4.1 Visualize Fractions

Learning objectives

Learning objectives

• 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 $\frac{1}{2}$ 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 $\frac{1}{4}$ of the pizza.

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 $\frac{1}{12}$ of the pizza.

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 $\frac{1}{3}$ of the circle. This type of model is called a fraction circle. Other shapes, such as rectangles, can also be used to model fractions.

What does the fraction $\frac{2}{3}$ represent? The fraction $\frac{2}{3}$ means two of three equal parts.

Example 4.1

Solution

1. Identify the circle's equal parts. For part (a), the circle is divided into eight equal regions, so the denominator of the shaded fraction must be $8$.
2. Count the shaded circle regions. Five of the eight regions in part (a) are shaded, so the numerator is $5$ and the fraction is $\frac{5}{8}$.
3. Identify the square's equal parts. For part (b), the square is divided into nine equal regions, so the denominator of the shaded fraction must be $9$.
4. Count the shaded square regions. Two of the nine regions in part (b) are shaded, so the numerator is $2$ and the fraction is $\frac{2}{9}$.
5. State both shaded fractions. The circle is $\frac{5}{8}$ shaded, and the square is $\frac{2}{9}$ shaded, because each numerator counts shaded parts and each denominator counts all equal parts.

Takeaway: To read a fraction from a shaded figure, put the number of shaded equal parts in the numerator and the total number of equal parts in the denominator, giving $\frac{5}{8}$ and $\frac{2}{9}$.

Example 4.2

Solution

1. Use the denominator to set the partitions. The target is $\frac{3}{4}$, so the circle must be divided into four equal parts so that each part represents one-fourth of the whole.
2. Divide the circle into equal regions. The four regions must have the same size, because a fraction model represents equal pieces and a different region size would not represent one-fourth each.
3. Use the numerator to select regions. Since the numerator is $3$, shade exactly three of the four equal regions and leave the remaining fourth region unshaded.
4. Check the completed model. The model contains three shaded fourths out of four equal fourths, so the shaded part represents $\frac{3}{4}$ of the circle.

Takeaway: To shade a fraction of a whole, divide the shape into as many equal parts as the denominator and shade the number of parts named by the numerator, so $\frac{3}{4}$ uses three of four parts.

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.

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

It takes $24$ twenty-fourths, so $\frac{24}{24}=1.$

This leads us to the Property of One.

Example 4.3

Solution

1. Identify the size of each piece. The pieces are fourths, fifths, and sixths, so one whole contains $4$ fourths, $5$ fifths, or $6$ sixths.
2. Build each whole. ⓐ Use $4$ fourths to fill one circle. ⓑ Use $5$ fifths to fill one circle. ⓒ Use $6$ sixths to fill one circle.
3. State the wholes made. The pieces make $1$ whole in ⓐ, $1$ whole in ⓑ, and $1$ whole in ⓒ, with no pieces left over.

Takeaway: A fraction circle makes exactly one whole when it contains as many equal pieces as the denominator of the unit fraction.

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

Example 4.4

Solution

1. Group the three halves. A whole requires two halves, and $3=1\cdot 2+1$, so the pieces make $1$ whole with $1$ half left over.
2. Group the eight fifths. A whole requires five fifths, and $8=1\cdot 5+3$, so the pieces make $1$ whole with $3$ fifths left over.
3. Group the seven thirds. A whole requires three thirds, and $7=2\cdot 3+1$, so the pieces make $2$ wholes with $1$ third left over.
4. Match each remainder to its size. The leftover pieces in the three groups are respectively $1$ half, $3$ fifths, and $1$ third because each remainder uses the denominator of that group.
5. State the three mixed numbers. Therefore the groups represent $1\frac{1}{2}$, $1\frac{3}{5}$, and $2\frac{1}{3}$, respectively.

Takeaway: To combine fractional pieces into wholes, divide the number of pieces by the denominator and retain the remainder, giving $1\frac{1}{2}$, $1\frac{3}{5}$, and $2\frac{1}{3}$.

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, $\frac{1}{5},$ so altogether you had eight fifths, which we can write as $\frac{8}{5}.$ The fraction $\frac{8}{5}$ is one whole, $1,$ plus three fifths, $\frac{3}{5},$ or $1\frac{3}{5},$ which is read as one and three-fifths.

The number $1\frac{3}{5}$ is called a mixed number. A mixed number consists of a whole number and a fraction.

