Uses of The numbers,,,, and are all fractions. A fraction is written with two whole numbers that are separated by a fraction bar. The top number is called the numerator. The bottom number is called the denominator. The numerator of a fraction can be any whole number. The denominator can be any whole number except. teaspoon When naming fractions, name the numerator first, then name the denominator. yard mile were invented thousands of years ago to name numbers between whole numbers. People needed these in-between numbers for making careful measurements. Here are some examples of measurements that use fractions: cup, hour, km, and lb. are also used to name parts of wholes. The whole might be one single thing, like a pizza. Or, the whole might be a collection of things, like all the students in a classroom. The whole is sometimes called the ONE. Name the whole, or ONE, for each statement. Derek ate of a pizza of the pizza. The whole is the entire pizza. The fraction,, names the part of the pizza that Derek ate. In Mrs. Blake s classroom, of the students are girls. The whole is the collection of all students in Mrs. Blake s classroom. The fraction,, names the part of that collection that are girls. In Everyday Mathematics, fractions are used in other ways that may be new to you. are used in the following ways: to show rates (such as cost per ounce) to compare (such as comparing the weights of two animals) to name percents ( is %) to show divisions ( can be written ) to show the scale of a map or a picture to show probabilities forty-two Mrs. Blake s Classroom of the students are girls. Arab mathematicians began to use the horizontal fraction bar around the year. They were the first to write fractions as we do today. Number Sense.; Number Sense.
Here are some other examples of uses of fractions: Study the recipe shown at the right. Many of the amounts listed in the recipe include fractions. A movie critic gave the film Finding Cosmo a rating of stars (on a scale of to stars). This spinner has of the circle colored red, colored blue, and colored green. If we spin the spinner many times, it will land on red about of the time. It will land on blue about of the time. And it will land on green about of the time. The probability that the spinner will land on a color that is not green is. If a map includes a scale, you can use the scale to estimate real-world distances. The scale on the map shown here is given as :,. This means that every distance on the map is, of the real-world distance. A centimeter distance on the map stands for a realworld distance of, centimeters ( meters). are often used to describe clothing sizes. For example, women s shoes come in sizes,,,, and so on, up to. Part of a size chart for women s shoes is shown at the right. It gives the recommended shoe size for women whose feet are between and inches long. Number Sense.; Number Sense. Size Chart for Women s Shoes Heel-to-toe length (in.) Size to to to to to to to forty-three
Understanding the many ways people use fractions will help you solve problems more easily. for Parts of a Whole are used to name a part of a whole thing that is divided into equal parts. For example, the circle at the right has been divided into equal parts. Each part is of the circle. Three of the parts are blue, so is blue. (three-eighths) of the circle In Everyday Mathematics, the whole thing that is divided into equal parts is called the ONE. To understand a fraction used to name part of a whole, you need to know what the ONE is. Sally ate half a pizza. Is that a lot? The answer depends on how big the pizza was. If the pizza was small, then is not a lot. If the pizza was large, then is a lot. for Parts of a Collection A fraction may be used to name part of a collection of things. Look at the collection of counters. What fraction of the counters is red? There are counters in all. Five of the counters are red. Five out of counters are red. This fraction shows what part of the collection is red. Name the fraction of counters that are each shape in the collection above.. Circles. Triangles. Squares Check your answers on page. forty-four Number Sense.; Number Sense.
