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Start thinking of math as a language , not a pile of numbers. Just like any other language, math can help us communicate thoughts and ideas with each other. An expression is a thought or idea communicated by language.

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Start thinking of math as a language, not a pile of numbers

Just like any other language, math can help us communicate thoughts and ideas with each other

An expression is a thought or idea communicated by language

In the same way, a mathematical expression can be considered a mathematical thought or idea communicated by the language of mathematics.

Mathematics is a language, and the best way to learn a new language is to immerse yourself in it.


Just like English has nouns, verbs, and adjectives,

mathematics has terms, factors, and coefficients.

Well, sort of.


are the pieces of the expression that are separated by plus or minus signs, except when those signs are within grouping symbols like parentheses, brackets, curly braces, or absolute value bars.

Every mathematical expression has at least one term.

Has two terms.

A term that has no variables is often called a constant because it never changes.

Within each term, there can be two or more factors.

The numbers and/or variables multiplied together.

Has two factors: 3 and x.

There are always at least two factors, though one of them may be the number 1, which isn't usually written.

Finally, a coefficient is a factor (usually numeric) that is multiplying a variable.

Using the example, the 3 in the first term is the coefficient of the variable x.

The order or degree of a mathematical expression is the largest sum of the exponents of the variables when the expression is written as a sum of terms.

Order is 1

Since the variable x in the first term has an exponent of 1 and there are no other terms with variables.

Order is 2

Order is 5

Now that we have our words, we can start putting them together and make expressions

Translate mathematical expressions into English

"the sum of 3 times a number and 2,"

"2 more than three times a number"

It's much easier to write the mathematical expression than to write it in English

just the facts order of operations and properties of real numbers

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
Important things toremember
  • 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
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

sample problem 1



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)

More division


sample problem



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


order of operations basics think pemdas please excuse my dear aunt sally
Order of Operations-BASICSThink: PEMDAS Please Excuse My Dear Aunt Sally
  • Parenthesis
  • Exponents
  • Multiplication
  • Division
  • Addition
  • Subtraction
lesson extension
Lesson Extension
  • Can you fill in the missing operations?
  • 2 - (3+5) + 4 = -2
  • 4 + 7 * 3 ÷ 3 = 11
  • 5 * 3 + 5 ÷ 2 = 10
properties of real numbers a listing
Properties of Real Numbers(A listing)
  • Associative Properties
  • Commutative Properties
  • Inverse Properties
  • Identity Properties
  • Distributive Property

All of these rules apply to Addition and Multiplication

associative properties associate group
Associative PropertiesAssociate = group

It doesn’t matter how you group (associate) addition or multiplication…the answer will be the same!


Associative Property of Addition

(a+b)+c = a+(b+c)

Associative Property of Multiplication

(ab)c = a(bc)


Associative Property of Addition

(1+2)+3 = 1+(2+3)

Associative Property of Multiplication

(2x3)4 = 2(3x4)

commutative properties commute travel move
Commutative PropertiesCommute = travel (move)

It doesn’t matter how you swap addition or multiplication around…the answer will be the same!


Commutative Property of Addition

a+b = b+a

Commutative Property of Multiplication

ab = ba


Commutative Property of Addition

1+2 = 2+1

Commutative Property of Multiplication

(2x3) = (3x2)

stop and think
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)?

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
Inverse PropertiesThink: Opposite


Inverse Property of Addition

a+(-a) = 0

Inverse Property of Multiplication

a(1/a) = 1

What is the opposite (inverse) of addition?

What is the opposite of multiplication?

Subtraction (add the negative)

Division (multiply by reciprocal)


Inverse Property of Addition


Inverse Property of Multiplication


identity properties
Identity Properties


Identity Property of Addition

a+0 = a

Identity Property of Multiplication

a(1) = a

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)


Identity Property of Addition


Identity Property of Multiplication


distributive property
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.


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