FRACTIONS: MULTIPLICATION 1

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1 . Fractions: Fractions Multiplication FRACTIONS: MULTIPLICATION In this lesson students extend the concepts and procedures for multiplication of whole numers to fractions. They interpret multiplication of a fraction y a whole numer as repeated addition. They use the area model to understand multiplication of two fractions or mixed numers. From diagrams and examples, the students oserve that the multiply a c ac across rule for multiplication of fractions = is a sensile procedure. d d This lesson falls near the end of a lesson cluster that focuses on fraction concepts and operations. In earlier lessons, students explored the meaning of parts and wholes; used visual models and numerical strategies to convert etween fractions, decimals, and percents; compared and located rational numers on a numer line; and developed understanding of fraction addition and sutraction. In the lessons ahead, the meaning of fraction division will e explored and students will practice operations for fluency. Math Goals (Standards for posting in old) Extend concepts of whole numer multiplication to fraction multiplication. (Gr NS.; Gr7 NS.; Gr7 MR.; Gr7 MR.) Develop fraction multiplication concepts and procedures. (Gr5 NS.0; Gr6 NS.; Gr6 NS.) Summative Assessment Future Week Week 7: Fractions: Multiplication and Division (Gr6 NS.; Gr7 NS.) Unit : Expressions and Equations (Teacher Pages) Week TP8

2 . Fractions: Fractions Multiplication PLANNING INFORMATION Student Pages Estimated Time: 5 60 Minutes Materials Reproduciles * SP: Ready, Set, Go * SP: Fraction Multiplication: Repeated Addition * SP: Fraction Multiplication: Area Model * SP: Fraction Multiplication: Area Model (continued) * SP5: More Fraction Multiplication SP6: Fraction Multiplication Practice Homework SP6: Fraction Multiplication Practice Prepare Ahead Management Reminders Assessment Strategies for English Learners Strategies for Special Learners * SP5: Knowledge Check R5-5: Knowledge Challenge A5: Weekly Quiz Write vocaulary words on the oard along with pictorial examples. proper fraction improper fraction 7 mixed numer 5 Compare conversational and academic meanings of proper, improper, and mixed. Link multiplication of fractions to addition and multiplication of whole numers. This will increase concept development and reduce the numer of procedures that students need to rememer. * Recommended transparency: Blackline masters for overheads 0- and 7 can e found in the Teacher Resource Binder. Unit : Expressions and Equations (Teacher Pages) Week TP9

3 . Fractions: Fractions Multiplication THE WORD BANK area model proper fraction The area model for multiplication is a pictorial way of representing multiplication. In the area model, the length and width of a rectangle represent factors, and the area of the rectangle represents their product. A proper fraction is a fraction of the form m n, where m < n. Example: The fractions,, and 5 6 are proper fractions. improper fraction An improper fraction is a fraction of the form m n where m n > 0. Example: The fractions, 7, and 6 are improper fractions. mixed numer r A mixed numer is an expression of the form p for the numer q where p and r q is a proper fraction. We may regard r areviation for r p +. q p as an q r p +, q Example: The improper fraction 7 = + can e represented y the mixed numer ( four and one fourth ). This should not e confused with the product =. multiplicative identity property The multiplicative identity property states that a = a = a for all numers a. In other words, is a multiplicative identity. The multiplicative identity property is sometimes called the multiplication property of. Example: =, (-5) = -5 multiplicative inverse For 0, the multiplicative inverse of is the numer, denoted y, that satisfies ( ) as the reciprocal of. =. The multiplicative inverse of is also referred to Example: The multiplicative inverse of is, since =. Unit : Expressions and Equations (Teacher Pages) Week TP0

4 . Fractions: Fractions Multiplication MATH BACKGROUND Extending Multiplication to Fractions Two important mathematical ideas that were developed in previous lessons are used here to explain multiplication of fractions.. Multiplication y a whole numer can e interpreted as repeated addition. For whole numers, this means that groups of can e written as = + = 6. When extended to fractions, this means that groups can e written as of 9 = + + = =. groups of groups of Math Background Preview/Warmup Introduce. An area model can e used to explain multiplication ecause the product of the ase and height of a rectangle equals the area of the rectangle. When the ase and height of the rectangle (i.e. factors) are written in an expanded form, the smaller rectangles inside of the large rectangle represent partial products. The sum of these partial products is the product of the factors. This diagram shows that 57 =, =,. The reasonaleness of the multiplication rule for fractions is evident in products displayed with an area model. This diagram shows that = 6 The numerator of the product represents the numer of parts in the shaded region (). The denominator of the product represents the numer of parts in the whole square (6). Multiplication of mixed numers can e illustrated using an area model as well. This approach applies the distriutive property to an expanded from of fractions as shown here: 6 = = 6 = , x 57, , Area model rectangles are not drawn to scale. Unit : Expressions and Equations (Teacher Pages) Week TP