To understand a fraction that is used to name part of a collection, you need to know how big the whole collection is. Only half of Sam s cousins can come to his party. Is that many people? It depends on how many cousins Sam has. If Sam has only cousins, then cousins are coming; that s not many people. But if Sam has cousins, then cousins are coming. That s many people. in Measuring are used to make more careful measurements. Think about the inch scale on a ruler. Suppose the spaces between the whole-inch marks are left unmarked. With a ruler like this, you can measure only to the nearest inch. Now suppose the -inch spaces are divided into quarters by -inch and -inch marks. With this ruler, you can measure to the nearest inch or to the nearest inch. To understand a fraction used in a measurement, you need to know what the unit is. To say, Susan lives from here makes no sense. Susan might live half a block away or half a mile. The unit in measurement is like the ONE when fractions are used to name a part of a whole. Ruler has inch marks only. You can measure to the nearest inch. Ruler has - and -inch marks. You can measure to the nearest inch or to the nearest inch. in Probability A fraction may tell the chance that an event will happen. This chance, or probability, is always a number from to. An impossible event has a probability of ; it has no chance of happening. An event with a probability of is sure to happen. An event with a probability of has an equal chance of happening or not happening. When you pick a ball out of this jar without looking, the chance of getting a red ball is. The chance of getting a blue ball is. probability of picking a red ball number of red balls total number of balls probability of picking a blue ball number of blue balls total number of balls Number Sense.; Number Sense. forty-five
and Division Division problems can be written using a slash / instead of the division symbol. For example, can be written /. / Division problems can also be written as fractions. One of the many uses of fractions is to show divisions. The example below shows that can be written as the fraction. Show that This is the whole, or ONE. and also. This is the whole after dividing it into equal parts. Each part is of the whole. The picture below shows that thirds make wholes. So,. less than can also be thought of as divisions. Show that. Think of as an equal-sharing problem. Suppose friends want to share oranges. They could cut or divide each orange into equal parts. The word fraction is derived from the Latin word frangere, which means to break. are sometimes called broken numbers. Each person gets of an orange. So,. You can rename any fraction by dividing on your calculator. To rename, think of it as a division problem and divide: Press. The answer in the display will show, which is another name for. forty-six Number Sense.; Number Sense.
in Rates and Ratios are often used to name rates and ratios. Rate A rate compares two numbers with different units. For example, miles per hour is a rate that compares distance with miles time. It can be written as hour. A ratio is like a rate, but it compares two quantities that have the same unit. Example speed (jogging) distance blocks time minutes price cost quantity erasers conversion of units distance in yards yard distance in feet feet Ratio Example won / lost record games (won) games (lost) rainy days compared to total days (rainy) days (total) days Other Uses of are used to compare distances on maps to distances in the real world, and to describe size changes. Find the real-world distance from Clay St. to S. Lake St. Measure this distance on the map. It is cm. Each distance on the map is, of the real-world distance. So, the real-world distance equals, times the map distance. The real-world distance, * cm, cm. cm m. So,, cm m and, cm, m. The distance from Clay St. to S. Lake St. is, m. A length of centimeters on the original will be centimeters on the copy. copy size original size Number Sense.; Number Sense. forty-seven
Mixed Numbers Numbers like,, and are called mixed numbers. A mixed number has a whole-number part and a fraction part. In the mixed number, the whole-number part is and the fraction part is. A mixed number is equal to the sum of the whole-number part and the fraction part:. Mixed numbers are used in many of the same ways that fractions are used. Mixed numbers can be renamed as fractions. For example, if a circle is the ONE, then names whole circles and of another circle. If you divide the whole circles into fifths, then you can see that. To rename a mixed number as a fraction, first rename as a fraction with the same denominator as the fraction part. Then add all of the fractions. For example, to rename as a fraction, first rename as. Then. like and are called improper fractions. An improper fraction is a fraction that is greater than or equal to. In an improper fraction, the numerator is greater than or equal to the denominator. A proper fraction is a fraction that is less than. In a proper fraction, the numerator is less than the denominator. Note Even though they are called improper, there is nothing wrong about improper fractions. Do not avoid them. Write a mixed number for each picture... Write an improper fraction for each mixed number.... Check your answers on page. forty-eight Number Sense.; Number Sense.
