Lesson 5 NYS COMMON CORE MATHEMATICS CURRICULUM 7β’3 Lesson 5: Using the Identity and Inverse to Write Equivalent Expressions Student Outcomes ο§ Students recognize the identity properties of 0 and 1 and the existence of inverses (opposites and reciprocals) to write equivalent expressions. Classwork Opening Exercise (5 minutes) Students work independently to rewrite numerical expressions recalling the definitions of opposites and reciprocals. Opening Exercise a. In the morning, Harrison checked the temperature outside to find that it was βππ°π . Later in the afternoon, the temperature rose ππ°π . Write an expression representing the temperature change. What was the afternoon temperature? βππ + ππ; the afternoon temperature was π°π . b. Rewrite subtraction as adding the inverse for the following problems and find the sum. i. πβπ π + (βπ) = π ii. βπ β (βπ) (βπ) + π = π MP.8 iii. The difference of π and π π β π = π + (βπ) = π iv. πβπ π + (βπ) = π c. What pattern do you notice in part (a) and (b)? The sum of a number and its additive inverse is equal to zero. d. Add or subtract. i. ππ + π ππ Lesson 5: Using the Identity and Inverse to Write Equivalent Expressions 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 73 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 5 NYS COMMON CORE MATHEMATICS CURRICULUM ii. 7β’3 πβπ π + (βπ) = βπ iii. βπ + π βπ iv. π+π π v. What pattern do you notice in parts (i) through (iv)? The sum of any quantity and zero is equal to the value of the quantity. MP.8 e. Your younger sibling runs up to you and excitedly exclaims, βIβm thinking of a number. If I add it to the number π ten times, that is, π + my number + my number + my number, and so on, then the answer is π. What is my number?β You almost immediately answer, βzero,β but are you sure? Can you find a different number (other than zero) that has the same property? If not, can you justify that your answer is the only correct answer? No, there is no other number. On a number line, π can be represented as a directed line segment that starts at π, ends at π, and has length π. Adding any other (positive or negative) number π to π is equivalent to attaching another directed line segment with length |π| to the end of the first line segment for π: If π is any number other than π, then the directed line segment that represents π will have to have some length, so π + π will have to be a different number on the number line. Adding π again just takes the new sum further away from the point π on the number line. Discussion (5 minutes) Discuss the following questions and conclude the Opening Exercise with definitions of opposite, additive inverse, and the identity property of zero. ο§ In Problem 1, what is the pair of numbers called? οΊ ο§ What is the sum of a number and its opposite? οΊ ο§ It is always equal to 0. In Problem 5, what is so special about 0? οΊ ο§ Opposites or additive inverses Zero is the only number that when added with another number, the result is that other number. This property makes zero special among all the numbers. Mathematicians have a special name for zero, called the additive identity; they call the property the additive identity property of zero. Lesson 5: Using the Identity and Inverse to Write Equivalent Expressions 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 74 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 5 NYS COMMON CORE MATHEMATICS CURRICULUM 7β’3 Example 1 (5 minutes) As a class, write the sum, and then write an equivalent expression by collecting like terms and removing parentheses when possible. State the reasoning for each step. Example 1 Write the sum, and then write an equivalent expression by collecting like terms and removing parentheses. a. ππ and βππ + π ππ + (βππ + π) b. (ππ + (βππ)) + π Associative property, collect like terms π+π Additive inverse π Additive identity property of zero ππ β π and the opposite of ππ ππ + (βπ) + (βππ) c. ππ + (βππ) + (βπ) Commutative property, associative property π + (βπ) Additive inverse βπ Additive identity property of zero The opposite of (ππ β π) and ππ β(ππ β π) + ππ βπ(ππ β π) + ππ Taking the opposite is equivalent to multiplying by β π. βππ + π + ππ Distributive property (βππ + ππ) + π Commutative property, any order property π+π Additive inverse π Additive identity property of zero Exercise 1 (10 minutes) In pairs, students will take turns dictating how to write the sums while partners write what is being dictated. Students should discuss any discrepancies and explain their reasoning. Dialogue is encouraged. Exercise 1 With a partner, take turns alternating roles as writer and speaker. The speaker verbalizes how to rewrite the sum and properties that justify each step as the writer writes what is being spoken without any input. At the end of each problem, discuss in pairs the resulting equivalent expressions. Write the sum, and then write an equivalent expression by collecting like terms and removing parentheses whenever possible. a. βπ and ππ + π βπ + (ππ + π) (βπ + π) + ππ Any order, any grouping π + ππ Additive inverse ππ Additive identity property of zero Lesson 5: Using the