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Start thinking of math as a language , not a pile of numbers PowerPoint PPT Presentation


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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

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


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

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.


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

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.


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

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


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

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


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

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


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

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


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

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)

More division

Subtraction


Sample problem

SAMPLE PROBLEM

Exponents

Parenthesis

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

Order of Operations-BASICSThink: PEMDAS Please Excuse My Dear Aunt Sally

  • Parenthesis

  • Exponents

  • Multiplication

  • Division

  • Addition

  • Subtraction


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

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


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

Practice 1.2 Order of Operations and Evaluating Expression


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

Practice 1.2 Order of Operations and Evaluating Expression


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

Practice 1.2 Order of Operations and Evaluating Expression


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


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

Practice 1.3 Real Number and the Number Line


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

Practice 1.3 Real Number and the Number Line


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

Practice 1.3 Real Number and the Number Line


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!

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)


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!

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)


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

Rules:

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)

Samples:

Inverse Property of Addition

3+(-3)=0

Inverse Property of Multiplication

2(1/2)=1


Identity properties

Identity Properties

Rules:

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)

Samples:

Identity Property of Addition

3+0=3

Identity Property of Multiplication

2(1)=2


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.

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


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

Practice 1.4 Properties of Real Numbers


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

Practice 1.5 Adding and Subtracting Real Numbers


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

Practice 1.6 Multiplying and Dividing Real Numbers


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

Practice 1.7 Distributive Property


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

Practice 1.7 Distributive Property


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

Practice 1.8 An Introduction to Equations


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

Practice 1.8 An Introduction to Equations


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