Transcript Just the facts: Order of Operations and Properties of real

Just the facts:
Order of Operations and
Properties of real numbers
A GEMS/ALEX Submission
Submitted by: Elizabeth Thompson, PhD
Summer, 2008
Important things to remember
• Parenthesis – anything grouped… including information
above or below a fraction bar.
• Exponents – anything in the same family as a ‘power’…
this includes radicals (square roots).
• Multiplication- this includes distributive property
(discussed in detail later).
Some items are grouped!!!
• Multiplication and Division are GROUPED from left to
right (like reading a book- do whichever comes first.
• Addition and Subtraction are also grouped from left to
right, do whichever comes first in the problem.
So really it looks like this…..
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Parenthesis
Exponents
Multiplication and Division In order from left to right
Addition and Subtraction In order from left to right
SAMPLE PROBLEM #1
16  4(3  1)  22 11 
3
16  4(2)  22 11 
3
Parenthesis
Exponents
16  4(8)  22  11 
4(8)  22  11 
This one is tricky!
Remember: Multiplication/Division are grouped from left to right…what comes 1st?
Division did…now do the multiplication (indicated by parenthesis)
32  22 11 
32  2 
More division
Subtraction
30
SAMPLE PROBLEM
3(5)  65
3(2  3)  65


2
2
2
2
Exponents
Parenthesis
75  65 10
3(25)  65



2
2
2
Remember the division
symbol here is grouping
everything on top, so
work everything up there
first….multiplication
Subtraction
Division –
because all the
work is done
above and
below the line
5
Order of Operations-BASICS
Think: PEMDAS
Please Excuse My Dear Aunt Sally
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Parenthesis
Exponents
Multiplication
Division
Addition
Subtraction
Take time to practice
Assignment #1
(When all assigned problems are finished –
do for Homework as needed)
• Remember PEMDAS and “Please Excuse
My Dear Aunt Sally”?
• Make up your own acronym for PEMDAS
and post it on the class wiki.
• Write it on White Paper and Illustrate your
acronym.
• Make sure it is school appropriate.
Lesson Extension
• Can you fill in the missing operations?
1. 2 - (3+5) + 4 = -2
2. 4 + 7 * 3 ÷ 3 = 11
3. 5 * 3 + 5 ÷ 2 = 10
Assignment #2
Create a Puzzle Greeting
• Fold a piece of paper (white or colored)
like a greeting card.
• On the cover: Write an equation with
missing operations (like the practice slide)
• In the middle: Write the equation with the
correct operations
• On the back: Put your name as you would
find a companies name on the back of a
greeting card.
Part 2:
Properties of Real Numbers
(A listing)
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Associative Properties
Commutative Properties
Inverse Properties
Identity Properties
Distributive Property
All of these rules apply to Addition and Multiplication
Associative Properties
Associate = group
It doesn’t matter how you group (associate) addition
or multiplication…the answer will be the same!
Rules:
Samples:
Associative Property of Addition
Associative Property of Addition
(a+b)+c = a+(b+c)
(1+2)+3 = 1+(2+3)
Associative Property of Multiplication
Associative Property of Multiplication
(ab)c = a(bc)
(2x3)4 = 2(3x4)
Commutative Properties
Commute = travel (move)
It doesn’t matter how you swap addition or
multiplication around…the answer will be the same!
Rules:
Samples:
Commutative Property of Addition
Commutative Property of Addition
a+b = b+a
1+2 = 2+1
Commutative Property of Multiplication
Commutative Property of Multiplication
ab = ba
(2x3) = (3x2)
Stop and think!
• Does the Associative Property hold true
for Subtraction and Division?
Is (5-2)-3 = 5-(2-3)?
Is (6/3)-2 the same as 6/(3-2)?
• Does the Commutative Property hold true
for Subtraction and Division?
Is 5-2 = 2-5?
Is 6/3 the same as 3/6?
Properties of real numbers are only for Addition and Multiplication
Inverse Properties
Think: Opposite
What is the opposite (inverse) of addition?
What is the opposite of multiplication?
Rules:
Inverse Property of Addition
a+(-a) = 0
Subtraction (add the negative)
Division (multiply by reciprocal)
Samples:
Inverse Property of Addition
3+(-3)=0
Inverse Property of Multiplication
Inverse Property of Multiplication
a(1/a) = 1
2(1/2)=1
Identity Properties
What can you add to a number & get the same number back? 0 (zero)
What can you multiply a number by and get the number back? 1 (one)
Rules:
Identity Property of Addition
a+0 = a
Samples:
Identity Property of Addition
3+0=3
Identity Property of Multiplication
Identity Property of Multiplication
a(1) = a
2(1)=2
Distributive Property
If something is sitting just outside a set of parenthesis, you can
distribute it through the parenthesis with multiplication and
remove the parenthesis.
Rule:
a(b+c) = ab+bc
Samples:
4(3+2)=4(3)+4(2)=12+8=20
• 2(x+3) = 2x + 6
• -(3+x) = -3 - x
Take time to practice
Homework
Log on to class wiki / discussion thread
Follow the directions given:
• Give an example of each of the properties
discussed in class, do not duplicate a
previous entry.