Mathematics
Multiplication and Division of Fractions
Multiplication of fractions involves multiplying the numerators together to get the new numerator and multiplying the denominators together to get the new denominator. Division of fractions is done by multiplying the first fraction by the reciprocal of the second fraction. In both operations, it's important to simplify the fractions by canceling out common factors.
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11 Key excerpts on "Multiplication and Division of Fractions"
eBook - PDF- Robert Reys, Mary Lindquist, Diana V. Lambdin, Nancy L. Smith, Anna Rogers, Audrey Cooke, Bronwyn Ewing, Kylie Robson(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
Therefore, the solution is clearly 2 5 . In contrast, children who follow a rule (e.g. convert the two wholes to ten-fifths, subtract the eight-fifths, gives two-fifths) might make the problem harder, and they are unlikely to have the flexibility of procedural fluency, an essential part of being mathematically proficient. Children who have a firm understanding of fractions will find ways to add and subtract fractions without having to rely solely on the traditional procedures. Multiplication The procedure for multiplication of fractions is one of the simplest rules that can be learned, but doing so does not build understanding of fractions and can lead to children becoming confused with addition processes. Learning the rule by rote can happen in minutes (followed by forgetting it in seconds), but does not give children insight into why it works or when to use it. Rather, it is suggested that learning multiplication of fractions needs to happen meaningfully by building up a strong conceptual knowledge of multiplication, particularly because the size of an answer needs to make sense. In the following sections, three different cases of multiplication are explained. These are: multiples of a fraction; fraction of an amount (fraction times a whole number); and a fraction of a fraction. In each case, multiplication with fractions is connected to multiplicative thinking. For example, knowing that one meaning of multiplication is equal groups (3 × 4 is three groups of four) and that another meaning is the area of a rectangle (three rows by-four columns), is very helpful. In the primary years, it is more important to build conceptual understanding of the operations than to apply procedures. CHAPTER 12 Fractions and decimals: meanings and operations 421 Multiples of a fraction One way to gain an understanding of multiplying a whole number with a fraction is to use a region model. For example, you have three serving plates, each with 4 5 of a pizza.
eBook - PDFUnpacking Fractions
Classroom-Tested Strategies to Build Students' Mathematical Understanding
- Monica Neagoy(Author)
- 2017(Publication Date)
- ASCD(Publisher)
FIGURE 6.5 Meanings of Multiplication and Division Multiplication Division 1. Adding equal groups 1A. Equal or fair sharing (partitive concept) 2. Increasing or reducing quantities 3. Moving from factors to product or product to factors 1B. Equal grouping or segmenting (quotative, measurement, or repeated subtraction concept) Adding equal groups multiplication. The CCSS for 4th grade math state that students will “apply and extend previous understandings of multiplication to multiply a fraction by a whole number” (Common Core State Standards Initiative, 2010). Students know that 6 × 4 can be interpreted as six equal groups of four, or 4 + 4 + 4 . . . (six times), or even six hops of four units on the number line, starting at 0. Similarly, 6 × 2 _ 8 means six copies or iterations of the quantity two-eighths. This was Eric’s reasoning when he drew six copies of two-eighths of a pizza and counted them by repeated addition. 202 // Unpacking Fractions The CCSS for 5th grade math (Number and Operations—Fractions) state that students will “apply and extend previous understandings of mul-tiplication to multiply a fraction or whole number by a fraction” (Common Core State Standards Initiative, 2010). Samantha’s choice of the commuted factors is a good example: 2 _ 8 × 6, or 1 _ 4 × 6. We’d like students to think of 1 _ 4 × 6 as “ 1 _ 4 of 6,” as Samantha did, but too often they memorize “ of means times ,” because we tell them that. The equal groups of meaning of multipli-cation helps make the connection: if 3 × 6 means three groups of six, and 1 × 6 means one group of six, then 1 _ 4 × 6 likewise means one-fourth of a group of six. Division as the inverse of equal-groups multiplication. Since the two factors in equal-groups multiplication play different roles, and since divi-sion is the inverse of multiplication, two different division interpretations emerge: division by multiplier and division by multiplicand.
