using the identity & inverse to write equivalent expressions & proofs engage ny: lesson 5...

USING THE IDENTITY & INVERSE TO WRITE EQUIVALENT

EXPRESSIONS & PROOFSEngage NY: Lesson 5Pink Packet pages 19-22

WRITE AN EXPRESSION TO REPRESENT THE TEMPERATURE CHANGE. In the morning, Harrison checked the temperature outside to find that it was -12°F. Later in the afternoon, the temperature rose 12°F. What was the afternoon temperature?

Write an expression to represent the temperature change

WHAT IS THE TEMPERATURE CHANGE? In the morning, Harrison checked the temperature outside to find that it was -12°F. Later in the afternoon, the temperature rose 12°F. What was the afternoon temperature?

What is the temperature change?

REWRITE SUBTRACTION AS ADDING THE INVERSE & FIND THE SUM A) – 2

2 + (-2) = 0

REWRITE SUBTRACTION AS ADDING THE INVERSE & FIND THE SUM B) -4 – (-4)

(-4) + 4 = 0

REWRITE SUBTRACTION AS ADDING THE INVERSE & FIND THE SUM D) g – g

g + (-g) = 0

THE SUM OF ADDITIVE INVERSES EQUALS___?

ZERO

ADD OR SUBTRACT

A) 16 + 0

16 + 0 = 16

ADD OR SUBTRACT

B) 0 – 7 =

0 + (-7) = -7

ADD OR SUBTRACT

C) -4 + 0 =

-4 + 0 = -4

ADD OR SUBTRACT

D) 0 + d

0 + d = d

WHAT PATTERNS DO YOU NOTICE IN 4A – 4D?

The sum of any quantity and zero is equal to the value of

the quantity.

WHAT IS ANOTHER WORD FOR OPPOSITES?

Additive Inverse

WHAT IS THE ADDITIVE INVERSE OR OPPOSITE OF -12.5? -12.5 + 12.5 = 0

WHAT IS THE SUM OF A NUMBER & ITS OPPOSITE?

ZERO

GUESS MY NUMBER…

Your younger sibling runs up to you and excitedly exclaims, “I’m thinking of a number. If I add it to the number ten times, that is, my number my numbermy number… and so on, then the answer is. What is my number?”

Justify your answer.

ADDITIVE IDENTITY PROPERTY OF ZEROZero is the only number that when summed with another number, results in that number again. This property makes zero special among all the numbers, so special in fact, that mathematicians have a special name for zero, called the “additive identity”; they call that property the “Additive Identity Property of Zero.”8 + 0 = 8

PROOFS…

1. Write down the problem

2. Work out each problem, showing every SINGLE LITTLE step

3. Write what property goes with each step you take

4. Use your pink packet vocabulary to help you.

Additive Identity Property of Zero- The sum of any number and zero = ITSELF0 + (-5) = -5

Additive Inverse- Opposites added together that have a sum of ZERO2x + (-2x) = 0Used when adding negative integers or subtracting integers

Associative Property- Any Grouping with parenthesis(3x + 4) + 6 = 3x + (4 + 6)

Commutative Property- Any Order- Switch places of numbers3x + 4 = 4 + 3x

Distributive Property- Expanded or standard form 4 (x + 6) = 4x + 24

ADDITION & SUBTRACTION PROPERTIES

WRITE THE SUM & THEN WRITE AN EQUIVALENT EXPRESSION BY COLLECTING LIKE TERMS. WRITE IN PROOF FORM. 2x and -2x + 3 Always write down original problem

Then, write problem with parenthesis.

Associative property, collect like-terms

Additive inverse

Additive identity property of zero

(Green is not part of the proof)

2X – 7 AND THE OPPOSITE OF 2X

Subtraction as adding the inverse

Commutative property, associative property

Additive inverse


THE OPPOSITE OF (5X – 1) AND 5X

Taking the opposite is equivalent to multiplying by

Distributive property

Commutative property, any grouping property

Additive inverse


-4 AND 4B + 4

Any order, any grouping

Additive inverse


COMPLETE- PACKET PAGE 19

1

𝑾𝒉𝒂𝒕𝒂𝒓𝒆𝒕𝒉𝒆𝒔𝒆𝒑𝒂𝒊𝒓𝒔 𝒐𝒇 𝒏𝒖𝒎𝒃𝒆𝒓𝒔 𝒄𝒂𝒍𝒍𝒆𝒅 ?Reciprocal

𝑾𝒉𝒂𝒕 𝒊𝒔 𝒂𝒏𝒐𝒕𝒉𝒆𝒓 𝒕𝒆𝒓𝒎 𝒇𝒐𝒓 𝒓𝒆𝒄𝒊𝒑𝒓𝒐𝒄𝒂𝒍?Multiplicative Inverse

What happens to the sign of the expression when converting it to its multiplicative inverse?

No change to the sign

WHAT IS THE PRODUCT OF A MULTIPLICATIVE INVERSE?

ONE

MULTIPLICATIVE IDENTITY PROPERTY OF ONEOne is the only number that when multiplied with another number, results in that number again.

This property makes special among all the numbers, so special, in fact, that mathematicians have a special name for one, called the “multiplicative identity”

They call that property the “Multiplicative Identity Property of One.”

=

-1… IS A SPECIAL NUMBER

has the property that multiplying a number by it is the same as taking the opposite of the number. -1●5 =-(5) = -1(5)=

-(-5) = -1(-5)=

-5-55

Additive Identity Property of Zero- The sum of any number and zero = ITSELF0 + (-5) = -5

Multiplicative Identity Property of One- The product of any number and its reciprocal = ITSELF

(-5) ● 1 = -5

Additive Inverse- Opposites added together that have a sum of ZERO2x + (-2x) = 0Used when adding negative integers or subtracting integers

Multiplicative Inverse- Opposites multiplied together that have a product of ONE2x + (-2x) = 0Used when adding negative integers or subtracting integers

Associative Property- Any Grouping with parenthesis(3x + 4) + 6 = 3x + (4 + 6)

Commutative Property- Any Order- Switch places of numbers3x + 4 = 4 + 3x

Distributive Property- Expanded or standard form 4 (x + 6) = 4x + 24

PROPERTIES

WRITE THE PRODUCT AND THEN WRITE THE EXPRESSION IN STANDARD FORM BY REMOVING PARENTHESES AND COMBINING LIKE TERMS. JUSTIFY EACH STEP.

and Original Problem

Multiplicative inverse



THE OPPOSITE OF 4X AND -5 + 4X


Additive inverse


The multiplicative inverse of and


Multiplicative inverses, multiplication

Multiplicative identity property of one

THE MULTIPLICATIVE INVERSE OF AND




THE OPPOSITE OF (– 7 – 4V) AND – 4V

Taking the opposite is equivalent to multiplying by


Any grouping, additive inverse


3X + (1 – 3X)

Subtraction as adding the inverse


Additive inverse


THE OPPOSITE OF -10T AND T-10T


Additive inverse


THE RECIPROCAL OF 3 AND -6Y – 3X Rewrite subtraction as an addition problem





Rewrite subtraction as an addition problem



Multiplicative identity property ofone


Rewrite subtraction as an addition problem




using the identity & inverse to write equivalent expressions & proofs engage ny: lesson 5...

Documents

identity inverse

additive inverse opposites

integersassociative

afternoon temperature

distributive property

commutative property

sum of additive inverses

special number