# 4.1 Visualize fractions  (Page 7/11)

 Page 7 / 11

Locate and label each of the given fractions on a number line:

$\frac{2}{3},-\frac{2}{3},2\frac{1}{4},-2\frac{1}{4},\frac{3}{2},-\frac{3}{2}$ Locate and label each of the given fractions on a number line:

$\frac{3}{4},-\frac{3}{4},1\frac{1}{2},-1\frac{1}{2},\frac{7}{3},-\frac{7}{3}$ ## 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.

Order each of the following pairs of numbers, using $<$ or $>:$

1. $-\frac{2}{3}____-1$
2. $-3\frac{1}{2}____-3$
3. $-\frac{3}{7}____-\frac{3}{8}$
4. $-2____\frac{-16}{9}$

## Solution

$-\frac{2}{3}>-1$ $-3\frac{1}{2}<-3$ $-\frac{3}{7}\phantom{\rule{0.2em}{0ex}}\text{<}\phantom{\rule{0.2em}{0ex}}-\frac{3}{8}$ $-2<\frac{-16}{9}$ Order each of the following pairs of numbers, using $<$ or $>:$

1. $-\frac{1}{3}\underset{__}{\phantom{\rule{1em}{0ex}}}-1$
2. $-1\frac{1}{2}\underset{__}{\phantom{\rule{1em}{0ex}}}-2$
3. $-\frac{2}{3}\underset{__}{\phantom{\rule{1em}{0ex}}}-\frac{1}{3}$
4. $-3\underset{__}{\phantom{\rule{1em}{0ex}}}-\frac{7}{3}$

1. >
2. >
3. <
4. <

Order each of the following pairs of numbers, using $<$ or $>:$

1. $-3__-\frac{17}{5}$
2. $-2\frac{1}{4}__-2$
3. $-\frac{3}{5}\underset{__}{\phantom{\rule{1em}{0ex}}}-\frac{4}{5}$
4. $-4\underset{__}{\phantom{\rule{1em}{0ex}}}-\frac{10}{3}$

1. >
2. <
3. >
4. <

## Key concepts

• Property of One
• Any number, except zero, divided by itself is one.
$\frac{a}{a}=1$ , where $a\ne 0$ .
• Mixed Numbers
• A mixed number consists of a whole number $a$ and a fraction $\frac{b}{c}$ where $c\ne 0$ .
• It is written as follows: $a\frac{b}{c}\phantom{\rule{2em}{0ex}}c\ne 0$
• Proper and Improper Fractions
• The fraction $ab$ is a proper fraction if $a and an improper fraction if $a\ge b$ .
• Convert an improper fraction to a mixed number.
1. Divide the denominator into the numerator.
2. Identify the quotient, remainder, and divisor.
3. Write the mixed number as quotient $\frac{\text{remainder}}{\text{divisor}}$ .
• Convert a mixed number to an improper fraction.
1. Multiply the whole number by the denominator.
2. Add the numerator to the product found in Step 1.
3. Write the final sum over the original denominator.
• Equivalent Fractions Property
• If $\mathrm{a, b,}$ and $c$ are numbers where $b\ne 0$ , $c\ne 0$ , then $\frac{a}{b}=\frac{a\cdot c}{b\cdot c}$ .

## Practice makes perfect

In the following exercises, name the fraction of each figure that is shaded. 1. $\frac{1}{4}$
2. $\frac{3}{4}$
3. $\frac{3}{8}$
4. $\frac{5}{9}$ In the following exercises, shade parts of circles or squares to model the following fractions.

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

$\frac{3}{4}$ $\frac{2}{5}$

$\frac{5}{6}$ $\frac{7}{8}$

$\frac{5}{8}$ $\frac{7}{10}$

In the following exercises, use fraction circles to make wholes, if possible, with the following pieces.

$3\phantom{\rule{0.2em}{0ex}}\text{thirds}$ $8\phantom{\rule{0.2em}{0ex}}\text{eighths}$

$7\phantom{\rule{0.2em}{0ex}}\text{sixths}$ $4\phantom{\rule{0.2em}{0ex}}\text{thirds}$

$7\phantom{\rule{0.2em}{0ex}}\text{fifths}$ $7\phantom{\rule{0.2em}{0ex}}\text{fourths}$

In the following exercises, name the improper fractions. Then write each improper fraction as a mixed number. 1. $\frac{5}{4}=1\frac{1}{4}$
2. $\frac{7}{4}=1\frac{3}{4}$
3. $\frac{11}{8}=1\frac{3}{8}$  1. $\frac{11}{4}=2\frac{3}{4}$
2. $\frac{19}{8}=2\frac{3}{8}$

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

$\frac{3}{3}$

$\frac{4}{4}$ $\frac{7}{4}$

$\frac{5}{3}$ $\frac{11}{6}$

$\frac{13}{8}$ $\frac{10}{3}$

$\frac{9}{4}$ In the following exercises, rewrite the improper fraction as a mixed number.