Equivalent Two or more fractions that name the same number are called equivalent fractions. The four circles below are the same size, but they are divided into different numbers of parts. The green areas are the same in each circle. These circles show different fractions that are equivalent to. equal parts part green of the circle is green. equal parts parts green of the circle is green. equal parts parts green of the circle is green. equal parts parts green of the circle is green. The fractions,, and are all equivalent. They are just different names for the part of the circle that is green. You can write: On Ms. Klein s bus route, she picks up students, boys and girls. equal groups Each group is of the total. equal groups Each group is of the total. equal groups Each group is of the total. groups of girls of the students are girls. groups of girls of the students are girls. group of girls of the students are girls.,, and write. The fractions You can are all equivalent. Number Sense.; Number Sense. forty-nine
Rules for Finding Equivalent Here are two shortcuts for finding equivalent fractions. Using Multiplication If the numerator and the denominator of a fraction are both multiplied by the same number (not ), the result is a fraction that is equivalent to the original fraction. Change to an equivalent fraction. Multiply the numerator and the denominator of In symbols, you can write *. by. is red. * So, is equivalent to. is red. Using Division If the numerator and the denominator of a fraction are both divided by the same number (not ), the result is a fraction that is equivalent to the original fraction. To understand why division works, use the example shown above. But start with fraction by : this time and divide both numbers in the The division by undoes the multiplication by that we did before. Dividing both numbers in by gives an equivalent fraction,. Find fractions that are equivalent to.. a. What fraction of this rectangle is shaded? b. Give two other fractions for the shaded part.. Name fractions that are equivalent to.. Name fractions that are equivalent to. Check your answers on page. fifty Number Sense.; Number Sense.
Table of Equivalent This table lists equivalent fractions. All the fractions in a row name the same number. For example, all the fractions in the last row are names for the number. Note Every fraction is either in simplest form or is equivalent to a fraction in simplest form. Every fraction in the first column is in simplest form. A fraction is in simplest form if there is no equivalent fraction with a smaller numerator and smaller denominator. Simplest Name Lowest terms means the same as simplest form. Equivalent Fraction Names (zero) (one). True or false? a. b. c. Under normal conditions, of the length of a telephone pole should be in the ground. d.. a. Use the table to find other fractions that are equivalent to. b. Add more equivalent fractions that are not in the table. Check your answers on page. Number Sense.; Number Sense. fifty-one
Equivalent on a Ruler Rulers marked in inches usually have tick marks of different lengths. The longest tick marks on the ruler below show the whole inches. The marks used to show half inches, quarter inches, and eighths of an inch become shorter and shorter. The shortest marks show the sixteenths of an inch. Every tick mark on this ruler can be named by a number of sixteenths. Some tick marks can also be named by eighths, fourths, halves, and ones. The picture below shows the pattern of fraction names for a part of the ruler. This pattern continues past inch, with mixed numbers naming the tick marks.. Name a fraction or mixed number for each mark labeled A, B, and C on the ruler above.. What is the length of this nail? a. in quarter inches c. in sixteenths of an inch b. in eighths of an inch Check your answers on page. fifty-two Number Sense.; Number Sense.
Comparing When you compare fractions, you have to pay attention to both the numerator and the denominator. Like Denominators are easy to compare when they have the same denominator. For example, to decide which is larger, or, think of them as eighths and eighths. Just as bananas is more than bananas, and dollars is more than dollars, eighths is more than eighths. or because. because. Like Numerators If the numerators of two fractions are the same, then the fraction with the smaller denominator is larger. Remember, a smaller denominator means the ONE has fewer parts and each part is bigger. For example, because fifths are bigger than eighths, so fifths is more than eighths. is less than is greater than is equal to Note To compare fractions that have the same denominators, just look at the numerators. The fraction with the larger numerator is larger. with like denominators have the same denominator. and have like denominators. with like numerators have the same numerator. and have like numerators. Use of the symbol for equal to dates back to. Use of the symbols and for greater than and less than dates back to. because halves are bigger than thirds. because eighths are smaller than fourths. Number Sense.. fifty-three
Unlike Numerators and Unlike Denominators Several strategies can help you compare fractions when both the numerators and the denominators are different. Comparing to Compare and. Notice that is more than is less. and So, Comparing to or than. Comparing fractions to or can also be helpful. For example, because is closer to. ( is away from but is away from. Since eighths are smaller than fourths, is closer to.) Using Equivalent One way to compare fractions that always works is to find equivalent fractions that have the same denominator. For example, to compare and, look at the table of equivalent fractions on page. The table shows that both fifths and eighths can be written as ths: know that Using Decimal Equivalents.. and Since, you Using decimal equivalents is another way to compare fractions that always works. For example, to compare and, use a calculator to change both fractions to decimals: : : Key in: Answer:. Key in: Answer:. Note Remember that fractions can be used to show division problems. a a b b Since.., you know that. Compare. Write,, or in each box..... Check your answers on page. fifty-four Number Sense.
