1.2: Visualize Fractions
- Page ID
- 152022
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- Understand the meaning of fractions
- Model improper fractions and mixed numbers
- Convert between improper fractions and mixed numbers
- Model equivalent fractions
- Find equivalent fractions
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 to mean one out of two parts.
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 of the pizza.
On Wednesday, the family invites some friends over for a pizza dinner. There are a total of people. If they share the pizza equally, each person would get of the pizza.
Fractions
A fraction is written where and are integers and In a fraction, is called the numerator and is called the denominator.
A fraction is a way to represent parts of a whole. The denominator, represents the number of equal parts the whole has been divided into, and the numerator, , represents how many parts are included. The denominator, cannot equal zero because division by zero is undefined.
In Figure 1.2.1, the circle has been divided into three parts of equal size. Each part represents of the circle. This type of model is called a fraction circle. Other shapes, such as rectangles, can also be used to model fractions.
What does the fraction represent? The fraction means two of three equal parts.
Example 1.2.1
Name the fraction of the shape that is shaded in each of the figures.
- Answer
-
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.
ⓐ
Five out of eight parts are shaded. Therefore, the fraction of the circle that is shaded is
ⓑ
Two out of nine parts are shaded. Therefore, the fraction of the square that is shaded is
Your Turn 1.2.1
Name the fraction of the shape that is shaded in each figure:
Example 1.2.2
Shade of the circle.
- Answer
-
The denominator is so we divide the circle into four equal parts ⓐ.
The numerator is so we shade three of the four parts ⓑ.
of the circle is shaded.
Your Turn 1.2.2
Shade of the circle.
In Example 1.2.1 and Example 1.2.2, we used circles and rectangles to model fractions. Fractions can also be modeled as manipulatives called fraction tiles, as shown in Figure 1.2.2. 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.
We’ll be using fraction tiles to discover some basic facts about fractions. Refer to Figure 1.2.2 to answer the following questions:
How many tiles does it take to make one whole tile? | It takes two halves to make a whole, so two out of two is |
How many tiles does it take to make one whole tile? | It takes three thirds, so three out of three is |
How many tiles does it take to make one whole tile? | It takes four fourths, so four out of four is |
How many tiles does it take to make one whole tile? | It takes six sixths, so six out of six is |
What if the whole were divided into equal parts? (We have not shown fraction tiles to represent this, but try to visualize it in your mind.) How many tiles does it take to make one whole tile? | It takes twenty-fourths, so |
This leads us to the Property of One.
Property of One
Any number, except zero, divided by itself is one.
What if we have more fraction pieces than we need for whole? We’ll look at this in the next example.
Example 1.2.3
Use fraction circles to make wholes using the following pieces:
- ⓐ halves
- ⓑ fifths
- ⓒ thirds
- Answer
-
ⓐ halves make whole with half left over.
ⓑ fifths make whole with fifths left over.
ⓒ thirds make wholes with third left over.
Your Turn 1.2.3
Use fraction circles to make wholes with the following pieces: thirds.
Model Improper Fractions and Mixed Numbers
In Example 1.2.3 (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, so altogether you had eight fifths, which we can write as The fraction is one whole, plus three fifths, or which is read as one and three-fifths.
The number 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 and a fraction
Fractions such as and 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 and are proper fractions.
Proper and Improper Fractions
The fraction
Example 1.2.4
Name the improper fraction modeled. Then write the improper fraction as a mixed number.
- Answer
-
Each circle is divided into three pieces, so each piece is of the circle. There are four pieces shaded, so there are four thirds or The figure shows that we also have one whole circle and one third, which is So,
Your Turn 1.2.4
Name the improper fraction. Then write it as a mixed number.
Example 1.2.5
Draw a figure to model
- Answer
-
The denominator of the improper fraction is Draw a circle divided into eight pieces and shade all of them. This takes care of eight eighths, but we have eighths. We must shade three of the eight parts of another circle.
So,
Your Turn 1.2.5
Draw a figure to model
Example 1.2.6
Use a model to rewrite the improper fraction as a mixed number.
- Answer
-
We start with sixths We know that six sixths makes one whole.
That leaves us with five more sixths, which is
So,
Your Turn 1.2.6
Use a model to rewrite the improper fraction as a mixed number:
Example 1.2.7
Use a model to rewrite the mixed number as an improper fraction.
- Answer
-
The mixed number means one whole plus four fifths. The denominator is so the whole is Together five fifths and four fifths equals nine fifths.
So,
Your Turn 1.2.7
Use a model to rewrite the mixed number as an improper fraction:
Convert between Improper Fractions and Mixed Numbers
In Example 1.2.6, we converted the improper fraction to the mixed number using fraction circles. We did this by grouping six sixths together to make a whole; then we looked to see how many of the pieces were left. We saw that made one whole group of six sixths plus five more sixths, showing that
The division expression (which can also be written as ) tells us to find how many groups of are in To convert an improper fraction to a mixed number without fraction circles, we divide.
