USING THE IDENTITY & INVERSE TO WRITE EQUIVALENT EXPRESSIONS & PROOFS Engage NY: Lesson 5 Pink...
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Transcript of USING THE IDENTITY & INVERSE TO WRITE EQUIVALENT EXPRESSIONS & PROOFS Engage NY: Lesson 5 Pink...
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?
WHAT PATTERNS DO YOU NOTICE IN 4A – 4D?
The sum of any quantity and zero is equal to the value of
the quantity.
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
Additive identity property of zero
THE OPPOSITE OF (5X – 1) AND 5X
Taking the opposite is equivalent to multiplying by
Distributive property
Commutative property, any grouping property
Additive inverse
Additive identity property of zero
What happens to the sign of the expression when converting it to its multiplicative inverse?
No change to the sign
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
Distributive property
Multiplicative inverse
THE OPPOSITE OF 4X AND -5 + 4X
Any order, any grouping
Additive inverse
Additive identity property of zero
The multiplicative inverse of and
Distributive property
Multiplicative inverses, multiplication
Multiplicative identity property of one
THE MULTIPLICATIVE INVERSE OF AND
Distributive property
Multiplicative inverse
Multiplicative identity property of one
THE OPPOSITE OF (– 7 – 4V) AND – 4V
Taking the opposite is equivalent to multiplying by
Distributive property
Any grouping, additive inverse
Additive identity property of zero
3X + (1 – 3X)
Subtraction as adding the inverse
Any order, any grouping
Additive inverse
Additive identity property of zero
THE OPPOSITE OF -10T AND T-10T
Any order, any grouping
Additive inverse
Additive identity property of zero
THE RECIPROCAL OF 3 AND -6Y – 3X Rewrite subtraction as an addition problem
Distributive property
Multiplicative inverse
Multiplicative identity property of one
The multiplicative inverse of and
Rewrite subtraction as an addition problem
Distributive property
Multiplicative inverse
Multiplicative identity property ofone