Adding and Subtracting Like Denominators Adding or subtracting fractions that have the same denominator is easy: Just add or subtract the numerators, and keep the same denominator. You can use division to put the answer in simplest form. Unlike Denominators When you are adding and subtracting fractions that have unlike denominators, you must be especially careful. One way is to model the problem with pattern blocks. Remember that different denominators mean the ONE is divided into different numbers (and different sizes) of parts.? If the hexagon is ONE, then the rhombus is and the triangle is. When you put one rhombus and one triangle together, you will find that they make a trapezoid. If the hexagon is ONE, then the trapezoid is. So,.? If the hexagon is ONE, then is triangles and To take away ( rhombuses) from need to take away triangles. Then there would be triangle or So, is rhombuses. ( triangles), you would left.. Number Sense. fifty-five
Clock A clock face can be used to model fractions with,,,,,,,,,, or in the denominator. Note hour minutes hour minutes hour minute Thousands of years ago, the ancient Babylonians divided the day into hours, the hour into minutes, and the minute into seconds. This system for keeping time is a good model for working with many fractions. hour minutes A clock face can help in solving simple fraction addition and subtraction problems.? hour minutes hour minutes? hour minutes hour minutes Using a Calculator Some calculators can add and subtract fractions. Key in:? ; or Answer: Solve. Use pattern blocks or clock faces to help you.. fifty-six. Check your answers on page... Number Sense.
Sometimes tools like pattern blocks or clock faces are not helpful for solving a fraction addition or subtraction problem. Here is a method that always works. Using a Like Denominator To add or subtract fractions that have different denominators, first rename them as fractions with a like denominator. A quick like denominator to use is the product of the denominators.? A furlong is a unit of distance, equal to mile. It is often used to measure distances in horse and dog races. A quick way to find a like denominator for these fractions is to multiply the denominators: *. Rename and as ths: * * * So, * To add a distance in furlongs (eighths of a mile) and a distance given in tenths of a mile, you could rename the fractions using * as a like denominator..? A like denominator for these fractions is *. Rename and as ths: * * * So, *. If two fractions are renamed so that they have the same denominator, that denominator is called a common denominator. Add or subtract..... Check your answers on page. Number Sense. fifty-seven
Multiplying and Whole Numbers There are several ways to think about multiplying a whole number and a fraction. Using a Number Line One way to multiply a whole number and a fraction is to think about hops on a number line. The whole number tells how many hops to make, and the fraction tells how long each hop should be. For example, to solve *, imagine taking hops on a number line, each unit long. * Using Addition You can use addition to multiply a fraction and a whole number. For example, to find *, draw models of. Then add up all of the fractions. * Using Fraction of an Area You can think of multiplying with a fraction as finding the fraction of an area. For example, to solve * (which is the same as * ), find of an area that is square units. The rectangle on the left has an area of square units. The shaded area of the rectangle on the right has an area of square units ( small rectangles, each with an area of.) So, of the rectangle area the shaded area squares * *. = + + + of means of means of means *. *. *. The word of in problems like these means multiplication. * Use any method to solve these problems.. * fifty-eight. * *. Check your answers on page.. *. * Number Sense.; Number Sense.
