Lesson 6 NYS COMMON CORE MATHEMATICS CURRICULUM 7β’3 Lesson 6: Collecting Rational Number Like Terms Student Outcomes ο§ Students rewrite rational number expressions by collecting like terms and combining them by repeated use of the distributive property. Classwork Opening Exercise (5 minutes) Students work in pairs to write the expressions in standard form. Reconvene as a class and review the steps taken to rewrite the expressions, justifying each step verbally as you go. Opening Exercise Solve each problem, leaving your answers in standard form. Show your steps. a. Terry weighs ππ π€π . Janice weighs π π π ππ + (ππ β π ) = ππ β π β b. π π π π π βπ β π π€π less than Terry. What is their combined weight? π π π π π = ππ β = ππ . Their combined weight is ππ π€π . π π π π π π π π π β β π π π ππ ππ ππ β β ππ ππ ππ ππ ππ c. π π + (βπ) βπ d. π π π π π( ) π π ( ) π π ππ π π π π Lesson 6: Collecting Rational Number Like Terms This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G7-M3-TE-1.3.0-08.2015 85 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 6 NYS COMMON CORE MATHEMATICS CURRICULUM π π 7β’3 π π Mr. Jackson bought π π₯π. of beef. He cooked of it for lunch. How much does he have left? e. π π π π If he cooked of it for lunch, he had π π π π of the original amount left. Since (π ) ( ) = π π π π β = , he had π₯π. π π π π left. Teachers: You can also show your students how to write the answer as one expression. π π (π ) (π β ) π π ο§ How is the process of writing equivalent expressions by combining like terms in this Opening Exercise different from the previous lesson? οΊ There are additional steps to find common denominators, convert mixed numbers to improper numbers (in some cases), and convert back. Example 1 (4 minutes) Scaffolding: Students who need a challenge could tackle the following problem: Example 1 Rewrite the expression in standard form by collecting like terms. π π π π πβ π+ π+π π π π π π ππ ππ π π πβ π+ π+π π ππ ππ ππ ππ π+ ππ π π ππ ο§ 1 2 1 1 1 3 3 π+2 π+ β πβ1 π+ + πβ4 2 3 5 4 2 5 4 4 β π 5 11 16 πβ . 30 5 What are various strategies for adding, subtracting, multiplying, and dividing rational numbers? οΊ Find common denominators; change from mixed numbers and whole numbers to improper fractions, and then convert back. Exercise 1 (4 minutes) Walk around as students work independently. Have students check their answers with a partner. Address any unresolved questions. Lesson 6: Collecting Rational Number Like Terms This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G7-M3-TE-1.3.0-08.2015 86 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 6 NYS COMMON CORE MATHEMATICS CURRICULUM 7β’3 Exercise 1 For the following exercises, predict how many terms the resulting expression will have after collecting like terms. Then, write the expression in standard form by collecting like terms. π a. π πβ π π βπ+ π ππ πβ π π There will be two terms. π π π π π β ππ + πβ β π ππ π π π π π π ( β π + )π β ( + ) π ππ π π β π ππ πβ ππ ππ π π π π π π π π π + ππ β π + π + π β π + π b. There will be two terms. π π π π π + π β π + π β π + ππ + π π π π π π π π π ( + + (βπ)) π + (π β + π + ) π π π π π β π π π+π π ππ ππ Example 2 (5 minutes) Read the problem as a class and give students time to set up their own expressions. Reconvene as a class to address each expression. Example 2 At a store, a shirt was marked down in price by $ππ. ππ. A pair of pants doubled in price. Following these changes, the price of every item in the store was cut in half. Write two different expressions that represent the new cost of the items, using π for the cost of each shirt and π for the cost of a pair of pants. Explain the different information each one shows. For the cost of a shirt: π π π π (π β ππ) The cost of each shirt is π of the quantity of the original cost of the shirt, minus ππ. π π β π The cost of each shirt is half off the original price, minus π, since half of ππ is π. For the cost of a pair of pants: π π (ππ) The cost of each pair of pants is half off double the price. π The cost of each pair of pants is the original cost because ο§ π π is the multiplicative inverse of π. Describe a situation in which either of the two expressions in each case would be more useful. οΊ 1 Answers may vary. For example, π would be more useful than (2π) because it is converted back to an 2 isolated variable, in this case the original cost. Lesson 6: Collecting Rational Number Like Terms This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G7-M3-TE-1.3.0-08.2015 