# Just the facts: Order of Operations and Properties of real numbers

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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. Click to extend information

So really it looks like this…..
Parenthesis Exponents Multiplication and Division Addition and Subtraction In order from left to right In order from left to right Click to extend information

SAMPLE PROBLEM #1 Parenthesis Exponents 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) Click and the operation / next step will appear…keep clicking until the end. More division Subtraction

SAMPLE PROBLEM Exponents Parenthesis
Click and the operation / next step will appear…keep clicking until the end. Remember the division symbol here is grouping everything on top, so work everything up there first….multiplication Division – because all the work is done above and below the line Subtraction

Order of Operations-BASICS Think: PEMDAS Please Excuse My Dear Aunt Sally
Parenthesis Exponents Multiplication Division Addition Subtraction Click to extend information

Take time to practice Assign problems from text and/or worksheet.
Work some problems with the students, allow time for questions.

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. This is a silly but fun assignment. Students get to be creative, all the time thinking of “Order of Operations”. Example: Purple elephants may dance and sing (draw a picture of a singing/dancing purple elephant) Sample 2: People enjoy my daily apple slices (a stick person smiling near apple slices).

Lesson Extension Can you fill in the missing operations?
2 - (3+5) = -2 * 3 ÷ 3 = 11 5 * ÷ 2 = 10 Teachers: One location for worksheets with ‘blank’ operations is For an extension of this slide, have each student fold a paper like a card. On the cover of the card, they create a problem with blanks (like the slide) on the inside of the card is the solution. On the back of the card is their name (like you would find a label on a real card). This can be done as homework or a classroom extension.

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. Make a sample for students to see as needed.

Part 2: Properties of Real Numbers (A listing)
Associative Properties Commutative Properties Inverse Properties Identity Properties Distributive Property Click for examples of each 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: Associative Property of Addition (a+b)+c = a+(b+c) Associative Property of Multiplication (ab)c = a(bc) Samples: Associative Property of Addition (1+2)+3 = 1+(2+3) Associative Property of Multiplication (2x3)4 = 2(3x4) Later we will discuss how these do not apply to subtraction and division.

Commutative Properties Commute = travel (move)
It doesn’t matter how you swap addition or multiplication around…the answer will be the same! Rules: Commutative Property of Addition a+b = b+a Commutative Property of Multiplication ab = ba Samples: Commutative Property of Addition 1+2 = 2+1 Commutative Property of Multiplication (2x3) = (3x2) The next slide will discuss how these do not apply to subtraction and division.

Is (5-2)-3 = 5-(2-3)? Is (6/3)-2 the same as 6/(3-2)?
Stop and think! Does the Associative Property hold true for Subtraction and Division? Does the Commutative Property hold true for Subtraction and Division? Is (5-2)-3 = 5-(2-3)? Is (6/3)-2 the same as 6/(3-2)? Leave time to discuss prior to clicking examples. 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? Subtraction (add the negative) Division (multiply by reciprocal) Rules: Inverse Property of Addition a+(-a) = 0 Inverse Property of Multiplication a(1/a) = 1 Samples: Inverse Property of Addition 3+(-3)=0 Inverse Property of Multiplication 2(1/2)=1

Identity Properties Rules: a+0 = a a(1) = a Samples: 3+0=3 2(1)=2
What can you add to a number & get the same number back? What can you multiply a number by and get the number back? 0 (zero) 1 (one) Rules: Identity Property of Addition a+0 = a Identity Property of Multiplication a(1) = a Samples: Identity Property of Addition 3+0=3 Identity Property of Multiplication 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 Discuss/illustrate how arrows can help a student stay on track

Take time to practice Assign problems from text and/or worksheet.
Work some problems with the students, allow time for questions.

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.

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