### Lesson 5: Using the Identity and Inverse to Write Equivalent

```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 βππ°π. Later in the afternoon,
the temperature rose ππ°π. Write an expression representing the temperature change. What was the
afternoon temperature?
βππ + ππ; the afternoon temperature was π°π.
b.
Rewrite subtraction as adding the inverse for the following problems and find the sum.
i.
πβπ
π + (βπ) = π
ii.
βπ β (βπ)
(βπ) + π = π
MP.8
iii.
The difference of π and π
π β π = π + (βπ) = π
iv.
πβπ
π + (βπ) = 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.
i.
ππ + π
ππ
Lesson 5:
Using the Identity and Inverse to Write Equivalent Expressions
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ii.
7β’3
πβπ
π + (βπ) = βπ
iii.
βπ + π
βπ
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, βIβm 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, βzero,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
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?
οΊ
ο§
What is the sum of a number and its opposite?
οΊ
ο§
It is always equal to 0.
In Problem 5, what is so special about 0?
οΊ
ο§
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
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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.
ππ and βππ + π
ππ + (βππ + π)
b.
(ππ + (βππ)) + π
Associative property, collect like terms
π+π
π
ππ β π and the opposite of ππ
ππ + (βπ) + (βππ)
c.
ππ + (βππ) + (βπ)
Commutative property, associative property
π + (βπ)
βπ
The opposite of (ππ β π) and ππ
β(ππ β π) + ππ
βπ(ππ β π) + ππ
Taking the opposite is equivalent to multiplying by β π.
βππ + π + ππ
Distributive property
(βππ + ππ) + π
Commutative property, any order property
π+π
π
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.
βπ and ππ + π
βπ + (ππ + π)
(βπ + π) + ππ
Any order, any grouping
π + ππ
ππ
Lesson 5:
Using the Identity and Inverse to Write Equivalent Expressions
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b.
7β’3
ππ and π β ππ
ππ + (π β ππ)
c.
ππ + (π + (βππ))
(ππ + (βππ)) + π
Any order, any grouping
π+π
π
The opposite of ππ and βπ + ππ
βππ + (βπ + ππ)
d.
(βππ + ππ) + (βπ)
Any order, any grouping
π + (βπ)
βπ
The opposite of βπππ and π β πππ
πππ + (π β πππ)
e.
(πππ + (βπππ)) + π
Any order, any grouping
π+π
π
The opposite of (βπ β ππ) and βππ
β(βπ β ππ) + (βππ)
βπ(βπ β ππ) + (βππ)
Taking the opposite is equivalent to multiplying by β π.
π + ππ + (βππ)
Distributive property
π+π
π
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 ) × βπ = π
ο§
(β ) × (β ) = π
π
π
π
π
Lesson 5:
Using the Identity and Inverse to Write Equivalent Expressions
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ο§
What are these pairs of numbers called?
οΊ
ο§
Reciprocals
What is another term for reciprocal?
οΊ
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 β2 is (β ). The negative
οΊ
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.
οΊ
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?
οΊ
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; β1 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 (ππ β )
π
π
π (ππ β )
π(ππ) β π β
π
π
πππ β π
b.
Distributive property
Multiplicative inverses
The multiplicative inverse of π and (ππ + π)
π
π
( ) (ππ + π)
π
π
π
π
( ) (ππ) + ( ) (π)
Distributive property
ππ + π
Multiplicative inverses, multiplication
π+π
Multiplicative identity property of one
Lesson 5:
Using the Identity and Inverse to Write Equivalent Expressions
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c.
The multiplicative inverse of (
(ππ + π) β
π
π
π+
π
π
) and
ππ+π
π
π
π
π
π
ππ ( ) + π ( )
ππ +
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 βππ β ππ
π
π
( ) (βππ + (βππ))
π
π
b.
Rewrite subtraction as an addition problem
π
π
( ) (βππ) + ( ) (βππ)
Distributive property
βππ β ππ
Multiplicative inverse
βππ β π
Multiplicative identity property of one
The multiplicative inverse of π and ππ β ππ
π
π
( ) (ππ + (βππ))
π
π
c.