5 . Fractions: Fractions Multiplication MATH BACKGROUND (continued) The Multiply Across Rule for the Multiplication of Fractions The plausiility of the multiply across rule for multiplication of fractions has een estalished using the area model (see Math Background ). Here we provide a mathematical justification of the multiply across rule on the asis of definitions and properties of arithmetic. As a warmup, we justify two asic identities. a First identity (the ig one identity): = a for a 0 By the definition of fraction notation, Hence, a a a = a =. a = a, where is the multiplicative inverse of. Second identity (multiplicative inverse of a product): = for, d 0. d d To see that is the multiplicative inverse of d, we simply multiply it y d : d Math Background Teacher Mathematical Insight ( d) = d = = d. d Here we have used the commutative property of multiplication, the multiplicative identity property, and the definition of as the multiplicative inverse of e. e a c a c d d Multiply across rule: = for, d 0. Note that the second identity aove is a special case of the multiply across rule, where the numerators are oth equal to. For the general case, we use the associative and commutative properties of multiplication and the definitions to otain: a c = a c d d = a c d = a c d a c = d (definition of a and c d ) (commutativity) (formula two proved aove) (definition of fraction notation) Unit : Expressions and Equations (Teacher Pages) Week TP

6 . Fractions: Fractions Multiplication MATH BACKGROUND (continued) The Multiplicative Inverse of a Fraction Math Background Teacher Mathematical Insight It is intuitively clear that the multiplicative inverse of a is. This can e justified easily y a using the multiply across rule, commutativity, and the ig one identity. a a a = = = a a a Another way to express this, using the notation for division, is that the multiplicative inverse of a is a. In other words, a = a. Preview of the Multiply-y-the-Reciprocal Rule for Division The multiply-y-the-reciprocal rule for division is that a c = a d for, c, d 0. d c Math Background Teacher Mathematical Insight This rule follows immediately from the rule for division (division y c is defined to e d multiplication y the multiplicative inverse of c ), and the fact that the multiplicative inverse d of c d is d c (proved in math ackground ). We will return to the multiply-y-the-reciprocal rule in. when we take up division of fractions. TEACHING TIPS Group Work: Questioning, Assessing, Accountaility Group work offers many enefits to students. Group work requires active participation (written and veral) among learners, which increases the chance that more students will learn more mathematics. English language learners are more likely to participate in group discussions than whole class discussions ecause they are less intimidating. Teaching Tip Explore Some questions and statements that promote thinking and keep students on task are: I notice that Jose wrote while Jesse wrote. Are they oth correct? Beth, explain to Wanda why you wrote that answer. You all have the same answers. If I called on your group to explain how you got this, are you sure you would e ale to convince the rest of the class that you are right? It appears that Terry has een doing the majority of the work in this group. The rest of you will e the ones to come to the overhead to explain this work in 5 to 0 minutes. Unit : Expressions and Equations (Teacher Pages) Week TP

7 . Fractions: Fractions Multiplication PREVIEW / WARMUP Whole Class SP* Ready, Set, Go Math Background Introduce the goals and standards of the lesson. Discuss important vocaulary as relevant. Students practice the previously learned skills of whole numer multiplication; represented as repeated addition and with an area model; and fraction addition. All three are important su-skills for this lesson. Discuss as needed. INTRODUCE Whole Class SP* Fraction Multiplication: Repeated Addition SP* Fraction Multiplication: Area Model SP* Fraction Multiplication: Area Model (continued) SP5* More Fraction Multiplication (start) Math Background Guide students through the various exercises designed to uild understanding of fraction multiplication. Looking at prolems #-7 on SP, how is multiplying a whole numer y a proper fraction like whole numer multiplication? They can oth e performed as repeated addition. Looking at prolems #-7 on SP-, how is multiplying a proper fraction y a proper fraction like whole numer multiplication? The dimensions of the rectangles represent the factors. For proper fractions, the product of the numerators represents the numer of parts in the shaded region (area), and the product of the denominators represents the numer of parts in one whole square unit. a c ac =, seen in This diagram suggests the plausiility of the multiply across rule ( ) prolem #8. d d Looking at prolems #9-0 on SP and prolems #- on SP5, how can the area model e used to multiply a whole numer y a mixed numer? Write the mixed numer in an expanded form. Then use the distriutive property (write numerically or display visually with an area diagram) to find the partial products, and add them together. Repeated addition can e used to find the partial product of the whole numer and fraction. ( ) ( ) ( ) () = () = () = 6 = 7 EXPLORE Small Groups/Pairs SP5* More Fraction Multiplication (finish) Students do the next prolems on the page as recommended. Invite students to come to the overhead to demonstrate their understanding of these methods for multiplying mixed numers. Teaching Tip Unit : Expressions and Equations (Teacher Pages) Week TP

