Dividing Rational Numbers LAUNCH (7 MIN) Before What are the rules for dividing two integers with the same sign? With different signs? During What is one way you can sort these expressions? After How would you write an expression for division of fractions as an equivalent multiplication expression? PART 1 (8 MIN) How do you write the reciprocal of a fraction? How can you switch the numerator and denominator of an integer? Jay Says (Screen ) Use the Jay Says button to introduce reciprocals and help students remember what they are. How do you know which sign the reciprocal should have? Why do you first write a mixed number as an improper fraction when finding its reciprocal? Why is multiplicative inverse a good name for a reciprocal? PART 2 (8 MIN) How is dividing by a negative fraction different from dividing by a positive fraction? Jay Says (Screen ) Use the Jay Says button to emphasize that it is the divisor, the number after the division symbol, that is rewritten when you change division to multiplication. How would you simplify the multiplication expressions? PART (7 MIN) How do you know which operation to use when solving a one-step equation? While solving the problem Why do you represent the distance with a negative number? Why would dividing the distance by the rate give the same solution as solving the equation!2 2 =! 4 5 t? KEY CONCEPT ( MIN) Use the Key Concept to summarize that finding a reciprocal is easiest when the rational number is written as a fraction because you can flip the numerator and denominator. Why is the reciprocal of a negative rational number also negative? CLOSE AND CHECK (7 MIN) How are a number and its reciprocal related? How do you use multiplication to divide two rational numbers? How are the signs of a dividend and divisor related to the sign of the quotient when you divide positive and negative rational numbers?
Dividing Rational Numbers LESSON OBJECTIVES 1. Apply and extend previous understandings of multiplication and division of fractions to multiply and divide rational numbers. 2. Interpret quotients of rational numbers by describing real-world contexts. FOCUS QUESTION How does the relationship between multiplication and division help you divide rational numbers? MATH BACKGROUND Students have previously learned the inverse relationship between multiplication and division and used the relationship to divide fractions and divide integers. They learned algorithms to divide positive fractions and decimals. Students have also used this inverse relationship to connect the rules for multiplying integers with dividing integers. They will combine these techniques and rules to divide rational numbers in this lesson. In this lesson, students divide rational numbers by multiplying by the reciprocal (multiplicative inverse) of the divisor. They learn that the product of a number and its multiplicative inverse is 1 and practice finding reciprocals in preparation to divide rational numbers. Because the focus of this lesson is on multiplying by reciprocals, decimals are used sparingly in this lesson. Students will later use their knowledge of dividing rational numbers to help them simplify rational expressions with all four operations and to help them understand complex fractions. Understanding how to divide rational numbers enables students to solve real-world problems and equations involving rational numbers and multiple operations. LAUNCH (7 MIN) Objective: Connect the rules of dividing with integers to dividing with rational numbers. Students connect the rules for dividing with integers to division involving a mixed number. Students should also connect multiplication and division to what they did when dividing integers. Instructional Design You can move the tiles around on the screen. This gives you some options as students explain their reasoning on how to sort them. Students should realize that the quotients are different based on the signs of the dividend and divisor, and they should know the rules for determining the sign of each quotient as they sort tiles. Before What are the rules for dividing two integers with the same sign? With different signs? [Sample answer: The quotient is positive when the signs are the same and negative when the signs are different.] During What is one way you can sort these expressions? [Sample answer: Put the positive quotients together and put the negative quotients together.]