Identity and Inverse to Write Equivalent Expressions 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 75 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 5 NYS COMMON CORE MATHEMATICS CURRICULUM b. 7β’3 ππ and π β ππ ππ + (π β ππ) c. ππ + (π + (βππ)) Subtraction as adding the inverse (ππ + (βππ)) + π Any order, any grouping π+π Additive inverse π Additive identity property of zero The opposite of ππ and βπ + ππ βππ + (βπ + ππ) d. (βππ + ππ) + (βπ) Any order, any grouping π + (βπ) Additive inverse βπ Additive identity property of zero The opposite of βπππ and π β πππ πππ + (π β πππ) e. (πππ + (βπππ)) + π Any order, any grouping π+π Additive inverse π Additive identity property of zero The opposite of (βπ β ππ) and βππ β(βπ β ππ) + (βππ) βπ(βπ β ππ) + (βππ) Taking the opposite is equivalent to multiplying by β π. π + ππ + (βππ) Distributive property π+π Any grouping, additive inverse π Additive identity property of zero Example 2 (5 minutes) Students should complete the first five problems independently and then discuss as a class. Example 2 π π π π ο§ ( )×( )= π ο§ π× ο§ π π π =π π ×π=π π π ο§ (β ) × βπ = π ο§ (β ) × (β ) = π π π π π Lesson 5: Using the Identity and Inverse to Write Equivalent Expressions 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 76 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 5 NYS COMMON CORE MATHEMATICS CURRICULUM ο§ What are these pairs of numbers called? οΊ ο§ Reciprocals What is another term for reciprocal? οΊ ο§ 7β’3 The multiplicative inverse What happens to the sign of the expression when converting it to its multiplicative inverse? 1 2 There is no change to the sign. For example, the multiplicative inverse of β2 is (β ). The negative οΊ sign remains the same. ο§ What can you conclude from the pattern in the answers? The product of a number and its multiplicative inverse is equal to 1. οΊ ο§ Earlier, we saw that 0 is a special number because it is the only number that when added to another number, results in that number again. Can you explain why the number 1 is also special? οΊ ο§ One is the only number that when multiplied with another number, results in that number again. This property makes 1 special among all the numbers. Mathematicians have a special name for 1, the multiplicative identity; they call the property the multiplicative identity property of one. As an extension, ask students if there are any other special numbers that they have learned. Students should respond: Yes; β1 has the property that multiplying a number by it is the same as taking the opposite of the number. Share with students that they are going to learn later in this module about another special number called pi. As a class, write the product, and then write an equivalent expression in standard form. State the properties for each step. After discussing questions, review the properties and definitions in the Lesson Summary emphasizing the multiplicative identity property of one and the multiplicative inverse. Write the product, and then write the expression in standard form by removing parentheses and combining like terms. Justify each step. a. The multiplicative inverse of π π π π and (ππ β ) π π π (ππ β ) π(ππ) β π β π π πππ β π b. Distributive property Multiplicative inverses The multiplicative inverse of π and (ππ + π) π π ( ) (ππ + π) π π π π ( ) (ππ) + ( ) (π) Distributive property ππ + π Multiplicative inverses, multiplication π+π Multiplicative identity property of one Lesson 5: Using the Identity and Inverse to Write Equivalent Expressions 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 77 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 5 NYS COMMON CORE MATHEMATICS CURRICULUM c. The multiplicative inverse of ( (ππ + π) β π π π+ π π ) and ππ+π π π π π π ππ ( ) + π ( ) ππ + 7β’3 Distributive property π π Multiplicative inverse π π Multiplicative identity property of one Exercise 2 (10 minutes) As in Exercise 1, have students work in pairs to rewrite the expressions, taking turns being the speaker and writer. Exercise 2 Write the product, and then write the expression in standard form by removing parentheses and combining like terms. Justify each step. a. The reciprocal of π and βππ β ππ π π ( ) (βππ + (βππ)) π π b. Rewrite subtraction as an addition problem π π ( ) (βππ) + ( ) (βππ) Distributive property βππ β ππ Multiplicative inverse βππ β π Multiplicative identity property of one The multiplicative inverse of π and ππ β ππ π π ( ) (ππ + (βππ)) π π c. Rewrite subtraction as an addition problem π π ( ) (ππ) + ( ) (βππ) Distributive property ππ + (βπ) Multiplicative inverse πβπ Multiplicative identity property of one π π π π The multiplicative inverse of β and π β π π π (βπ) (π + (β π)) Rewrite subtraction as an addition problem π π (βπ)(π) + (βπ) (β π) Distributive property βππ + ππ Multiplicative inverse βππ + π Multiplicative identity property of one Lesson 5: Using the Identity and Inverse to Write Equivalent Expressions 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 78 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 5 NYS COMMON CORE MATHEMATICS CURRICULUM 7β’3 Closing (3 minutes) ο§ What