Finding a Fraction of a Set You can think of multiplication with fractions as finding a fraction of a set. For example, think of the problem * as What is of? One way to solve this problem is first to find of, and then use that answer to find of. of means *. *? Think of the problem as What is of? Step : Find of. To do this, divide the pennies into equal groups. Then count the number of pennies in one group., so of is. equal groups, with in each group Step : Next find of. Since of is, of is *. * *? Think of the problem as What is of? of Step : Find of. Divide pennies into equal groups., so of is. equal groups, with in each group Step : Next find of. Since of is, of is *. * of Find each answer.. *?. of?. *?. Rita and Hunter earned $ raking lawns. Since Rita did most of the work, she got of the money. How much did each person get? Check your answers on page. Number Sense. fifty-nine
Negative Numbers and Rational Numbers People have used counting numbers (,,, and so on) for thousands of years. Long ago people found that the counting numbers did not meet all of their needs. They needed numbers for in-between measures such as inches and hours. were invented to meet these needs. can also be renamed as decimals and percents. Most of the numbers you have seen are fractions or can be renamed as fractions. Rename as fractions:,,.,., and %... % However, even fractions did not meet every need. For example, problems such as and have answers that are less than and cannot be named as fractions. (, by the way they are defined, can never be less than.) This led to the invention of negative numbers. Negative numbers are numbers that are less than. The numbers,., and are negative numbers. The number is read negative. Negative numbers serve several purposes: To express locations such as temperatures below zero on a thermometer and depths below sea level Note Every whole number (,,, and so on) can be renamed as a fraction. For example, can be written as. And can be written as. Note Numbers like. and may not look like negative fractions, but they can be renamed as negative fractions. To show changes such as yards lost in a football game., and To extend the number line to the left of zero To calculate answers to many subtraction problems The opposite of every positive number is a negative number, and the opposite of every negative number is a positive number. The number is neither positive nor negative; is also its own opposite. The diagram at the right shows this relationship. The rational numbers are all the numbers that can be written or renamed as fractions or as negative fractions. sixty Number Sense.
, Decimals, and Percents, decimals, and percents are different ways to write numbers. Sometimes it is easier to work with a fraction instead of a decimal or a percent. Other times it is easier to work with a decimal or a percent. Renaming a Fraction as a Decimal You can rename a fraction as a decimal if you can find an equivalent fraction with a denominator of,, or,. This only works for certain fractions.. and % (decimal and percent rounded to digits) *. * * *. Note Remember that a a b is true for b a any fraction. b Another way to rename a fraction as a decimal is to divide the numerator by the denominator. You can use a calculator for this division. : : The U.S. Constitution did not take effect until of the original states had approved it. Key in: Answer:. Key in: Answer:. : Key in: : Key in: Renaming a Decimal as a Fraction To change a decimal to a fraction, write the decimal as a fraction with a denominator of,, or,. Then you can rename the fraction in simplest form. Write each decimal as a fraction. For., the rightmost digit is, which is in the ths place. So,., or. For., the rightmost digit is, which is in the,ths place. So,.,. For., the rightmost digit is, which is in the ths place. So,. (a fraction) or or (mixed numbers). Number Sense. Answer:. Answer:. Note This method will work for most of the decimal numbers you see. But it will not work for every decimal number. For example,. cannot be written as a fraction with a denominator of,,,, or any other power of. sixty-one
Renaming a Decimal as a Percent To rename a decimal as a percent, try to write the decimal as a fraction with a denominator of. Then use the meaning of percent (number of hundredths) to rename the fraction as a percent. Rename each decimal as a percent... %. %.. % Renaming a Percent as a Decimal To rename a percent as a decimal, try to rename it as a fraction with a denominator of. Then rename the fraction as a decimal. Rename each percent as a decimal. %. %., or. %. Renaming a Percent as a Fraction To rename a percent as a fraction, try to write it as a fraction with a denominator of. Rename each percent as a fraction in simplest form. % % % % Renaming a Fraction as a Percent To rename a fraction as a percent, try to rename it as a fraction with a denominator of. Then rename the fraction as a percent. Rename each fraction as a percent.. %. %...% Write each number as a fraction, a decimal, and a percent..... %. Check your answers on page. sixty-two Number Sense.