87 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 6 NYS COMMON CORE MATHEMATICS CURRICULUM 7β’3 Exercise 2 (3 minutes) Exercise 2 Write two different expressions that represent the total cost of the items if tax was π ππ of the original price. Explain the different information each shows. For the cost of a shirt: π π (π β ππ) + π ππ π The cost of each shirt is π π of the quantity of the original cost of the shirt, minus ππ, plus π ππ of the cost of the shirt. π π π β π The cost of each shirt is π π π π of the original price (because it is π + π π π= π), minus π, since half of ππ ππ ππ is π. For the cost of a pair of pants: π π π (ππ) + π ππ π The cost of each pair of pants is half off double the price plus π ππ of the cost of a pair of pants. π π π π π The cost of each pair of pants is π (because ππ + π = π π) times the number of pair of pants. ππ ππ ππ ππ Example 3 (4 minutes) Example 3 Write this expression in standard form by collecting like terms. Justify each step. π π π π π β (π ) ( π β ) π π π π ππ π ππ π + (β π π ) (π π) + (β π π π π + (β π) + π π β π+( π π β π+ ο§ ππ ππ π π ) (β π ) Write mixed numbers as improper fractions, then distribute. Any grouping (associative) and arithmetic rules for multiplying rational numbers ππ π + ) π π Commutative property and associative property of addition, collect like terms ππ π Apply arithmetic rule for adding rational numbers 1 3 1 3 A student says he created an equivalent expression by first finding this difference: 5 β 3 . Is he correct? Why or why not? οΊ Although they do appear to be like terms, taking the difference would be incorrect. In the 1 3 1 2 1 4 1 3 expression (3 ) ( π₯ β ), 3 must be distributed before applying any other operation in this problem. Lesson 6: Collecting Rational Number Like Terms This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G7-M3-TE-1.3.0-08.2015 88 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 6 NYS COMMON CORE MATHEMATICS CURRICULUM ο§ 7β’3 1 3 How should 3 be written before being distributed? οΊ The mixed number can be rewritten as an improper fraction 10 3 . It is not necessary to convert the mixed number, but it makes the process more efficient and increases the likelihood of getting a correct answer. Exercise 3 (5 minutes) Walk around as students work independently. Have students check their answers with a partner. Address any unresolved questions. Exercise 3 Rewrite the following expressions in standard form by finding the product and collecting like terms. a. π π π π π π βπ β ( + π) π π π π βπ + (β ) ( ) + (β ) π π π π π π π π βπ + (β ) + (β π) π π π π π π β π β (π + ) π π π π π π β π β (π + ) π ππ ππ π π β πβπ π ππ b. π π + π π ( π π πβπ π π ) π π π π π + ( π) + (β ) π π π π π π π π + πβ π ππ π π π π π+( β ) ππ π π π π π+ ππ π Example 4 (5 minutes) Example 4 Model how to write the expression in standard form using rules of rational numbers. π ππ π + π ππ β π + + + ππ π π ππ Lesson 6: Collecting Rational Number Like Terms This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G7-M3-TE-1.3.0-08.2015 89 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. ο§ What are other equivalent expressions of οΊ ο§ What about οΊ ο§ Other expressions include 1 20π₯ 1π₯ 20 and π₯ 20 1 20 7β’3 Lesson 6 NYS COMMON CORE MATHEMATICS CURRICULUM ? How do you know? π₯ because of the arithmetic rules of rational numbers. ? How do you know? It is not equivalent because if π₯ = 2, the value of the expression is 1 40 , which does not equal 1 10 . How can the distributive property be used in this problem? 1 οΊ For example, it can be used to factor out οΊ Or, for example, it can be used to distribute 20 from each term of the expression. 1 10 : 3π₯β1 10 = 1 10 (3π₯ β 1) = 3π₯ 10 β 1 . 10 Below are two solutions. Explore both with the class. π π(ππ) ππ(π + π) π(ππ β π) + + + ππ ππ ππ ππ π π π π π π π+ π+ π+ + πβ ππ π π π ππ ππ π + ππ + πππ + ππ + ππ β π ππ π π π π π π ( + + + )π + ( β ) ππ π π ππ π ππ πππ + π ππ π π ππ π π π ( + + + )π + ( β ) ππ ππ ππ ππ ππ ππ π π π+ π π π π π+ π π Ask students to evaluate the original expression and the answers when π₯ = 20 to see if they get the same number. Evaluate the original expression and the answers when π = ππ. Do you get the same number? π ππ π + π ππ β π + + + ππ π π ππ π π π+ π π ππ π(ππ) ππ + π π(ππ) β π + + + ππ π π ππ π π (ππ) + π π π+π+ π+ ππ ππ + π ππ πππ ππ + ππ ππ π+ ππ π π π π πππ ππ π + ππ ππ Lesson 6: ππ + π ππ π π Collecting Rational