8, will be between −1 and −2 on the number line.A number line is shown. It shows the numbers negative 2, negative 1, 0, 1, and 2. Between negative 2 and negative 1, negative 1 and 7 eighths is labeled and marked with a red dot. The distance between negative 1 and 7 eighths and 0 is marked as 15 eighths units. Between 1 and 2, 1 and 7 eighths is labeled and marked with a red dot. The distance between 0 and 1 and 7 eighths is marked as 15 eighths units.

Order Fractions and Mixed Numbers

We can use the inequality symbols to order fractions. Remember that a>ba>b means that aa is to the right of bb on the number line. As we move from left to right on a number line, the values increase.

Section 4.1 Exercises

Practice Makes Perfect

In the following exercises, name the fraction of each figure that is shaded.1.

In part “a”, a circle is divided into 4 equal pieces. 1 piece is shaded. In part “b”, a circle is divided into 4 equal pieces. 3 pieces are shaded. In part “c”, a circle is divided into 8 equal pieces. 3 pieces are shaded. In part “d”, a circle is divided into 8 equal pieces. 5 pieces are shaded.

2.

In part “a”, a circle is divided into 12 equal pieces. 7 pieces are shaded. In part “b”, a circle is divided into 12 equal pieces. 5 pieces are shaded. In part “c”, a square is divided into 9 equal pieces. 4 of the pieces are shaded. In part “d”, a square is divided into 9 equal pieces. 5 pieces are shaded.

In the following exercises, shade parts of circles or squares to model the following fractions.

  1. 12
  2. 13
  3. 34
  4. 25
  5. 56
  6. 78
  7. 58
  8. 710
    In the following exercises, use fraction circles to make wholes using the following pieces.
  9. 3 thirds
  10. 8 eighths
  11. 7 sixths
  12. 4 thirds
  13. 7 fifths
  14. 7 fourths
    In the following exercises, name the improper fractions. Then write each improper fraction as a mixed number.

17.

In part “a”, two circles are shown. Each is divided into 4 equal pieces. The circle on the left has all 4 pieces shaded. The circle on the right has 1 piece shaded. In part “b”, two circles are shown. Each is divided into 4 equal pieces. The circle on the left has all 4 pieces shaded. The circle on the right has 3 pieces shaded. In part “c”, two circles are shown. Each is divided into 8 equal pieces. The circle on the left has all 8 pieces shaded. The circle on the right has 3 pieces shaded.

18.

In part “a”, 2 circles are shown. Each is divided into 8 equal pieces. The circle on the left has all 8 pieces shaded. The circle on the right has 1 piece shaded. In part “b”, two squares are shown. Each is divided into 4 equal pieces. The square on the left has all 4 pieces shaded. The circle on the right has 1 piece shaded. In part “c”, two squares are shown. Each is divided into 9 equal pieces. The square on the left has all 9 pieces shaded. The square on the right has 2 pieces shaded.

19.

In part “a”, 3 circles are shown. Each is divided into 4 equal pieces. The first two circles have all 4 pieces shaded. The third circle has 3 pieces shaded. In part “b”, 3 circles are shown. Each is divided into 8 equal pieces. The first two circles have all 8 pieces shaded. The third circle has 3 pieces shaded.

In the following exercises, draw fraction circles to model the given fraction.

  1. 33
  2. 44
  3. 74
  4. 53
  5. 116
  6. 138
  7. 103
  8. 94
    In the following exercises, rewrite the improper fraction as a mixed number.
  9. 32
  10. 53
  11. 114
  12. 135
  13. 256
  14. 289
  15. 4213
  16. 4715
    In the following exercises, rewrite the mixed number as an improper fraction.
  17. 123
  18. 125
  19. 214
  20. 256
  21. 279
  22. 257
  23. 347
  24. 359
    In the following exercises, use fraction tiles or draw a figure to find equivalent fractions.
  25. How many sixths equal one-third?
  26. How many twelfths equal one-third?
  27. How many eighths equal three-fourths?
  28. How many twelfths equal three-fourths?
  29. How many fourths equal three-halves?
  30. How many sixths equal three-halves?
    In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.
  31. 14
  32. 13
  33. 38
  34. 56
  35. 27
  36. 59
    In the following exercises, plot the numbers on a number line.
  37. 23,54,125
  38. 13,74,135
  39. 14,95,113
  40. 710,52,138,3
  41. 213,−213
  42. 134,−135
  43. 34,−34,123,−123,52,−52
  44. 25,−25,134,−134,83,−83
    In the following exercises, order each of the following pairs of numbers, using < or >.
  45. −1__−14
  46. −1__−13
  47. −212__−3
  48. −134__−2
  49. −512__−712
  50. −910__−310
  51. −3__−135
  52. −4__−236

Everyday Math

  1. Music Measures A choreographed dance is broken into counts. A 11 count has one step in a count, a 12 count has two steps in a count and a 13 count has three steps in a count. How many steps would be in a 15 count? What type of count has four steps in it?
  2. Music Measures Fractions are used often in music. In 44 time, there are four quarter notes in one measure.
    ⓐ How many measures would eight quarter notes make?
    ⓑ The song “Happy Birthday to You” has 25 quarter notes. How many measures are there in “Happy Birthday to You?”
  3. Baking Nina is making five pans of fudge to serve after a music recital. For each pan, she needs 12 cup of walnuts.
    ⓐ How many cups of walnuts does she need for five pans of fudge?
    ⓑ Do you think it is easier to measure this amount when you use an improper fraction or a mixed number? Why?

Writing Exercises

75

Give an example from your life experienc (outside of school) where it was important to understand fractions.76. 

Explain how you locate the improper fraction 21/4 on a number line on which only the whole numbers from 00 through 1010 are marked.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.