Fractions such as $\frac{5}{4},\frac{3}{2},\frac{5}{5},$ and $\frac{7}{3}$ 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 $\frac{1}{2},\frac{3}{7},$ and $\frac{11}{18}$ are proper fractions.

Example 4.5

Solution

1. Identify the size of each shaded piece. Every circle is divided into three equal pieces, and each piece represents one-third of one whole circle.
2. Count all four shaded pieces. The model contains four shaded thirds, so its improper fraction has numerator $4$ and denominator $3$, giving $\frac{4}{3}$.
3. Separate the complete group. Three of the shaded pieces fill one whole circle, leaving one additional third outside that complete group.
4. Write the mixed number. The complete group contributes $1$ whole and the remaining piece contributes $\frac{1}{3}$, so $\frac{4}{3}=1\frac{1}{3}$.

Takeaway: An improper fraction whose numerator is the denominator plus a remainder can be rewritten by dividing to obtain the whole-number part and keeping the remainder over the original denominator, so $\frac{4}{3}=1\frac{1}{3}$.

Example 4.6

Solution

1. Determine the unit-piece size. The denominator $8$ tells us that each circle should be divided into eight equal parts, making every part one-eighth of a whole circle.
2. Model the first complete group. Shade all eight parts of the first circle to represent $8$ eighths, which is exactly $1$ whole because $\frac{8}{8}=1$.
3. Model the remaining pieces. The fraction $\frac{11}{8}$ has three more eighths than the first group because $11-8=3$, so shade three parts of a second circle.
4. Combine the two groups. The complete circle gives $1$ whole and the partial circle gives $\frac{3}{8}$, so the model represents $\frac{11}{8}=1\frac{3}{8}$.

Takeaway: To model an improper fraction, fill one whole with denominator-many pieces and use a second figure for the remaining numerator-minus-denominator pieces, so $\frac{11}{8}=1\frac{3}{8}$.

Example 4.7

Solution

1. Interpret the numerator as unit pieces. The improper fraction $\frac{11}{6}$ represents eleven sixths, so begin with a model containing eleven equal pieces of size one-sixth.
2. Remove one complete group. Six sixths make exactly one whole because $\frac{6}{6}=1$, so remove six pieces from the original group.
3. Identify the leftover pieces. After removing six pieces, $11-6=5$ pieces remain, and those five pieces represent the fraction $\frac{5}{6}$.
4. Combine the whole and remainder. The original group therefore contains one whole and five sixths, giving $\frac{11}{6}=1\frac{5}{6}$.

Takeaway: Converting $\frac{11}{6}$ to a mixed number means removing one whole of six sixths and keeping the remaining five sixths, so $\frac{11}{6}=1\frac{5}{6}$.

Example 4.8

Solution

1. Rewrite the whole as unit pieces. The mixed number $1\frac{4}{5}$ means one whole plus four fifths, and one whole equals $\frac{5}{5}$.
2. Add like-sized pieces. The two parts together are $\frac{5}{5}+\frac{4}{5}=\frac{9}{5}$.
3. State the improper fraction. Therefore $1\frac{4}{5}=\frac{9}{5}.$

Takeaway: To convert a mixed number to an improper fraction, use the original denominator and add the fractional numerator to the whole times that denominator.

Convert between Improper Fractions and Mixed Numbers

In Example 4.7, we converted the improper fraction $\frac{11}{6}$ to the mixed number $1\frac{5}{6}$ using fraction circles. We did this by grouping six sixths together to make a whole; then we looked to see how many of the $11$ pieces were left. We saw that $\frac{11}{6}$ made one whole group of six sixths plus five more sixths, showing that $\frac{11}{6}=1\frac{5}{6}.$

The division expression $\frac{11}{6}$ (which can also be written as $611$) tells us to find how many groups of $6$ are in $11.$ To convert an improper fraction to a mixed number without fraction circles, we divide.

Example 4.9

Solution

1. Divide the numerator by the denominator. Set up $11\div 6$; the figure labels $6$ as the divisor because the denominator determines the size of each fractional part.
2. Read the quotient and remainder. Since $6\cdot 1=6$ and $11-6=5$, the quotient is $1$ and the remainder is $5$, with $5<6$.
3. Write the mixed number. The quotient $1$ gives the whole-number part, and the remainder $5$ goes over the original divisor $6$, so $\frac{11}{6}=1\frac{5}{6}$.