Example 1.2.8
Convert to a mixed number.
- Answer
-
Divide the denominator into the numerator. Remember means . Identify the quotient, remainder and divisor. Write the mixed number as quotient remainder divisor .remainder divisor So,
Your Turn 1.2.8
Convert the improper fraction to a mixed number:
How To
Convert an improper fraction to a mixed number.
- Divide the denominator into the numerator.
- Identify the quotient, remainder, and divisor.
- Write the mixed number as quotient
remainder divisor .remainder divisor
Example 1.2.9
Convert the improper fraction to a mixed number.
- Answer
-
Divide the denominator into the numerator. Remember, means . Identify the quotient, remainder, and divisor. Write the mixed number as quotient remainder divisor .remainder divisor So,
Your Turn 1.2.9
Convert the improper fraction to a mixed number:
In Example 1.2.7, we changed to an improper fraction by first seeing that the whole is a set of five fifths. So we had five fifths and four more fifths.
Where did the nine come from? There are nine fifths—one whole (five fifths) plus four fifths. Let us use this idea to see how to convert a mixed number to an improper fraction.
Example 1.2.10
Convert the mixed number to an improper fraction.
- Answer
-
Multiply the whole number by the denominator. The whole number is 4 and the denominator is 3. Simplify. Add the numerator to the product. The numerator of the mixed number is 2. Simplify. Write the final sum over the original denominator. The denominator is 3.
Your Turn 1.2.10
Convert the mixed number to an improper fraction:
How To
Convert a mixed number to an improper fraction.
- Multiply the whole number by the denominator.
- Add the numerator to the product found in Step 1.
- Write the final sum over the original denominator.
Example 1.2.11
Convert the mixed number to an improper fraction.
- Answer
-
Multiply the whole number by the denominator. The whole number is 10 and the denominator is 7. Simplify. Add the numerator to the product. The numerator of the mixed number is 2. Simplify. Write the final sum over the original denominator. The denominator is 7.
Your Turn 1.2.11
Convert the mixed number to an improper fraction:
Model Equivalent Fractions
Let’s think about Andy and Bobby and their favorite food again. If Andy eats of a pizza and Bobby eats of the pizza, have they eaten the same amount of pizza? In other words, does We can use fraction tiles to find out whether Andy and Bobby have eaten equivalent parts of the pizza.
Equivalent Fractions
Equivalent fractions are fractions that have the same value.
Fraction tiles serve as a useful model of equivalent fractions. You may want to use fraction tiles to do the following activity. Or you might make a copy of Figure 1.2.2 and extend it to include eighths, tenths, and twelfths.
Start with a tile. How many fourths equal one-half? How many of the tiles exactly cover the tile?
Since two tiles cover the tile, we see that is the same as or
How many of the tiles cover the tile?
Since three tiles cover the tile, we see that is the same as
So, The fractions are equivalent fractions.
Example 1.2.12
Use fraction tiles to find equivalent fractions. Show your result with a figure.
- ⓐ How many eighths equal one-half?
- ⓑ How many tenths equal one-half?
- ⓒ How many twelfths equal one-half?
- Answer
-
ⓐ It takes four tiles to exactly cover the tile, so
ⓑ It takes five tiles to exactly cover the tile, so
ⓒ It takes six tiles to exactly cover the tile, so
Suppose you had tiles marked How many of them would it take to equal Are you thinking ten tiles? If you are, you’re right, because
We have shown that and are all equivalent fractions.
Your Turn 1.2.12
Use fraction tiles to find equivalent fractions: How many eighths equal one-fourth?
Find Equivalent Fractions
We used fraction tiles to show that there are many fractions equivalent to
We can show this with pizzas, too. Figure 1.2.3(a) shows a single pizza, cut into two equal pieces with shaded. Figure 1.2.3(b) shows a second pizza of the same size, cut into eight pieces with shaded.
This is another way to show that is equivalent to
How can we use mathematics to change into 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:
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.
Equivalent Fractions Property
If and are numbers where and then
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.
So, we say that and are equivalent fractions.
Example 1.2.13
Find three fractions equivalent to
- Answer
-
To find a fraction equivalent to we multiply the numerator and denominator by the same number (but not zero). Let us multiply them by and
So, and are equivalent to
Your Turn 1.2.13
Find three fractions equivalent to
Example 1.2.14
Find a fraction with a denominator of that is equivalent to
- Answer
-
To find equivalent fractions, we multiply the numerator and denominator by the same number. In this case, we need to multiply the denominator by a number that will result in
Since we can multiply by to get we can find the equivalent fraction by multiplying both the numerator and denominator by
Your Turn 1.2.14
Find a fraction with a denominator of that is equivalent to