Number Like Terms This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G7-M3-TE-1.3.0-08.2015 90 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 6 NYS COMMON CORE MATHEMATICS CURRICULUM 7β’3 Important: After students evaluate both expressions for π₯ = 20, ask them which expression was easier. (The expression in standard form is easier.) Explain to students: When you are asked on a standardized test to simplify an expression, you must put the expression in standard form because standard form is often much simpler to evaluate and read. This curriculum is specific and will often tell you the form (such as standard form) it wants you to write the expression in for an answer. Exercise 4 (3 minutes) Allow students to work independently. Exercise 4 Rewrite the following expression in standard form by finding common denominators and collecting like terms. ππ π π β π β + π π π π(ππ) π(π) π(π β π) β + ππ ππ ππ πππ β ππ + ππ β ππ ππ πππ β ππ ππ ππ π πβ ππ π Example 5 (Optional, 5 minutes) Give students a minute to observe the expression and decide how to begin rewriting it in standard form. Example 5 Rewrite the following expression in standard form. π(ππ β π) ππ + π β π π ο§ How can we start to rewrite this problem? οΊ There are various ways to start rewriting this expression, including using the distributive property, renaming 2 6 , rewriting the subtraction as an addition, distributing the negative in the second term, rewriting each term as a fraction (e.g., 2 6 (3π₯ β 4) β ( 5π₯ 8 + 2 )), or finding the lowest common 8 denominator. Lesson 6: Collecting Rational Number Like Terms This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G7-M3-TE-1.3.0-08.2015 91 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 6 NYS COMMON CORE MATHEMATICS CURRICULUM Method 1: Method 2a: π(ππ β π) π(ππ + π) β ππ π (πππ β ππ β πππ β π) ππ ππ β ππ ππ ππ ππ β ππ ππ π ππ πβ π ππ ((πππ β ππ) β (πππ + π)) ππ (πππ β ππ β πππ β π) ππ ππ β ππ ππ ππ ππ β ππ ππ π ππ πβ π ππ Finding the lowest common denominator would keep the number in integer form; this is important because working with the terms would be more convenient. Is one method better than the rest of the methods? οΊ ο§ π π π π πβ β πβ π π π π π ππ π πβ β β π π π π π π ππ β π β β π π π π ππ π πβ β π ππ ππ π ππ πβ π ππ π ππ π (ππ β π) β ( + ) π π π π π π πβ β πβ π π π π π π ππ β π β β π π π π ππ π πβ β π ππ ππ π ππ πβ π ππ Which method(s) keep(s) the numbers in the expression in integer form? Why would this be important to note? οΊ ο§ Method 3: ππ β π ππ + π β π π π(ππ β π) ππ + π β π π π(ππ β π) π(ππ + π) β ππ ππ ο§ Method 2b: 7β’3 No, it is by preference; however, the properties of addition and multiplication must be used properly. Are these expressions equivalent: οΊ 3 8 π₯, 3π₯ 8 , and 3 8π₯ ? How do you know? The first two expressions are equivalent, but the third one, 3 8π₯ , is not. If you substitute a value other 2 than zero or one (such as π₯ = 2), the values of the first expressions are the same, . The value of the third expression is ο§ 3 16 3 . What are some common errors that could occur when rewriting this expression in standard form? οΊ Some common errors may include distributing only to one term in the parentheses, forgetting to multiply the negative sign to all the terms in the parentheses, incorrectly reducing fractions, and/or adjusting the common denominator but not the numerator. Lesson 6: Collecting Rational Number Like Terms This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G7-M3-TE-1.3.0-08.2015 92 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 6 NYS COMMON CORE MATHEMATICS CURRICULUM 7β’3 Exercise 5 (Optional, 3 minutes) Allow students to work independently. Have students share the various ways they started to rewrite the problem. Exercise 5 Write the following expression in standard form. ππ β ππ π(π β π) β π ππ π(ππ β ππ) π β π(π β π) β ππ ππ (πππ β ππ) β π(π β π) ππ πππ β ππ β ππ + ππ ππ ππ β ππ ππ π π πβπ π ππ Closing (2 minutes) ο§ Jane says combining like terms is much harder to do when the coefficients and constant terms are not integers. Why do you think Jane feels this way? οΊ There are usually more steps, including finding common denominators, converting mixed numbers to improper fractions, etc. Exit Ticket (5 minutes) Lesson 6: Collecting Rational Number Like Terms This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G7-M3-TE-1.3.0-08.2015 93 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 6 NYS COMMON CORE MATHEMATICS CURRICULUM Name ___________________________________________________ 7β’3 Date____________________ Lesson 6: Collecting Rational Number Like Terms Exit Ticket 1 1 5 10 For the problem π β βπ+1 3 10 πβ 1 10 , Tyson created an equivalent expression using the following steps. 