eBook - PDFMathematics for Elementary Teachers
A Contemporary Approach
- Gary L. Musser, Blake E. Peterson, William F. Burger(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
These approaches provide a meaningful way of learn- ing fraction division. Such approaches are a departure from simply memorizing the rote procedure of “invert and multiply,” which offers no insight into fraction division. 238 Chapter 6 Fractions By using common denominators, division of fractions can be viewed as an exten- sion of whole-number division. For example, 6 7 2 7 ÷ is just a measurement division problem where we ask the question, “How many groups of size 2 7 are in 6 7 ?” The answer to this question is the equivalent measurement division problem of 6 2 ÷ , where the question is asked, “How many groups of size 2 are in 6?” Since there are three 2s in 6, there are three 2 7 s in 6 7 . Figure 6.24 illustrates this visually. In general, the division of fractions in which the divisor is not a whole number can be viewed as a measurement division problem. On the other hand, if the division problem has a divisor that is a whole number, then it should be viewed as a sharing division problem. Figure 6.24 Reflection from Research Until students understand opera- tions with fractions, they often have misconceptions that multi- plication always results in a larger answer and division in a smaller answer (Greer, 1988). Find the following quotients. a. 12 13 4 13 ÷ b. 6 17 3 17 ÷ c. 16 19 2 19 ÷ SOLUTION a. 12 13 4 13 3 ÷ = , since there are three 4 13 in 12 13 . b. 6 17 3 17 2 ÷ = , since there are two 3 17 in 6 17 . c. 16 19 2 19 8 ÷ = , since there are eight 2 19 in 16 19 . Notice that the answers to all three of these problems can be found simply by dividing the numerators in the correct order. ■ In the case of 12 13 5 13 ÷ , we ask ourselves, “How many 5 13 make 12 13 ?” But this is the same as asking, “How many 5s (including fractional parts) are in 12?” The answer is 2 and 2 5 fives or 12 5 fives. Thus 12 13 5 13 12 5 ÷ = . Generalizing this idea, fraction division is defined as follows.
eBook - PDF- Tom Bassarear, Meg Moss(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
218 CHAPTER 4 Understanding Multiplication and Division CLASSROOM CONNECTION Grade 4 From Big Ideas Math Modeling Real Life Grade 4 Volume 1 by Ron Larson and Laurie Boswell. Reprinted by permission. Copyright © 2019 Big ideas Learning, LLC. All rights reserved. oksanika/Shutterstock.com stockcam/Getty Images Section 4.3 Understanding Multiplication and Division of Fractions 219 Just as we decomposed and recomposed numbers in working with whole numbers, we need to decompose this (shaded) amount in such a way that we can recompose the amount as a set of equal-size pieces. We can create equal-size pieces (needed to name a fraction) by dividing the square first into fourths (with vertical lines) and then into thirds (with horizontal lines). If we put the two diagrams together and shade in the area enclosed by 3 4 times 2 3 , we have the figure on the right in Figure 4.18. How can you determine the answer now from the diagram? How can you justify that answer? Think and then read on. . . . Because our unit ( 1 square mile) has been divided into 12 equal-size regions, each region has a value of 1 12 square mile. The plot of land covers 6 of these rectangles, so its value is 6 12 or 1 2 square mile. This model helps to explain the procedure of “multiply straight across” that is often used for multiplying fractions. The shaded rectangle is 2 by 3 because of the numerators. The entire unit is 3 by 4 because of the denominators. There is a difference between multiplying whole numbers and multiplying fractions that bears noting, for it often baffles children when they encounter fraction computations for the first time. One notion of multiplication that many people (often unconsciously) get from working with whole numbers is that “multiplication makes bigger and division makes smaller,” but just the opposite is true when we multiply proper fractions! C.