Dividing Rational Numbers continued After How would you write an expression for division of fractions as an equivalent multiplication expression? [Sample answer: I would multiply by the reciprocal of the divisor.] Students should use what they know about the rules of signed numbers, but they do not have to find the actual quotients to sort the expressions into groups. Since the absolute values of the dividends and divisors are the same for every expression, the absolute values of the quotients will also be the same only the signs will be different. Students should also already be familiar with dividing fractions; explore this concept further by asking them how to write the expressions as multiplication, and have students explain why they can be written this way. Although it is not necessary to find the quotients to sort the tiles, some students may find the value of the expressions before grouping. These students may express the positive quotients as 5 2 or 2 1 2, and the negative quotients as! 5 2 or!2 1 2. Connect Your Learning Move to the Connect Your Learning Screen. Having just completed the lesson on dividing integers, students should be aware that the rules for multiplying signed numbers help you divide signed numbers. Ask students how what they know about multiplying rational numbers will help them divide rational numbers. PART 1 (8 MIN) Objective: Extend previous understandings of multiplication of fractions to write reciprocals of rational numbers. ELL Support Beginning After students read the problem statement, ask if the submarine moved up or down [down]. Discuss the definition of descend. Use the following sentence frames to help students understand the word descend and its different forms: As the submarine was moving down, it was [descending]. Yesterday, the submarine [descended] to 2 2 yesterday. miles below sea level. The submarine began its [descent] Intermediate After students read the problem statement, have them work with a partner to write sentences that describe the submarine s movement using different forms of the word descend. Introduce the word ascend to students and challenge them to include it in one of their descriptions. They may wish to draw a picture of a submarine to illustrate the submarine ascending and descending to use in their descriptions. Advanced Have students work with a partner to make sentences using different forms of the words ascend and descend to describe the movement of the submarine. Challenge them to use each word in four different forms: descend, descending, descended, and descent, for example. Students explore reciprocals of rational numbers and determine whether each rational number is a reciprocal of another rational number. This problem prepares them for dividing rational numbers.
Instructional Design Dividing Rational Numbers continued Use the Intro to relate the reciprocal of a positive fraction to the reciprocal of a negative fraction. Focus on reciprocals as having a product of 1. Multiplicative inverse and reciprocal are different terms for the same concept and may be used interchangeably. You may want to draw a parallel to these two terms and the terms opposites and additive inverses. In the Example, you can call volunteers to the whiteboard to cross out numbers that are not reciprocals or multiply rational numbers to determine whether they are reciprocals. Have students explain their reasoning as they choose. How do you write the reciprocal of a fraction? [Sample answer: Switch the numerator and the denominator.] How can you switch the numerator and denominator of an integer? [An integer has an implied denominator of 1.] Jay Says (Screen 2) Use the Jay Says button to introduce reciprocals and help students remember what they are. How do you know which sign the reciprocal should have? [Sample answer: The number and its reciprocal have the same sign.] Why do you first write a mixed number as an improper fraction when finding its reciprocal? [Sample answer: I need the number in a form in which the number has only a numerator and a denominator to find its reciprocal.] Why is multiplicative inverse a good name for a reciprocal? [Sample answers: When you multiply a number by its multiplicative inverse, the product is 1, the identity. Division is the inverse of multiplication.] Encourage students to multiply the number and the potential reciprocals to check each answer. Students may multiply and then simplify, or they may simplify immediately. Students may assume that reciprocals can also have a product of 1. Differentiated Instruction For struggling students: To make sure that students remember an important property of division, ask students what the reciprocal of 0 is. Students should understand that zero does not have a reciprocal because zero cannot be in the denominator. You cannot split a value into 0 groups. For advanced students: Show students how to find a reciprocal by solving an equation. For example, you can find the reciprocal of! 7 4 by solving! 7 x = 1. The solution 4 is! 4 7. Got It Notes While the simplest form of the reciprocal of! 7 is, is not one of the answer 21 choices. The idea here is for students to find the easiest reciprocal, the one obtained by switching the numerator and denominator. If you show answer choices, consider the following possible student errors:
Dividing Rational Numbers continued Students who choose A may think that the goal is to make the fraction a positive whole number. If students choose B, they are simplifying the fraction to find the reciprocal and but aren t aware of the sign. Students who select D found the opposite, not the reciprocal. Got It 2 Notes Students should understand that zero does not have a reciprocal because zero cannot be in the denominator. You cannot split a value into 0 groups. PART 2 (8 MIN) Objective: Apply previous understandings of division of fractions to write division by a rational number as multiplication by the reciprocal. Students review writing division as multiplication of a reciprocal and match a division expression to its equivalent multiplication expression. Students are familiar with the rule for dividing by a positive fraction and extend this rule to include all rational numbers. Instructional Design This Intro is intended to remind students why dividing by a fraction is equivalent to multiplying by the reciprocal of the fraction. Compare dividing a quantity into and finding 1 of a quantity. The Intro uses color to highlight using the reciprocal of the divisor. On Screen 2, have volunteers come to the whiteboard and circle the correct responses while explaining their reasoning. Have them explain why the other expressions are not equivalent. How is dividing by a negative fraction different from dividing by a positive fraction? [Sample answer: The process of multiplying by the reciprocal is the same. The sign of the quotient will be different from the sign of the dividend.] Jay Says (Screen 2) Use the Jay Says button to emphasize that it is the divisor, the number after the division symbol, that is rewritten when you change division to multiplication. How would you simplify the multiplication expressions? [Sample answer : Multiply the numerators, multiply the denominators, and then simplify. You can also simplify before you multiply.] Extend this problem by having students use what they learned previously in this topic to identify the sign of the quotient or even carry out the multiplication. Differentiated Instruction You may want to use the Multiplication mode of the Area Models tool to reinforce that dividing by a rational number is equivalent to multiplying by its reciprocal.