Rewrite subtraction as an addition problem
π
π
( ) (ππ) + ( ) (βππ)
Distributive property
ππ + (βπ)
Multiplicative inverse
πβπ
Multiplicative identity property of one
π
π
π
π
The multiplicative inverse of β and π β π
π
π
(βπ) (π + (β π))
Rewrite subtraction as an addition problem
π
π
(βπ)(π) + (βπ) (β π)
Distributive property
βππ + ππ
Multiplicative inverse
βππ + π
Multiplicative identity property of one
Lesson 5:
Using the Identity and Inverse to Write Equivalent Expressions
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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?
οΊ
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?
οΊ
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
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Lesson 5
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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
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7β’3
Exit Ticket Sample Solutions
1.
Find the sum of ππ + ππ and the opposite of ππ. Write an equivalent expression in standard form. Justify each
step.
(ππ + ππ) + (βππ)
2.
ππ + (ππ + (βππ))
ππ + π
ππ
For ππ + ππ and the multiplicative inverse of π, write the product and then write the expression in standard form, if
possible. Justify each step.
π
π
(ππ + ππ) ( )
π
π
π
π
(ππ) ( ) + ππ ( )
Distributive property
ππ + 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 ππ β π and the opposite of ππ
(ππ β π) + (βππ)
(ππ + (βπ)) + (βππ)
ππ + (βππ) + (βπ)
Regrouping/any order (or commutative property of addition)
π + (βπ)
βπ
b.
The product of βππ + ππ and the multiplicative inverse of βπ
π
π
(βππ + ππ) (β )
π
π
π
π
(βππ) (β ) + (ππ) (β )
ππ + (βπ)
ππ β π
πβπ
2.
Distributive property
Multiplicative inverse, multiplication
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 + (βπ) + π
π+π
π
Lesson 5:
Using the Identity and Inverse to Write Equivalent Expressions
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b.
7β’3
πππ + π and βπ
(πππ + π) + (βπ)
πππ + (π + (βπ))
πππ + π
πππ
c.
βπ β ππ and the opposite of βππ
(βπ + (βππ)) + ππ
βπ + ((βππ) + (ππ))
βπ + π
βπ
d.
The opposite of ππ and π + ππ
(βππ) + (π + ππ)
((βππ) + ππ) + π
π+π
π
e.
ππ and the opposite of (π β ππ)
ππ + (β(π β ππ))
ππ + (βπ) + ππ
ππ + ππ + (βπ)
ππ + (βπ)
ππ β π
3.
Write the product, and then rewrite the expression in standard form by removing parentheses and collecting like
terms.
a.
ππ β π and the multiplicative inverse of π
π
(ππ + (βπ)) ( )
π
π
π
( ) (ππ) + ( ) (βπ)
π
π
π
πβ
π
b.
The multiplicative inverse of βπ and πππ β π
π
(β ) (πππ β π)
π
π
π
(β ) (πππ) + (β ) (βπ)
π
π
βππ + π
Lesson 5:
Using the Identity and Inverse to Write Equivalent Expressions
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c.
7β’3
π β π and the multiplicative inverse of π
π
(π + (βπ)) ( )
π
π
π
( ) (π) + ( ) (βπ)
π
π
π
πβ π
π
d.
The multiplicative inverse of
π
π
and ππ β
π
π
π
π (ππ β )
π
π
π(ππ) + π (β )
π
πππ β π
e.
The multiplicative inverse of β
π
π
π
and
β
πππ
πππ ππ
π
π
(βπππ) (
β )
πππ ππ
π
π
(βπππ) (
) + (βπππ) (β )
πππ
ππ
βπ + π
4.
Write the expressions in standard form.
a.
π
π
(ππ + π)
π
π
(ππ) + (π)
π
π
π+π
b.
π
π
(π β π)
π
π
(π) + (βπ)
π
π
π
πβπ
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
Lesson 5
NYS COMMON CORE MATHEMATICS CURRICULUM
d.
π
π
7β’3
(ππ + π)
π
π
(ππ) + (π)
π
π
π
π
π+
π
π
e.
π
π
(ππ β π)
π
π
(ππ) + (βπ)
π
π
ππ
π
πβ
π
π
f.
π
π
(πππ β π) β π
π
π
(πππ) + (βπ) + (βπ)
π
π
ππ + (βπ) + (βπ)
ππ 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