Takeaway: In mixed-number division, the quotient gives the wholes and the remainder must be written over the original denominator.

Example 4.10

Solution

1. Divide the numerator by the denominator. Set up $33\div 8$, as represented by the long-division figure.
2. Find the whole part. Because $8\cdot 4=32$, the quotient is $4$ and $33-32=1$ is the remainder.
3. Write the mixed number. Use the remainder over the divisor, giving $\frac{33}{8}=4\frac{1}{8}.$

Takeaway: A remainder smaller than the divisor is already reduced, so it becomes the numerator of the fractional part directly.

In Example 4.8, we changed $1\frac{4}{5}$ 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 4.11

Solution

1. Replace the whole with third-sized pieces. Multiply the whole number by the denominator: $4\cdot 3=12$.
2. Add the fractional pieces. The remaining $2$ thirds give $12+2=14$ thirds, so the improper fraction is $\frac{14}{3}$.
3. Interpret the sequence of figures. The four diagrams show the product $4\cdot 3$, then $12+2$ in the numerator, and finally $14$ over $3$. Thus $4\frac{2}{3}=\frac{14}{3}.$

Takeaway: Converting a mixed number to an improper fraction means multiplying by the denominator, adding the numerator, and keeping the denominator unchanged.

Example 4.12

Solution

1. Replace the whole with seventh-sized pieces. Multiply the whole number by the denominator: $10\cdot 7=70$.
2. Add the fractional pieces. The remaining $2$ sevenths give $70+2=72$ sevenths.
3. Interpret the sequence of figures. The diagrams show $10\cdot 7$, then $70+2$, and finally $72$ over $7$. Therefore $10\frac{2}{7}=\frac{72}{7}.$

Takeaway: The numerator of an improper fraction representing a mixed number is the whole multiplied by the denominator plus the fractional numerator.

Model Equivalent Fractions

Let’s think about Andy and Bobby and their favorite food again. If Andy eats $\frac{1}{2}$ of a pizza and Bobby eats $\frac{2}{4}$ of the pizza, have they eaten the same amount of pizza? In other words, does $\frac{1}{2}=\frac{2}{4}?$ We can use fraction tiles to find out whether Andy and Bobby have eaten equivalent parts of the pizza.

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 4.3 and extend it to include eighths, tenths, and twelfths.

Start with a $\frac{1}{2}$ tile. How many fourths equal one-half? How many of the $\frac{1}{4}$ tiles exactly cover the $\frac{1}{2}$ tile?

Since two $\frac{1}{4}$ tiles cover the $\frac{1}{2}$ tile, we see that $\frac{2}{4}$ is the same as $\frac{1}{2},$ or $\frac{2}{4}=\frac{1}{2}.$

How many of the $\frac{1}{6}$ tiles cover the $\frac{1}{2}$ tile?

Since three $\frac{1}{6}$ tiles cover the $\frac{1}{2}$ tile, we see that $\frac{3}{6}$ is the same as $\frac{1}{2}.$

So, $\frac{3}{6}=\frac{1}{2}.$ The fractions are equivalent fractions.

Example 4.13

Solution

1. Match the eighth-sized tiles. Four tiles of size one-eighth cover a one-half tile exactly, so $4$ eighths equal one-half and $\frac{4}{8}=\frac{1}{2}$.
2. Match the tenth-sized tiles. Five tiles of size one-tenth cover a one-half tile exactly, so $5$ tenths equal one-half and $\frac{5}{10}=\frac{1}{2}$.
3. Match the twelfth-sized tiles. Six tiles of size one-twelfth cover a one-half tile exactly, so $6$ twelfths equal one-half and $\frac{6}{12}=\frac{1}{2}$.
4. Count the twentieth-sized tiles. Ten tiles of size one-twentieth cover a one-half tile exactly, because ten twentieths equal one whole, so $\frac{10}{20}=\frac{1}{2}$.
5. Confirm the equal values. The fractions $\frac{2}{4}$, $\frac{3}{6}$, $\frac{4}{8}$, $\frac{5}{10}$, $\frac{6}{12}$, and $\frac{10}{20}$ all represent the same quantity as $\frac{1}{2}$.
6. State the equivalent fractions. Each larger fraction has both numerator and denominator multiplied by the same nonzero factor, so all of the displayed fractions are equivalent.

Takeaway: Equivalent fractions use different-sized tiles but represent the same amount, as shown by $\frac{4}{8}=\frac{5}{10}=\frac{6}{12}=\frac{10}{20}=\frac{1}{2}$.