1 3 1 1 π + β1π + 1 π + β +β 5 10 10 10 4 1 β π+1 5 10 Is his final expression equivalent to the initial expression? Show how you know. If the two expressions are not equivalent, find Tysonβs mistake and correct it. Lesson 6: Collecting Rational Number Like Terms This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G7-M3-TE-1.3.0-08.2015 94 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 6 NYS COMMON CORE MATHEMATICS CURRICULUM 7β’3 Exit Ticket Sample Solutions π π π ππ For the problem π β βπ+π π ππ πβ π ππ , Tyson created an equivalent expression using the following steps. π π π π π + βππ + π π+β +β π ππ ππ ππ π π β π+π π ππ Is his final expression equivalent to the initial expression? Show how you know. If the two expressions are not equivalent, find Tysonβs mistake and correct it. No, he added the first two terms correctly, but he forgot the third term and added to the other like terms. If π = ππ, π π π π π + βππ + π π+β +β π ππ ππ ππ π π β π+π π ππ π π π π (ππ) + β (ππ) + βπ(ππ) + π +β π ππ ππ ππ π π β (ππ) + π π ππ π + (βππ) + ππ + (β π π ) ππ βπ + π π π βπ π ππ π ππ The expressions are not equal. He should factor out the π and place parentheses around the values using the distributive property in order to make it obvious which rational numbers need to be combined. π π π π π + βππ + π π+β +β π ππ ππ ππ π π π π ( π + βππ + π π) + (β +β ) π ππ ππ ππ π π π ( + βπ + π ) π + (β ) π ππ ππ π π π ( + ) π + (β ) ππ ππ π π π πβ π π Problem Set Sample Solutions 1. Write the indicated expressions. a. π π π π π inches in feet π× π ππ = Lesson 6: π ππ π. It is π ππ π ππ. Collecting Rational Number Like Terms This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G7-M3-TE-1.3.0-08.2015 95 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 6 NYS COMMON CORE MATHEMATICS CURRICULUM b. The perimeter of a square with π π π π π π π × π = π. The perimeter is c. π ππ¦ sides π π π ππ¦. The number of pounds in π π¨π³. π× d. 7β’3 π π π = . It is π₯π. ππ ππ ππ The average speed of a train that travels π miles in π π hour π π π« π = π. The average speed of the train is π miles per hour. π» π π π πΉ= ; π e. π π Devin is π years younger than Eli. April is π π as old as Devin. Jill is π years older than April. If Eli is π¬ years old, what is Jillβs age in terms of π¬? π π π« π π« π π« = π¬ β π , π¨ = , π¨ + π = π±, so π± = ( ) + π. π± = 2. π π π¬ π (π¬ β π ) + π. π± = + π . π π π π Rewrite the expressions by collecting like terms. a. π π πβ π π π b. π π πβ π π π π π π c. π π π π ππ π + ππ ππ ππ ππ + ππ ππ πππ ππ π π π π π π π π β πβ πβ + πβ π+ π d. π π βπ + π β π π π π π+ β π+π π ππ π π π π π π π π π β π+ πβ π+ πβ πβ π π π π π π π π π π π βπ β π + π π + π β π+ π π π ππ π π π π π β π+ πβ πβ π π π π π π π π π π β πβ π+π π+ πβ π+ π π π ππ ππ π π π π πβ πβ π π π ππ π π π+ π+ π ππ π π π π π π+ π+ π π π e. π π πβ π ππ ππ π πβ π ππ ππ π π ππ Lesson 6: f. ππ π β π π +π π π ππ ππ ππ β +π π π π ππ π π Collecting Rational Number Like Terms This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G7-M3-TE-1.3.0-08.2015 96 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 6 NYS COMMON CORE MATHEMATICS CURRICULUM 3. 7β’3 Rewrite the expressions by using the distributive property and collecting like terms. a. π π (πππ β π) b. π π π π π βπ( π β π ) e. π β π + ππ π g. π π π π π (π + π) + (π β π) π h. π π π π π (π + π) β π (π + π) β ππ π k. n. βπ π(ππ β π) π π π (πππ + π) β π f. π π π π (π + π) + π π (π β π) i. ππ + π q. π π π π (π + π) β π (π β π) π + πβπ ππ ππ + π π β π π + + π π ππ βπ βπ ππ π ππ πβπ ππ ππ π π (ππ β ππ) β ππ π π (π β π) β π π (ππ + π) l. π π π π π (π + π) + π (π β π) π π π ππ + π β π π ππ ππ πβ ππ ππ o. π π π β π π βπ β π π ππ β βπ ππ p. π π π π π β (ππ + π ) π π πβ ππ π ππ ππ ππ ππ πβ or πβ ππ ππ ππ π π π π+ π π m. c. π β π) ππ β π ππ π π+ ππ π j. ( π π π πβπ π πππ β π d. π π ππ β π ππ + ππ + π π ππ π or π π π π r. ππ π + ππ β π π β πβπ π πππ β ππ ππ ππ ππ πβ ππ ππ s. π+π π β π+π π + πβπ π ππ π β π ππ ππ Lesson 6: Collecting Rational Number Like Terms This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G7-M3-TE-1.3.0-08.2015 97 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
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