sic, and Ma u M, the nd u ma t o S ics Musicians make patterns of sound to create music. Mathematics can help us understand how both sound and music are created. Sound Every sound you hear begins with a vibration a back and forth motion. For musical instruments to produce sound, something must be set in motion. The sound of a drum starts when a person beats the drum head. When the drum head stops vibrating, the sound stops. The sound of a guitar starts when a person plucks or strums the strings. Each vibrating string moves back and forth at the same rate until it stops moving. When the strings stop vibrating, the sound stops. The sound of a flute starts when a person blows across the mouthpiece. A column of air moves back and forth inside the flute. When the player stops blowing, the column of air stops vibrating and the sound of the flute stops. The rate at which a string, a drum head, or a column of air vibrates is called the frequency. Higher frequency vibrations produce higher-pitched notes. Frequency is measured in Hertz (Hz), or vibrations per second. The human ear can hear vibrations from about Hz to, Hz. Measurement and Geometry.; Mathematical Reasoning. sixty-three
Instrument Length and Pitch Many instruments rely on a vibrating column of air to make sound. A longer column of air vibrates at a lower frequency and makes a deeper- or lower-pitched note. Shorter vibrating air columns make higher-pitched notes. Here are some instruments you may have heard, along with the frequency of the lowest note that can be played on the instrument. What happens to the frequency as the instruments get shorter? bassoon, Hz sixty-four clarinet, Hz oboe, Hz piccolo, Hz Measurement and Geometry.; Mathematical Reasoning.
The piccolo has a very short column of air within it, so it produces high-pitched notes. Piccolos produce notes in the range of about to, Hz, which humans can hear easily. This pan flute, from Peru, is played by blowing across the edges of hollow tubes of different lengths. Short tubes produce high-pitched notes, and long tubes produce low-pitched notes. The player slides the instrument from side to side to change notes. A recorder can play a range of pitches. By covering all of the finger holes on a recorder, the musician creates the longest possible column of air, and the lowest-pitched note. With all holes uncovered, a high note is produced. Because the alto saxophone is much longer than the piccolo, its sound is lower-pitched. Saxophones use a reed, which is a carefully-shaped piece of cane. The musician blows into the mouthpiece, which causes the reed to vibrate. This starts the vibration of the column of air. Measurement and Geometry.; Mathematical Reasoning. sixty-five
Percussion Instruments A drummer holds a West African talking drum, or donno, between the upper arm and the body. Squeezing the strings with the upper arm tightens the drumhead and raises the pitch of the drum. Releasing the strings loosens the drumhead and lowers the pitch of the drum. sixty-six In a trap set, the largest drum the base or kick drum produces the lowest-pitched notes. Each drum can be tuned up or down by tightening or loosening the heads. The steel drum, from the Caribbean island of Trinidad, is made by cutting off the top of a steel oil barrel. Each small rounded section of the drum head is shaped to play a different pitched note. The pitch of the instrument can be very high because the small metal sections vibrate rapidly. Drums are percussion instruments. The size of the instrument affects the pitch it can play. The size and tightness of the drum head and the materials that the drum head is made from also affect the pitch. Measurement and Geometry.; Mathematical Reasoning.
Stringed Instruments The pitch of the notes that a stringed instrument can play is related to the length, diameter, and tension of the strings. The violin, the smallest member of the string family, has short strings with small diameters. It is designed to play high-pitched notes. When a player presses down on a string, the vibrating part is shortened and the pitch becomes higher. When a musician winds a string tighter around its tuning peg, the string is tightened and the pitch becomes higher. This man is tuning his stringed instrument. Tightening a string raises the pitch. Loosening a string lowers the pitch. Compared to the violin, the cello has longer strings of greater diameters. It is designed to play low-pitched notes. Measurement and Geometry.; Mathematical Reasoning. sixty-seven
The Piano Looking closely at the way a piano works can help you see some of the mathematical relationships in music. A piano s sound begins when a player presses a key. This causes a felt-covered wooden hammer to hit the strings for that key. The strings then vibrate to produce sound. Each key produces a note with a different pitch. As you move to the right on the piano keyboard, the frequencies get higher. What patterns do you see in the frequencies? one octave C An octave begins and ends on a note with the same name. For example, the keys between Middle C and the C to the right of it represent octave. There are octaves on most pianos. The names of the white and black keys in an octave repeat eight times. Middle C This tuning fork vibrates times per second. A piano tuner tightens or loosens the A string until its pitch exactly matches the pitch of the vibrating tuning fork. Then all other strings are tightened or loosened based on that note. What patterns can you find in music? How have you seen mathematics used in music? sixty-eight Measurement and Geometry.; Mathematical Reasoning.