Find Equivalent Fractions

We used fraction tiles to show that there are many fractions equivalent to $\frac{1}{2}.$ For example, $\frac{2}{4},\frac{3}{6},$ and $\frac{4}{8}$ are all equivalent to $\frac{1}{2}.$ When we lined up the fraction tiles, it took four of the $\frac{1}{8}$ tiles to make the same length as a $\frac{1}{2}$ tile. This showed that $\frac{4}{8}=\frac{1}{2}.$ See Example 4.13.

We can show this with pizzas, too. Figure 4.4(a) shows a single pizza, cut into two equal pieces with $\frac{1}{2}$ shaded. Figure 4.4(b) shows a second pizza of the same size, cut into eight pieces with $\frac{4}{8}$ shaded.

This is another way to show that $\frac{1}{2}$ is equivalent to $\frac{4}{8}.$

How can we use mathematics to change $\frac{1}{2}$ into $\frac{4}{8}?$ 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.

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 $\frac{1}{2},\frac{2}{4},\frac{3}{6},$ and $\frac{10}{20}$ are equivalent fractions.

Example 4.14

Solution

1. Multiply by a factor of two. Multiplying the numerator and denominator by $2$ gives $2\cdot 2=4$ and $5\cdot 2=10$, so one equivalent fraction is $\frac{4}{10}$.
2. Multiply by a factor of three. Multiplying both parts by $3$ gives $2\cdot 3=6$ and $5\cdot 3=15$, so another equivalent fraction is $\frac{6}{15}$.
3. Multiply by a factor of five. Multiplying both parts by $5$ gives $2\cdot 5=10$ and $5\cdot 5=25$, so a third equivalent fraction is $\frac{10}{25}$.
4. Check that the factors agree. Each new numerator is $2$ times the chosen factor and each new denominator is $5$ times that same factor.
5. State the equivalent fractions. Therefore $\frac{4}{10}$, $\frac{6}{15}$, and $\frac{10}{25}$ are all equivalent to the original fraction $\frac{2}{5}$.

Takeaway: Multiplying both the numerator and denominator of $\frac{2}{5}$ by the same nonzero factor preserves its value and produces equivalent fractions such as $\frac{4}{10}$, $\frac{6}{15}$, and $\frac{10}{25}$.

Example 4.15

Solution

1. Compare the requested denominator. The original fraction is $\frac{2}{7}$, and the new denominator must be $21$, so the original denominator must be multiplied to reach $21$.
2. Find the common multiplier. Since $7\cdot 3=21$, the required factor is $3$, and this factor is nonzero so the fraction's value is preserved.
3. Multiply the numerator by the same factor. Applying the factor $3$ to the numerator gives $2\cdot 3=6$, which keeps the numerator and denominator in the same proportion.
4. Keep the target denominator. The equivalent fraction therefore has numerator $6$ and denominator $21$, giving $\frac{6}{21}$.
5. Verify the equivalence. Since both parts of $\frac{2}{7}$ were multiplied by $3$, the fraction $\frac{6}{21}$ represents the same value as the original fraction.

Takeaway: To build an equivalent fraction with denominator $21$, multiply the denominator $7$ by $3$ and multiply the numerator by $3$ as well, giving $\frac{6}{21}$.

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.

Let us locate $\frac{1}{5},\frac{4}{5},3,3\frac{1}{3},\frac{7}{4},\frac{9}{2},5,$ and $\frac{8}{3}$ on the number line.

We will start with the whole numbers $3$ and $5$ because they are the easiest to plot.

The proper fractions listed are $\frac{1}{5}$ and $\frac{4}{5}.$ We know proper fractions have values less than one, so $\frac{1}{5}$ and $\frac{4}{5}$ are located between the whole numbers $0$ and $1.$ The denominators are both $5,$ so we need to divide the segment of the number line between $0$ and $1$ into five equal parts. We can do this by drawing four equally spaced marks on the number line, which we can then label as $\frac{1}{5},\frac{2}{5},\frac{3}{5},$ and $\frac{4}{5}.$

Now plot points at $\frac{1}{5}$ and $\frac{4}{5}.$

The only mixed number to plot is $3\frac{1}{3}.$ Between what two whole numbers is $3\frac{1}{3}?$ Remember that a mixed number is a whole number plus a proper fraction, so $3\frac{1}{3}>3.$ Since it is greater than $3,$ but not a whole unit greater, $3\frac{1}{3}$ is between $3$ and $4.$ We need to divide the portion of the number line between $3$ and $4$ into three equal pieces (thirds) and plot $3\frac{1}{3}$ at the first mark.

