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

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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. A SSE 1

Just like English has nouns, verbs, and adjectives,
mathematics has terms, factors, and coefficients. Well, sort of. TERMS 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

Practice 1.1 Variables and Expressions A-SSE.A.1

Practice 1.1 Variables and Expressions A-SSE.A.1

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

Practice 1.2 Order of Operations and Evaluating Expression A-CED.1

Practice 1.2 Order of Operations and Evaluating Expression

Practice 1.2 Order of Operations and Evaluating Expression

Practice 1.2 Order of Operations and Evaluating Expression

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.

Practice 1.3 Real Number and the Number Line

Practice 1.3 Real Number and the Number Line

Practice 1.3 Real Number and the Number Line

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

Practice 1.4 Properties of Real Numbers

Practice 1.5 Adding and Subtracting Real Numbers

Practice 1.6 Multiplying and Dividing Real Numbers

Practice 1.7 Distributive Property

Practice 1.7 Distributive Property

Practice 1.8 An Introduction to Equations

Practice 1.8 An Introduction to Equations

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