eBook - PDF- Michael A. Calter, Paul A. Calter, Paul Wraight, Sarah White(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
9–1 Simplification of Fractions Parts of a Fraction A fraction has a numerator, a denominator, and a fraction line. fraction line a b numerator denominator Quotient A fraction is a way of indicating a quotient of two quantities. The fraction a/b can be read “a divided by b.” The two ways of writing a fraction, a b and a/b, are equally valid. Ratio We also speak of the quotient of two numbers or quantities as the ratio of those quantities. Thus the ratio of x to y is x y . 9 ◆◆◆ OBJECTIVES ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ When you have completed this chapter, you should be able to: • Simplify fractional expressions. • Multiply and divide fractional expressions. • Add and subtract fractional expressions. • Simplify complex fractional expressions. • Solve fractional equations. • Solve word problems using fractional equations. • Manipulate and work with literal equations and formulas. You already know about fractions with numbers. In algebra, however, the numbers are replaced with letters, coefficients, and even entire expressions. Many equations and formulas in science and technology are in the form of a fraction. Since the rules of working with the numerators and denominators of fractions must be applied to entire algebraic expressions, we’ll need to make much use of the factoring techniques of Chapter 8 in order to simplify them. As we work with formulas that include fractions, we must be careful: it’s very easy to make mistakes when we cross multiply. Remember that any- thing you do must be done to each term on both sides of the equation. Also, don’t be intimidated by complex fractions where a numerator or denominator might contain a fraction; use your skills and take it one step at a time. Not all of this material is new to us. Some was covered in Chapter 2, and we solved simple fractional equations in Chapter 3. Fractions and Fractional Equations
- Robert Brechner, Geroge Bergeman(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
This concept is used quite commonly in business. We may look at sales for 1 2 the year or reduce prices by 1 4 for a sale. A new production machine in your company may be 1 3 4 times faster than the old one, or you might want to cut 5 3 4 yards of fabric from a roll of material. Just like whole numbers, fractions can be added, subtracted, multiplied, divided, and even combined with whole numbers. This chapter introduces you to the various types of fractions and shows you how they are used in the business world. D ISTINGUISHING AMONG THE V ARIOUS T YPES OF F RACTIONS Technically, fractions express the relationship between two numbers set up as division. The numerator is the number on the top of the fraction. It represents the dividend in the division. The denominator is the bottom number of the fraction. It represents the divisor. The numera-tor and the denominator are separated by a horizontal or slanted line, known as the division line . This line means “divided by.” For example, the fraction 2/3 or 2 3 , read as “two-thirds,” means 2 divided by 3 , or 2 ÷ 3 . Numerator Denominator 2 3 Remember, fractions express parts of a whole unit. The unit may be dollars, feet, ounces, or anything else. The denominator describes how many total parts are in the unit. The numer-ator represents how many of the total parts we are describing or referring to. For example, an apple pie (the whole unit) is divided into eight slices (total equal parts, denominator). As a fraction, the whole pie would be represented as 8 8 . If five of the slices were eaten (parts referred to, numerator), what fraction represents the part that was eaten? The answer would be the fraction 5 8 , read “five-eighths.” Because five slices were eaten out of a total of eight, three slices, or 3 8 , of the pie is left. fractions A mathematical way of expressing a part of a whole thing. For example, 1 4 is a fraction expressing one part out of a total of four parts.
- Mark Zegarelli(Author)
- 2022(Publication Date)
- For Dummies(Publisher)
192 UNIT 4 Fractions Multiplying and Dividing Fractions One of the odd little ironies of life is that multiplying and dividing fractions is usually easier than adding or subtracting them. For this reason, I show you how to multiply and divide frac- tions before I show you how to add or subtract them. In fact, you may find multiplying fractions easier than multiplying whole numbers because the numbers you’re working with are usually small. More good news is that dividing fractions is nearly as easy as multiplying them. So I’m not even wishing you good luck — you don’t need it! Multiplying numerators and denominators straight across Everything in life should be as simple as multiplying fractions. All you need for multiplying fractions is a pen or pencil, something to write on, and a basic knowledge of the multiplication table. (See Chapter 3 for a multiplication refresher.) To multiply two fractions, multiply the numerators (the numbers on top) to get the numer- ator of the answer, then multiply the denominators (the numbers on the bottom) to get the denominator of the answer. For example, here’s how to multiply 2 5 3 7 × : 2 5 3 7 2 3 5 7 6 35 When multiplying fractions, you can often make your job easier by canceling out equal factors in the numerator and denominator. Canceling out equal factors makes the numbers that you’re multiplying smaller and easier to work with, and it also saves you the trouble of reducing at the end. Here’s how it works: » When the numerator of one fraction and the denominator of the other are the same, change both of these numbers to 1s. (See the nearby sidebar for why this works.) » When the numerator of one fraction and the denominator of the other are divisible by the same number, factor this number out of both. In other words, divide the numerator and denominator by that common factor. (For more on how to find factors, see Chapter 9.) For example, suppose you want to multiply the following two numbers: 5 13 13 20 × .