Finally, look at the improper fractions $\frac{7}{4},\frac{9}{2},$ and $\frac{8}{3}.$ Locating these points will be easier if you change each of them to a mixed number.

Here is the number line with all the points plotted.

Example 4.16

Solution

1. Set up equal unit intervals. Mark the consecutive integers on the number line and divide each relevant interval into the number of equal parts named by the denominator.
2. Locate the proper fraction. The value $\frac{3}{4}$ lies between $0$ and $1$, so divide that interval into fourths and place the label three fourths of the way from $0$.
3. Convert and locate four thirds. Since $\frac{4}{3}=1\frac{1}{3}$, the point lies between $1$ and $2$, one-third of the way to the right of $1$.
4. Convert and locate five thirds. Since $\frac{5}{3}=1\frac{2}{3}$, the point lies between $1$ and $2$, two-thirds of the way to the right of $1$.
5. Locate the mixed number. The value $4\frac{1}{5}$ lies between $4$ and $5$, one-fifth of the way to the right of $4$ after that interval is divided into fifths.
6. Convert and locate seven halves. Since $\frac{7}{2}=3\frac{1}{2}$, the point lies halfway between $3$ and $4$, which completes all five labels.

Takeaway: To plot fractions and mixed numbers on a number line, identify the interval between consecutive integers, divide that interval into denominator-many equal parts, and mark the numerator's fractional part.

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 $7$ is $-7$ and the opposite of $-7$ is $7.$

Fractions have opposites, too. The opposite of $\frac{3}{4}$ is $-\frac{3}{4}.$ It is the same distance from $0$ on the number line, but on the opposite side of $0.$

Thinking of negative fractions as the opposite of positive fractions will help us locate them on the number line. To locate $-\frac{15}{8}$ on the number line, first think of where $\frac{15}{8}$ is located. It is an improper fraction, so we first convert it to the mixed number $1\frac{7}{8}$ and see that it will be between $1$ and $2$ on the number line. So its opposite, $-\frac{15}{8},$ will be between $-1$ and $-2$ on the number line.

Example 4.17

Solution

1. Prepare mirrored unit intervals. Mark the integers and, for each unit interval, divide it into the number of equal parts required by the fractional part of the value being located.
2. Locate the quarter pair. The value $\frac{1}{4}$ is one-fourth of the way from $0$ to $1$, and $-\frac{1}{4}$ is one-fourth of the way from $0$ to $-1$ on the opposite side.
3. Locate the one-third pair. The value $1\frac{1}{3}$ is one-third of the way from $1$ to $2$, and $-1\frac{1}{3}$ is one-third of the way from $-1$ to $-2$.
4. Locate the half pair. The value $\frac{5}{2}=2\frac{1}{2}$ is halfway from $2$ to $3$, and $-\frac{5}{2}=-2\frac{1}{2}$ is halfway from $-2$ to $-3$.
5. Check the opposite locations. Each negative label is the mirror image of its positive counterpart across $0$ and is the same distance from $0$ on the opposite side.

Takeaway: Opposite numbers lie the same distance from $0$ on opposite sides of the number line, so $-\frac{1}{4}$ mirrors $\frac{1}{4}$ and $-2\frac{1}{2}$ mirrors $2\frac{1}{2}$.

Order Fractions and Mixed Numbers

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

Example 4.18

Solution

1. Compare the first pair. Since $-\frac{2}{3}$ lies between $-1$ and $0$, it is greater than $-1$.
2. Compare the second pair. The mixed number $-3\frac{1}{2}$ lies one-half unit to the left of $-3$, so it is less than $-3$.
3. Compare the third pair. Both fractions are negative, and $\frac{3}{7}<\frac{3}{8}$, so $-\frac{3}{7}<-\frac{3}{8}$.
4. Compare the final pair. The negative fraction $-\frac{16}{9}$ lies between $-2$ and $-1$, so $-2<-\frac{16}{9}$.
5. State all orders. The results are $-\frac{2}{3}>-1$, $-3\frac{1}{2}<-3$, $-\frac{3}{7}<-\frac{3}{8}$, and $-2<-\frac{16}{9}.$

Takeaway: Negating an inequality reverses its direction, which is why the fraction with the smaller magnitude is larger when both numbers are negative.

