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Guides d'étude > Prealgebra

Using Models to Represent Fractions and Mixed Numbers

Learning Outcomes

  • Write fractions that represent portions of objects
  • Use fraction circles to make wholes given
  • Use models to visualize improper fractions and mixed numbers.
 

Representing Parts of a Whole as 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 12\frac{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 14\frac{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 1212 people. If they share the pizza equally, each person would get 112\frac{1}{12} of the pizza. An image of a round pizza sliced into twelve equal wedges. Each piece is labeled as one twelfth.

Fractions

A fraction is written ab\frac{a}{b}, where aa and bb are integers and b0b\ne 0. In a fraction, aa is called the numerator and bb is called the denominator.
A fraction is a way to represent parts of a whole. The denominator bb represents the number of equal parts the whole has been divided into, and the numerator aa represents how many parts are included. The denominator, bb, cannot equal zero because division by zero is undefined. In the image below, the circle has been divided into three parts of equal size. Each part represents 13\frac{1}{3} 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. Doing the Manipulative Mathematics activity Model Fractions will help you develop a better understanding of fractions, their numerators and denominators. What does the fraction 23\frac{2}{3} represent? The fraction 23\frac{2}{3} means two of three equal parts. A circle is divided into three equal wedges. Two of the wedges are shaded.

Example

Name the fraction of the shape that is shaded in each of the figures. In part Solution: We need to ask two questions. First, how many equal parts are there? This will be the denominator. Second, of these equal parts, how many are shaded? This will be the numerator. How many equal parts are there?There are eight equal parts.How many are shaded?Five parts are shaded.\begin{array}{cccc}\text{How many equal parts are there?}\hfill & & & \text{There are eight equal parts}\text{.}\hfill \\ \text{How many are shaded?}\hfill & & & \text{Five parts are shaded}\text{.}\hfill \end{array} Five out of eight parts are shaded. Therefore, the fraction of the circle that is shaded is 58\frac{5}{8}. How many equal parts are there?There are nine equal parts.How many are shaded?Two parts are shaded.\begin{array}{cccc}\text{How many equal parts are there?}\hfill & & & \text{There are nine equal parts}\text{.}\hfill \\ \text{How many are shaded?}\hfill & & & \text{Two parts are shaded}\text{.}\hfill \end{array} Two out of nine parts are shaded. Therefore, the fraction of the square that is shaded is 29\frac{2}{9}.

Example

Shade 34\frac{3}{4} of the circle. An image of a circle.

Answer: Solution The denominator is 44, so we divide the circle into four equal parts ⓐ. The numerator is 33, so we shade three of the four parts ⓑ. In 34\frac{3}{4} of the circle is shaded.

 

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Shade 68\frac{6}{8} of the circle. A circle is divided into eight equal pieces.

Answer: A circle is shown divided into 8 pieces, of which 6 are shaded.

  Shade 25\frac{2}{5} of the rectangle. A rectangle is divided vertically into five equal pieces.

Answer: A rectangle is divided into 5 sections, of which 2 are shaded.

Watch the following video to see more examples of how to write fractions given a model. https://youtu.be/c_yIA4OQ4qA   In earlier examples, we used circles and rectangles to model fractions. Fractions can also be modeled as manipulatives called fraction tiles, as shown in the image below. 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 We’ll be using fraction tiles to discover some basic facts about fractions. Refer to the fraction tiles above to answer the following questions:
How many 12\frac{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 22=1\frac{2}{2}=1.
How many 13\frac{1}{3} tiles does it take to make one whole tile? It takes three thirds, so three out of three is 33=1\frac{3}{3}=1.
How many 14\frac{1}{4} tiles does it take to make one whole tile? It takes four fourths, so four out of four is 44=1\frac{4}{4}=1.
How many 16\frac{1}{6} tiles does it take to make one whole tile? It takes six sixths, so six out of six is 66=1\frac{6}{6}=1.
What if the whole were divided into 2424 equal parts? (We have not shown fraction tiles to represent this, but try to visualize it in your mind.) How many 124\frac{1}{24} tiles does it take to make one whole tile? It takes 2424 twenty-fourths, so 2424=1\frac{24}{24}=1.
It takes 2424 twenty-fourths, so 2424=1\frac{24}{24}=1. This leads us to the Property of One.

Property of One

Any number, except zero, divided by itself is one. aa=1(a0)\frac{a}{a}=1\left(a\ne 0\right)
Doing the Manipulative Mathematics activity "Fractions Equivalent to One" will help you develop a better understanding of fractions that are equivalent to one

Example

Use fraction circles to make wholes using the following pieces:
  1. 44 fourths
  2. 55 fifths
  3. 66 sixths

Answer: Solution Three circles are shown. The circle on the left is divided into four equal pieces. The circle in the middle is divided into five equal pieces. The circle on the right is divided into six equal pieces. Each circle says

Try it

Use fraction circles to make wholes with the following pieces: 33 thirds.

Answer: A circle is shown. It is divided into 3 equal pieces. All 3 pieces are shaded.

  Use fraction circles to make wholes with the following pieces: 88 eighths.

Answer: A circle is divided into 8 sections, of which all are shaded.

    What if we have more fraction pieces than we need for 11 whole? We’ll look at this in the next example.