8 . Fractions: Fractions Multiplication SUMMARIZE Whole Class SP5* More Fraction Multiplication (finish) Invite students to the overhead to share alternative strategies for multiplying fractions. For prolems and 5, what are some ways to multiply two mixed numers? () Make an area diagram and write the numers in an expanded form. The partial products will include the product of a whole numer, the product of a whole numer and a fraction, and the product of two fractions. Then, add the partial products together. () Change oth mixed numers to improper fractions and use the rule illustrated in prolem #8 on SP. PRACTICE Individuals SP6 Fraction Multiplication Practice This page is appropriate for practice as class work or homework. CLOSURE Whole Class SP* Ready, Set Review the goals and standards for the lesson. Unit : Expressions and Equations (Teacher Pages) Week TP5

9 . Fractions: Fractions Multiplication SP Warmup (Go) SP Fraction Multiplication: Repeated Addition SP Fraction Multiplication: Area Model SP Fraction Multiplication: Area Model (continued) SP5 More Fraction Multiplication SP6 Fraction Multiplication Practice SELECTED SOLUTIONS = = 8 5., =,76. 5 = = = = = = 8 a c ac d d 8. = = = 6 6 = = 7. 6 The numerator of the product represents the numer of parts in the shaded region. The denominator of the product represents the numer of parts in the whole square = = = = = = = = Unit : Expressions and Equations (Teacher Pages) Week TP6

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11 STUDENT PAGES

12 . Fractions: Multiplication FRACTIONS: MULTIPLICATION Ready (Summary) We will use whole numer multiplication concepts to provide meaning for fraction multiplication. Set (Goals) Extend concepts of whole numer multiplication to fraction multiplication. Develop fraction multiplication concepts and procedures. Go (Warmup) Write each multiplication expression as a repeated addition expression. Then, find the sum.. 6(5) = (7) = + + = Compute each sum = + + Multiply using an area model (not drawn to scale). 5. 8(7) Unit : Expressions and Equations (Student Packet) Week SP

13 . Fractions: Multiplication FRACTION MULTIPLICATION: REPEATED ADDITION One way to interpret multiplication is using repeated addition. That is, groups of can e written as: = + = 6 This works for fractions as well: groups of can e written as: 9 = + + = = Write each statement as a product and as repeated addition. Then, draw a picture to represent the repeated addition.. groups of 8 can e written as: 8 = + + = Picture:. groups of 5 can e written as: = + = 5 Picture:. groups of 7 can e written as:. groups of can e written as: = + + = Multiply using repeated addition Unit : Expressions and Equations (Student Packet) Week SP

14 . Fractions: Multiplication FRACTION MULTIPLICATION: AREA MODEL An area model is another way to explain multiplication. This y rectangle has an area of = 6 An area model is also useful for multiplying proper fractions. This is a square whose side length is unit. A rectangle that is y is shaded inside of it. The shaded area shows that =. 6 Use an area model to find each product.. This is a unit y unit square. Mark the side lengths. Shade a rectangle that is y.. This is a unit y unit square. Mark the side lengths. Shade a rectangle that is y. = =. This is a unit y unit square. Mark the side lengths. Shade a rectangle that is y.. This is a unit y unit square. Mark the side lengths. Shade a rectangle that is y. = = Unit : Expressions and Equations (Student Packet) Week SP

15 . Fractions: Multiplication FRACTION MULTIPLICATION: AREA MODEL (continued) Multiply using an area model Use the results of the shaded rectangles to state a multiply across rule. a c d = The numerator of the product represents The denominator of the product represents We will call this the multiply across rule. Use an area model to find each product (rectangles are not drawn to scale) = 5 (0+ 6) 0. = () Answer: + = Answer: + = Unit : Expressions and Equations (Student Packet) Week SP

16 . Fractions: Multiplication Multiply. MORE FRACTION MULTIPLICATION To multiply mixed numers, use an area model diagram or use the multiply across rule (rectangles are not drawn to scale).. This rectangle is y. Find the area of each smaller rectangle. 5. To use the multiply across rule, change each mixed numer to an improper fraction and then multiply. + = = + = Rectangle is not drawn to scale. = This shows that = = Unit : Expressions and Equations (Student Packet) Week SP5

17 . Fractions: Multiplication FRACTION MULTIPLICATION PRACTICE Compute using an area model, repeated addition, or the multiply across rule Unit : Expressions and Equations (Student Packet) Week SP6