Section 4.1 Exercises

Section 4.1 Exercises

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

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

In the following exercises, use fraction circles to make wholes using the following pieces.

In the following exercises, name the improper fractions. Then write each improper fraction as a mixed number.

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

In the following exercises, rewrite the improper fraction as a mixed number.

In the following exercises, rewrite the mixed number as an improper fraction.

In the following exercises, use fraction tiles or draw a figure to find equivalent fractions.

In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.

In the following exercises, plot the numbers on a number line.

In the following exercises, order each of the following pairs of numbers, using $<$ or $\mathrm{>.}$

Everyday Math

Writing Exercises

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.

1. Name the fraction of the shape that is shaded in each figure:

2. Name the fraction of the shape that is shaded in each figure:

3. Name the improper fraction. Then write it as a mixed number.

1. Use a model to rewrite the mixed number as an improper fraction: $1\frac{3}{8}.$

2. Use a model to rewrite the mixed number as an improper fraction: $1\frac{5}{6}.$

3. Convert the improper fraction to a mixed number: $\frac{13}{7}.$

1. Convert the improper fraction to a mixed number: $\frac{14}{9}.$

2. Convert the improper fraction to a mixed number: $\frac{23}{7}.$

3. Convert the improper fraction to a mixed number: $\frac{48}{11}.$

1. Convert the mixed number to an improper fraction: $3\frac{5}{7}.$

2. Convert the mixed number to an improper fraction: $2\frac{7}{8}.$

3. Convert the mixed number to an improper fraction: $4\frac{6}{11}.$

1. Convert the mixed number to an improper fraction: $11\frac{1}{3}.$

2. Find three fractions equivalent to $\frac{3}{5}.$

3. Find three fractions equivalent to $\frac{4}{5}.$

1. Find a fraction with a denominator of $21$ that is equivalent to $\frac{6}{7}.$

2. Find a fraction with a denominator of $100$ that is equivalent to $\frac{3}{10}.$

3. Order each of the following pairs of numbers, using $<$ or $\mathrm{>:}$

1. ⓐ $-\frac{1}{3}\underset{\mathrm{\_\_}}{\,}-1$

2. ⓑ $-1\frac{1}{2}\underset{\mathrm{\_\_}}{\,}-2$

3. ⓒ $-\frac{2}{3}\underset{\mathrm{\_\_}}{\,}-\frac{1}{3}$

4. ⓓ $-3\underset{\mathrm{\_\_}}{\,}-\frac{7}{3}$

1. Order each of the following pairs of numbers, using $<$ or $\mathrm{>:}$

1. ⓐ $-3\mathrm{\_\_}-\frac{17}{5}$

2. ⓑ $-2\frac{1}{4}\mathrm{\_\_}-2$

3. ⓒ $-\frac{3}{5}\underset{\mathrm{\_\_}}{\,}-\frac{4}{5}$

4. ⓓ $-4\underset{\mathrm{\_\_}}{\,}-\frac{10}{3}$

2. $\frac{5}{3}$

3. $\frac{13}{5}$

1. $\frac{28}{9}$

2. $\frac{47}{15}$

3. $1\frac{2}{5}$

1. $2\frac{5}{6}$

2. $2\frac{5}{7}$

3. $3\frac{5}{9}$

1. $\frac{1}{3}$

2. $\frac{5}{6}$

3. $\frac{5}{9}$

1. $-1\underset{\mathrm{\_\_}}{\,}-\frac{1}{3}$

2. $-1\frac{3}{4}\underset{\mathrm{\_\_}}{\,}-2$

3. $-\frac{9}{10}\underset{\mathrm{\_\_}}{\,}-\frac{3}{10}$

1. Name the improper fraction. Then write it as a mixed number.

2. Use a model to rewrite the improper fraction as a mixed number: $\frac{9}{7}.$

3. Use a model to rewrite the improper fraction as a mixed number: $\frac{7}{4}.$

1. $-4\underset{\mathrm{\_\_}}{\,}-\frac{23}{6}$

2. Music Measures Fractions are used often in music. In $\frac{4}{4}$ time, there are four quarter notes in one measure.

1. ⓐ How many measures would eight quarter notes make?

2. ⓑ The song “Happy Birthday to You” has $25$ quarter notes. How many measures are there in “Happy Birthday to You?”

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