eBook - PDFIntroductory Mathematics
Concepts with Applications
- Charles P. McKeague(Author)
- 2013(Publication Date)
- XYZ Textbooks(Publisher)
OBJECTIVES 82 KEY WORDS Chapter 2 Multiplication and Division of Fractions fraction terms numerator denominator proper improper equivalent multiplication property for fractions division property for fractions scatter diagram line graph A Identify the numerator and denominator of a fraction. B Locate fractions on the number line. C Write fractions equivalent to a given fraction. 2.1 The Meaning and Properties of Fractions The information in the table below shows enrollment for Cal Poly State University in California. The pie chart was created from the table. Both the table and pie chart use fractions to specify how the students at Cal Poly are distributed among the different schools within the university. From the table, we see that 1 _ 9 (one-ninth) of the students are enrolled in the College of Business. This means one out of every nine students at Cal Poly is studying Business. The fraction 1 _ 9 tells us we have 1 part of 9 equal parts. That is, the students at Cal Poly could be divided into 9 equal groups, so that one of the groups contained all the business students and only business students. Figure 1 shows a rectangle that has been divided into equal parts, four different ways. The shaded area for each rectangle is 1 _ 2 the total area. Now that we have an intuitive idea of the meaning of fractions, here are the more formal definitions and vocabulary associated with fractions. FIGURE 1 is shaded a. 1 2 b. are shaded 2 4 c. are shaded 3 6 d.
- Alan Tussy, Diane Koenig(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
1. When a fraction is followed by the word of, such as 1 3 of, it indicates that we are to find a part of some quantity using . 2. The answer to a multiplication is called the . 3. To a fraction, we remove common factors of the numerator and denominator. 4. In the expression 1 1 4 2 3 , the is 1 4 and the is 3. 5. The of a triangle is the amount of surface that it encloses. 6. Label the base and the height of the triangle shown below. CONCEPTS 7. Fill in the blanks: To multiply two fractions, multiply the and multiply the . Then , if possible. 8. Use the following rectangle to find 1 3 ? 1 4 . a. Draw three vertical lines that divide the given rectangle into four equal parts and lightly shade one part. What fractional part of the rectangle did you shade? b. To find 1 3 of the shaded portion, draw two horizontal lines to divide the given rectangle into three equal parts and lightly shade one part. Into how many equal parts is the rectangle now divided? How many parts have been shaded twice? c. What is 1 3 ? 1 4 ? 9. Determine whether each product is positive or negative. You do not have to find the answer. a. 2 1 8 ? 3 5 b. 2 7 16 a2 2 21 b c. 2 4 5 a 1 3 ba2 1 8 b d. 2 3 4 a2 8 9 ba2 1 2 b 10. Translate each phrase to symbols. You do not have to find the answer. a. 7 10 of 4 9 b. 1 5 of 40 Fill in the blanks. 11. Area of a triangle 5 1 2 ( )( ) or A 5 12. Area is measured in units, such as in. 2 and ft 2 . NOTATION 13. Write each of the following as a fraction. a. 4 b. –3 c. x 14. Fill in the blanks: 1 1 2 2 2 represents the repeated multiplication ? . Fill in the blanks to complete each step. 15. Multiply and simplify: 5 8 ? 7 15 5 5 ? 8 ? 5 5 ? 7 ? 2 ? 2 ? ? 5 5 1 ? 7 2 ? 2 ? 2 ? 3 ? 1 5 7 Answers to Self Checks 1. a. 1 16 b. 10 27 2. 2 5 18 3. 7 3 4. 2 5 5. 3 26 6. a. 8 125 b. 9 16 c. 2 9 16 7. 64 votes 8. 216 in. 2 Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
eBook - PDFMathematical Practices, Mathematics for Teachers
Activities, Models, and Real-Life Examples
- Ron Larson, Robyn Silbey(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