Got It Notes Dividing Rational Numbers continued If you show answer choices, consider the following possible student errors: If students select A, they are using the reciprocal of both the dividend and divisor. Students who choose C are changing the operation sign to multiplication without using the reciprocal. If students choose D, they are rewriting the same expression they were given. PART (7 MIN) Objective: Use a quotient of rational numbers to solve a problem in a real-world context. Students solve a real-world problem using the quotient of rational numbers. They write an equation to solve the problem, and then solve the equation by dividing rational numbers. This problem encourages students to interpret the solution of an equation in its real-world context. How do you know which operation to use when solving a one-step equation? [I use the inverse operation that is in the equation.] While solving the problem Why do you represent the distance with a negative number? [Sample answer: The negative symbol represents the direction.] Why would dividing the distance by the rate give the same solution as solving the equation!2 2 =! 4 t? [Both equations result in the same operation, dividing 5!2 2 by! 4 5 (the distance divided by the rate).] Use the animated solution to step through the process of substituting into the equation and solving for t. Guide students through the division and reinforce the process. At the end of the animation, point out the reasoning behind rewriting the answer. Emphasize that stopping at t 10 does not make sense in the context of this problem. The time should be in hours and minutes. Use the number line art to help students understand the problem. Explain that the formula distance = rate time represents the distance (in miles) the submarine travels over a certain time (in hours) at a given rate (in mph). Have them compare the formula to d =! 4 5 t to understand that! 4 is the rate, or the constant of proportionality. 5 Discuss that the negative sign only indicates direction. Error Prevention Some students may get the negative symbol caught in the middle while trying to convert the mixed number to an improper fraction. Encourage them to factor out the negative symbol (for example,!2 2 =!(2 2 )) before converting.
Dividing Rational Numbers continued Got It Notes Before students solve this problem, discuss the visual of this problem statement and how it indicates the appropriate sign for the distance 1 1. The distance to the bottom of the canyon is negative in comparison to the rim of the canyon. If you show answer choices, consider the following possible student errors: If students choose B, they are using the reciprocal of the dividend instead of the divisor. Students who select C are converting 2 hour to 2 minutes. Point out that students can multiply 2 hour by 60 minutes per hour to get 40 minutes. KEY CONCEPT ( MIN) Teaching Tips for the Key Concept Use the Key Concept to summarize that finding a reciprocal is easiest when the rational number is written as a fraction because you can flip the numerator and denominator. Have students click on the radio buttons to first show reciprocals and then applying the definition of a reciprocal to dividing by a rational number. Why is the reciprocal of a negative rational number also negative? [The product of reciprocals is 1, a positive number. You must multiply a negative rational number by another negative number to get a positive product.] CLOSE AND CHECK (7 MIN) Focus Question Sample Answer Sample: Dividing by a rational number is equivalent to multiplying by its reciprocal. You can use this relationship to rewrite a quotient of rational numbers as a product of rational numbers. Focus Question Notes Listen for students to define a reciprocal as two numbers whose product is 1. Essential Question Connection This lesson addresses the portion of the Essential Question that asks what relationships help you divide positive and negative rational numbers. How are a number and its reciprocal related? [Sample answer: A number and its reciprocal have a product of 1. The reciprocal of the number p q is q p.] How do you use multiplication to divide two rational numbers? [Sample answer: You multiply the dividend by the reciprocal of the divisor.] How are the signs of a dividend and divisor related to the sign of the quotient when you divide positive and negative rational numbers? [Sample answer: When the dividend and divisor have the same sign, the quotient is positive. When the dividend and divisor have different signs, the quotient is negative.