$\frac{3}{2}$

$\frac{5}{3}$

$1\frac{2}{3}$

$\frac{11}{4}$

$\frac{13}{5}$

$2\frac{3}{5}$

$\frac{25}{6}$

$\frac{28}{9}$

$3\frac{1}{9}$

$\frac{42}{13}$

$\frac{47}{15}$

$3\frac{2}{15}$

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

$1\frac{2}{3}$

$1\frac{2}{5}$

$\frac{7}{5}$

$2\frac{1}{4}$

$2\frac{5}{6}$

$\frac{17}{6}$

$2\frac{7}{9}$

$2\frac{5}{7}$

$\frac{19}{7}$

$3\frac{4}{7}$

$3\frac{5}{9}$

$\frac{32}{9}$

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

How many sixths equal one-third?

How many twelfths equal one-third?

4

How many eighths equal three-fourths?

How many twelfths equal three-fourths?

9

How many fourths equal three-halves?

How many sixths equal three-halves?

9

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

$\frac{1}{4}$

$\frac{1}{3}$

Answers may vary. Correct answers include $\frac{2}{6},\frac{3}{9},\frac{4}{12}.$

$\frac{3}{8}$

$\frac{5}{6}$

Answers may vary. Correct answers include $\frac{10}{12},\frac{15}{18},\frac{20}{24}.$

$\frac{2}{7}$

$\frac{5}{9}$

Answers may vary. Correct answers include $\frac{10}{18},\frac{15}{27},\frac{20}{36}.$

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

$\frac{2}{3},\frac{5}{4},\frac{12}{5}$

$\frac{1}{3},\frac{7}{4},\frac{13}{5}$ $\frac{1}{4},\frac{9}{5},\frac{11}{3}$

$\frac{7}{10},\frac{5}{2},\frac{13}{8},3$ $2\frac{1}{3},-2\frac{1}{3}$

$1\frac{3}{4},-1\frac{3}{5}$ $\frac{3}{4},-\frac{3}{4},1\frac{2}{3},-1\frac{2}{3},\frac{5}{2},-\frac{5}{2}$

$\frac{2}{5},-\frac{2}{5},1\frac{3}{4},-1\frac{3}{4},\frac{8}{3},-\frac{8}{3}$ In the following exercises, order each of the following pairs of numbers, using $<$ or $>.$

$-1\underset{__}{\phantom{\rule{1em}{0ex}}}-\frac{1}{4}$

$-1\underset{__}{\phantom{\rule{1em}{0ex}}}-\frac{1}{3}$

<

$-2\frac{1}{2}\underset{__}{\phantom{\rule{1em}{0ex}}}-3$

$-1\frac{3}{4}\underset{__}{\phantom{\rule{1em}{0ex}}}-2$

>

$-\frac{5}{12}\underset{__}{\phantom{\rule{1em}{0ex}}}-\frac{7}{12}$

$-\frac{9}{10}\underset{__}{\phantom{\rule{1em}{0ex}}}-\frac{3}{10}$

<

$-3\underset{__}{\phantom{\rule{1em}{0ex}}}-\frac{13}{5}$

$-4\underset{__}{\phantom{\rule{1em}{0ex}}}-\frac{23}{6}$

<

## Everyday math

Music Measures A choreographed dance is broken into counts. A $\frac{1}{1}$ count has one step in a count, a $\frac{1}{2}$ count has two steps in a count and a $\frac{1}{3}$ count has three steps in a count. How many steps would be in a $\frac{1}{5}$ count? What type of count has four steps in it?

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?”

1. 8
2. 4

Baking Nina is making five pans of fudge to serve after a music recital. For each pan, she needs $\frac{1}{2}$ cup of walnuts.

1. How many cups of walnuts does she need for five pans of fudge?
2. Do you think it is easier to measure this amount when you use an improper fraction or a mixed number? Why?

## Writing exercises

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

Explain how you locate the improper fraction $\frac{21}{4}$ on a number line on which only the whole numbers from $0$ through $10$ 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. Who 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.

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