Example

Use fraction circles to make wholes using the following pieces:
  1. 33 halves
  2. 88 fifths
  3. 77 thirds

Answer: Solution 1. 33 halves make 11 whole with 11 half left over. Two circles are shown, both divided into two equal pieces. The circle on the left has both pieces shaded and is labeled as 2. 88 fifths make 11 whole with 22 fifths left over. Two circles are shown, both divided into five equal pieces. The circle on the left has all five pieces shaded and is labeled as 3. 77 thirds make 22 wholes with 22 thirds left over. Three circles are shown, all divided into three equal pieces. The two circles on the left have all three pieces shaded and are labeled with ones. The circle on the right has one piece shaded and is labeled as one third.

 

try it

Use fraction circles to make wholes with the following pieces: 55 thirds.

Answer: Two circles are shown. Each is divided into three sections. All of the first circle is shaded. 2 out of 3 sections of the second circle are shaded.

  Use fraction circles to make wholes with the following pieces: 55 halves.

Answer: Three circles are shown. Each is divided into two sections. The first two circles are completely shaded. Half of the third circle is shaded.

 

Model Improper Fractions and Mixed Numbers

In an earlier example, 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, 15\frac{1}{5}, so altogether you had eight fifths, which we can write as 85\frac{8}{5}. The fraction 85\frac{8}{5} is one whole, 11, plus three fifths, 35\frac{3}{5}, or 1351\frac{3}{5}, which is read as one and three-fifths. The number 1351\frac{3}{5} is called a mixed number. A mixed number consists of a whole number and a fraction.

Mixed Numbers

A mixed number consists of a whole number aa and a fraction bc\frac{b}{c} where c0c\ne 0. It is written as follows. abcc0a\frac{b}{c}\text{, }c\ne 0
Fractions such as 54,32,55\frac{5}{4},\frac{3}{2},\frac{5}{5}, and 73\frac{7}{3} are called improper fractions. In an improper fraction, the numerator is greater than or equal to the denominator, so its value is greater than or equal to one. When a fraction has a numerator that is smaller than the denominator, it is called a proper fraction, and its value is less than one. Fractions such as 12,37\frac{1}{2},\frac{3}{7}, and 1118\frac{11}{18} are proper fractions.

Proper and Improper Fractions

The fraction ab\frac{a}{b} is a proper fraction if a<ba<b and an improper fraction if aba\ge b.
Doing the Manipulative Mathematics activity "Model Improper Fractions" and "Mixed Numbers" will help you develop a better understanding of how to convert between improper fractions and mixed numbers.

Example

Name the improper fraction modeled. Then write the improper fraction as a mixed number. Two circles are shown, both divided into three equal pieces. The circle on the left has all three pieces shaded. The circle on the right has one piece shaded. Solution: Each circle is divided into three pieces, so each piece is 13\frac{1}{3} of the circle. There are four pieces shaded, so there are four thirds or 43\frac{4}{3}. The figure shows that we also have one whole circle and one third, which is 1131\frac{1}{3}. So, 43=113\frac{4}{3}=1\frac{1}{3}.
 

try it

[ohm_question]145976[/ohm_question] [ohm_question]145977[/ohm_question]
 

Example

Draw a figure to model 118\frac{11}{8}.

Answer: Solution: The denominator of the improper fraction is 88. Draw a circle divided into eight pieces and shade all of them. This takes care of eight eighths, but we have 1111 eighths. We must shade three of the eight parts of another circle. Two circles are shown, both divided into eight equal pieces. The circle on the left has all eight pieces shaded and is labeled as eight eighths. The circle on the right has three pieces shaded and is labeled as three eighths. The diagram indicates that eight eighths plus three eighths is one plus three eighths. So, 118=138\frac{11}{8}=1\frac{3}{8}.

 

Try it

Draw a figure to model 76\frac{7}{6}.

Answer: Two circles are shown. Each is divided into six sections. All of the first circle is shaded and one section of the second circle is shaded.

  Draw a figure to model 65\frac{6}{5}.

Answer: Two circles are shown. Each is divided into five sections. All of the first circle is shaded and one section of the second circle is shaded.

 

Example

Use a model to rewrite the improper fraction 116\frac{11}{6} as a mixed number.

Answer: Solution: We start with 1111 sixths (116)\left(\frac{11}{6}\right). We know that six sixths makes one whole. 66=1\frac{6}{6}=1 That leaves us with five more sixths, which is 56(11sixths minus6sixths is5sixths)\frac{5}{6}\left(11\text{sixths minus}6\text{sixths is}5\text{sixths}\right). So, 116=156\frac{11}{6}=1\frac{5}{6}. Two circles are shown, both divided into six equal pieces. The circle on the left has all six pieces shaded and is labeled as six sixths. The circle on the right has five pieces shaded and is labeled as five sixths. Below the circles, it says one plus five sixths, then six sixths plus five sixths equals eleven sixths, and one plus five sixths equals one and five sixths. It then says that eleven sixths equals one and five sixths.

 

Try it

[ohm_question]145982[/ohm_question]
In the next video we show another way to draw a model that represents a fraction.  You will see example of both proper and improper fractions shown. https://youtu.be/akyByv80Uzc

Example

Use a model to rewrite the mixed number 1451\frac{4}{5} as an improper fraction.

Answer: Solution: The mixed number 1451\frac{4}{5} means one whole plus four fifths. The denominator is 55, so the whole is 55\frac{5}{5}. Together five fifths and four fifths equals nine fifths. So, 145=951\frac{4}{5}=\frac{9}{5}. Two circles are shown, both divided into five equal pieces. The circle on the left has all five pieces shaded and is labeled as 5 fifths. The circle on the right has four pieces shaded and is labeled as 4 fifths. It then says that 5 fifths plus 4 fifths equals 9 fifths and that 9 fifths is equal to one plus 4 fifths.

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[ohm_question]145981[/ohm_question]

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