Recognize equivalent fractions and find simplest form. Use a real number line to graph and order fractions. 6.2 Adding and Subtracting Fractions (page 209) 39–84 Add fractions with like denominators. Subtract fractions with like nominators. Add and subtract fractions with unlike denominators. 6.3 Multiplying and Dividing Fractions (page 219) 85–132 Multiply fractions, whole numbers, and mixed numbers. Divide fractions, whole numbers, and mixed numbers. Estimate products and quotients of fractions. 6.4 Ratios and Proportion (page 231) 133–146 Write a ratio as a part-to-part, a part-to-whole, or a whole-to-part comparison. Write and simplify rates. Decide whether two ratios or two rates are proportional. Important Concepts and Formulas Simplified Fraction Adding Fractions Adding Fractions Subtracting Fractions A fraction is in simplest form when its numerator and denominator have no common factors other than 1. a — c + b — c = a + b — c Common denominator a — b + c — d = ad + bc — bd Unlike denominators a — c − b — c = a − b — c Common denominator Subtracting Fractions Multiplying Fractions Dividing Fractions Proportion a — b − c — d = ad − bc — bd Unlike denominators a — b ⋅ c — d = a ⋅ c — b ⋅ d a — b ÷ c — d = a — b ⋅ d — c Multiply by reciprocal. If a — b = c — d , then ad = bc. fraction (p. 197) numerator (p. 197) denominator (p. 197) area model (p. 198) set model (p. 198) linear model (p. 199) proper fraction (p. 200) improper fraction (p. 200) mixed number (p. 200) equivalent (p. 201) simplest form (p. 201) cross products (p. 202) cross-multiplication (p. 202) is less than (p. 203) benchmarks (p. 203) multiplicative inverse (p. 221) reciprocal (p. 221) complex fraction (p. 224) ratio (p. 231) part-to-part (p. 231) part-to-whole (p. 231) whole-to-part (p. 231) rate (p. 233) unit rate (p. 233) unit analysis (p. 233) proportion (p. 234) proportional (p. 234) Copyright 2014 Cengage Learning.
eBook - PDF- Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis(Authors)
- 2020(Publication Date)
- Openstax(Publisher)
When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, a, can be written as a 1 . So, 3 = 3 1 , for example. EXAMPLE 4.28 Multiply, and write the answer in simplified form: ⓐ 1 7 · 56 ⓑ 12 5 (−20x) Solution ⓐ 1 7 · 56 Write 56 as a fraction. 1 7 · 56 1 Determine the sign of the product; multiply. 56 7 Simplify. 8 304 Chapter 4 Fractions This OpenStax book is available for free at http://cnx.org/content/col30939/1.6 ⓑ 12 5 (−20x) Write −20x as a fraction. 12 5 ⎛ ⎝ −20x 1 ⎞ ⎠ Determine the sign of the product; multiply. − 12 · 20 · x 5 · 1 Show common factors and then remove them. Multiply remaining factors; simplify. −48x TRY IT : : 4.55 Multiply, and write the answer in simplified form: ⓐ 1 8 · 72 ⓑ 11 3 (−9a) TRY IT : : 4.56 Multiply, and write the answer in simplified form: ⓐ 3 8 · 64 ⓑ 16x · 11 12 Find Reciprocals The fractions 2 3 and 3 2 are related to each other in a special way. So are − 10 7 and − 7 10 . Do you see how? Besides looking like upside-down versions of one another, if we were to multiply these pairs of fractions, the product would be 1. 2 3 · 3 2 = 1 and − 10 7 ⎛ ⎝ − 7 10 ⎞ ⎠ = 1 Such pairs of numbers are called reciprocals. Reciprocal The reciprocal of the fraction a b is b a , where a ≠ 0 and b ≠ 0, A number and its reciprocal have a product of 1. a b · b a = 1 To find the reciprocal of a fraction, we invert the fraction. This means that we place the numerator in the denominator and the denominator in the numerator. To get a positive result when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign. To find the reciprocal, keep the same sign and invert the fraction. The number zero does not have a reciprocal. Why? A number and its reciprocal multiply to 1. Is there any number r so that 0 · r = 1? No. So, the number 0 does not have a reciprocal. Chapter 4 Fractions 305
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