are the other terms for opposites and reciprocals, and what are the general rules of their sums and products? οΊ ο§ Additive inverse and multiplicative inverse; the sum of additive inverses equals 0; the product of multiplicative inverses equals 1. What do the additive identity property of zero and the multiplicative identity property of one state? οΊ The additive identity property of zero states that zero is the only number that when added to another number, the result is again that number. The multiplicative identity property of one states that one is the only number that when multiplied with another number, the result is that number again. Exit Ticket (5 minutes) Lesson 5: Using the Identity and Inverse to Write Equivalent Expressions 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 79 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 5 NYS COMMON CORE MATHEMATICS CURRICULUM Name ___________________________________________________ 7β’3 Date____________________ Lesson 5: Using the Identity and Inverse to Write Equivalent Expressions Exit Ticket 1. Find the sum of 5π₯ + 20 and the opposite of 20. Write an equivalent expression in standard form. Justify each step. 2. For 5π₯ + 20 and the multiplicative inverse of 5, write the product and then write the expression in standard form, if possible. Justify each step. Lesson 5: Using the Identity and Inverse to Write Equivalent Expressions 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 80 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 5 NYS COMMON CORE MATHEMATICS CURRICULUM 7β’3 Exit Ticket Sample Solutions 1. Find the sum of ππ + ππ and the opposite of ππ. Write an equivalent expression in standard form. Justify each step. (ππ + ππ) + (βππ) 2. ππ + (ππ + (βππ)) Associative property of addition ππ + π Additive inverse ππ Additive identity property of zero For ππ + ππ and the multiplicative inverse of π, write the product and then write the expression in standard form, if possible. Justify each step. π π (ππ + ππ) ( ) π π π π (ππ) ( ) + ππ ( ) Distributive property ππ + π Multiplicative inverses, multiplication π+π Multiplicative identity property of one Problem Set Sample Solutions 1. Fill in the missing parts. a. The sum of ππ β π and the opposite of ππ (ππ β π) + (βππ) (ππ + (βπ)) + (βππ) ππ + (βππ) + (βπ) Regrouping/any order (or commutative property of addition) π + (βπ) Additive inverse βπ b. Rewrite subtraction as addition Additive identity property of zero The product of βππ + ππ and the multiplicative inverse of βπ π π (βππ + ππ) (β ) π π π π (βππ) (β ) + (ππ) (β ) ππ + (βπ) ππ β π πβπ 2. Distributive property Multiplicative inverse, multiplication Adding the additive inverse is the same as subtraction Multiplicative identity property of one Write the sum, and then rewrite the expression in standard form by removing parentheses and collecting like terms. a. π and π β π π + (π β π) π + (βπ) + π π+π π Lesson 5: Using the Identity and Inverse to Write Equivalent Expressions 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 81 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 5 NYS COMMON CORE MATHEMATICS CURRICULUM b. 7β’3 πππ + π and βπ (πππ + π) + (βπ) πππ + (π + (βπ)) πππ + π πππ c. βπ β ππ and the opposite of βππ (βπ + (βππ)) + ππ βπ + ((βππ) + (ππ)) βπ + π βπ d. The opposite of ππ and π + ππ (βππ) + (π + ππ) ((βππ) + ππ) + π π+π π e. ππ and the opposite of (π β ππ) ππ + (β(π β ππ)) ππ + (βπ) + ππ ππ + ππ + (βπ) ππ + (βπ) ππ β π 3. Write the product, and then rewrite the expression in standard form by removing parentheses and collecting like terms. a. ππ β π and the multiplicative inverse of π π (ππ + (βπ)) ( ) π π π ( ) (ππ) + ( ) (βπ) π π π πβ π b. The multiplicative inverse of βπ and πππ β π π (β ) (πππ β π) π π π (β ) (πππ) + (β ) (βπ) π π βππ + π Lesson 5: Using the Identity and Inverse to Write Equivalent Expressions 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 82 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 5 NYS COMMON CORE MATHEMATICS CURRICULUM c. 7β’3 π β π and the multiplicative inverse of π π (π + (βπ)) ( ) π π π ( ) (π) + ( ) (βπ) π π π πβ π π d. The multiplicative inverse of π π and ππ β π π π π (ππ β ) π π π(ππ) + π (β ) π πππ β π e. The multiplicative inverse of β π π π and β πππ πππ ππ π π (βπππ) ( β ) πππ ππ π π (βπππ) ( ) + (βπππ) (β ) πππ ππ βπ + π 4. Write the expressions in standard form. a. π π (ππ + π) π π (ππ) + (π) π π π+π b. π π (π β π) π π (π) + (βπ) π π π πβπ π c. π π (π + π) π π (π) + (π) π π π π π+ π π Lesson 5: Using the Identity and Inverse to Write Equivalent Expressions 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 83 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 5 NYS COMMON CORE MATHEMATICS CURRICULUM d. π π 7β’3 (ππ + π) π π (ππ) + (π) π π π π π+ π π e. π π (ππ β π) π π (ππ) + (βπ) π π ππ π πβ π π f. π π (πππ β π) β π π π (πππ) + (βπ) + (βπ) π π ππ + (βπ) + (βπ) ππ β π Lesson 5: Using the Identity and Inverse to Write Equivalent Expressions 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 84 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
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