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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

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Just like English has nouns, verbs, and adjectives, mathematics has terms, factors, and coefficients. Well, sort of. TERMS A term that has no variables is often called a constant because it never changes. 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.

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Within each term, there can be two or more factors. There are always at least two factors, though one of them may be the number 1, which isn't usually written. The numbers and/or variables multiplied together. Has two factors: 3 and x. 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.

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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

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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

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

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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

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Important things toremember 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.

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So really it looks like this….. PParenthesis EExponents MDMultiplication and Division ASAddition and Subtraction In order from left to right

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SAMPLE PROBLEM #1 Parenthesis Exponents This one is tricky! Remember: Multiplication/Division are grouped from left to right…what comes 1 st ? Division did…now do the multiplication (indicated by parenthesis) More division Subtraction

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SAMPLE PROBLEM Subtraction Exponents Remember the division symbol here is grouping everything on top, so work everything up there first….multiplication Parenthesis Division – because all the work is done above and below the line

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Order of Operations-BASICS Think: PEMDAS Order of Operations-BASICS Think: PEMDAS Please Excuse My Dear Aunt Sally PParenthesis EExponents MMultiplication DDivision AAddition SSubtraction

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

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Practice 1.2 Order of Operations and Evaluating Expression

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Lesson Extension Can you fill in the missing operations? (3+5) + 4 = * 3 ÷ 3 = * ÷ 2 = 10

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Practice 1.3 Real Number and the Number Line

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Properties of Real Numbers (A listing) Associative PropertiesAssociative Properties Commutative PropertiesCommutative Properties Inverse PropertiesInverse Properties Identity PropertiesIdentity Properties Distributive PropertyDistributive Property All of these rules apply to Addition and Multiplication

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Associative Properties Associative Properties Associate = group Rules: Associative Property of Addition (a+b)+c = a+(b+c) Associative Property of Multiplication (ab)c = a(bc) It doesnt matter how you group (associate) addition or multiplication…the answer will be the same! Samples: Associative Property of Addition (1+2)+3 = 1+(2+3) Associative Property of Multiplication (2x3)4 = 2(3x4)

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Commutative Properties Commutative Properties Commute = travel (move) Rules: Commutative Property of Addition a+b = b+a Commutative Property of Multiplication ab = ba It doesnt matter how you swap addition or multiplication around…the answer will be the same! Samples: Commutative Property of Addition 1+2 = 2+1 Commutative Property of Multiplication (2x3) = (3x2)

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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 = 2-5? Is 6/3 the same as 3/6? Is (5-2)-3 = 5-(2-3)? Is (6/3)-2 the same as 6/(3-2)? Properties of real numbers are only for Addition and Multiplication

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Inverse Properties Think: Opposite 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 What is the opposite (inverse) of addition? What is the opposite of multiplication? Subtraction (add the negative) Division (multiply by reciprocal)

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Identity Properties 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 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)

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Distributive Property 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 If something is sitting just outside a set of parenthesis, you can distribute it through the parenthesis with multiplication and remove the parenthesis.

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Practice 1.4 Properties of Real Numbers

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Practice 1.5 Adding and Subtracting Real Numbers

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Practice 1.6 Multiplying and Dividing Real Numbers

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Practice 1.7 Distributive Property

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Practice 1.8 An Introduction to Equations

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