Cashill AAH 1.1 and 1.2

1.1 Real Numbers & Number Operations : 

1.1 Real Numbers & Number Operations Mr. Cashill Verona High School

What is a Real Number? : 

What is a Real Number? Real Numbers are all of the numbers you have used in your previous math classes. There are 4 types of Real Numbers: Whole Numbers Integers Irrational Numbers Rational Numbers

Set of Real Numbers : 

Set of Real Numbers

Examples of Real Numbers : 

Examples of Real Numbers Whole Numbers: 0, 1, 2, 3 (counting #s) Integers: -2, -1, 0, 1, 2 (+ & - whole #s) Rational Numbers: A number that can be written as a fraction. When written as a decimal, they terminate or repeat--½, 1/3, 4/5, 7/9 Irrational Numbers: Real numbers that are not rational such as π or √3. Decimals that do not terminate or repeat.

Properties of Multiplication & Addition : 

Properties of Multiplication & Addition (a, b & c are real numbers) Addition Multiplication Closure a+b is real a*b is real Commutative a+b=b+a ab=ba Associative (a+b)+c=a+(b+c) (ab)c=a(bc) Identity a+0=a, 0+a=a a*1=a, 1*a=a Inverse a+(-a)=0 a*(1/a)=1, a≠0 Distributive a(b+c)=ab+ac (a+b)c=ac+bc

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Additive Inverse (Opposite) examples: a and –a or -5 and 5 Multiplicative Inverse (Reciprocal) examples: a and 1/a or -1/3 and -3 Remember: Difference means subtract Quotient means divide

Unit Analysis Examples : 

Unit Analysis Examples 685ft + 225ft = 910ft (2.25h) = 135km 3. 45 mi/h

1.2 Algebraic Expressions & Models : 

1.2 Algebraic Expressions & Models Mr. Cashill Verona High School

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Base ? Exponent ? Power ?

Evaluating Powers : 

Evaluating Powers (-2)6 = (-2)*(-2)*(-2)*(-2)*(-2)*(-2) = 64 2. -26 = -(2*2*2*2*2*2) = -64

Order of OperationsKnow these!! : 

Order of OperationsKnow these!! Please Excuse My Dear Aunt Sally ( ) Exponents Multiply/Divide (L to R) Add/Subtract (L to R)

Evaluating Algebraic Expressions : 

Evaluating Algebraic Expressions To evaluate an algebraic expression means to: Replace the variable(s) in the expression with numeric values that are assigned to them, and Simplify the resulting numeric expression by using the ORDER OF OPERATION. The end result is the numeric value of the expression when the variable(s) take on the assigned values.

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Evaluating Algebraic Expressions Example: Find the value of the following algebraic expressions if a = – 2, b = 5 a + b = – 2a + b = – 2 + 5 = – 2 (– 2) + 5 = 3 4 + 5 = 9

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It is helpful when solving real-life problems… To first write a “word” equation before you write it in mathematical symbols. This “word equation” is called a verbal model. The verbal model is then used to write a mathematical statement which is called an algebraic model. Write a verbal model. Assign labels. Write an algebraic model Solve the algebraic Model Answer the question.

Slide 15: 

Writing and Using a Simple Model A water saving faucet has a flow rate of at most 9.6 cubic inches per second. To test whether your faucet meets this standard, you time how long it takes the faucet to fill a 470 cubic inch pot, obtaining a time of 35 seconds. Find your faucet’s flow rate. Does it meet the standard for water conservation? Solution Verbal Model Volume of Pot = Flow rate of faucet * Time to fill pot Labels Volume of pot = 470 (cubic inches) Flow rate of faucet = r (cubic inches per second) Time to fill pot = 35 (seconds) Algebraic Model 470 = r (35) The flow rate is about 13.4 cubic inches/sec, which does not meet the standard.

Simplifying Algebraic Expression : 

Simplifying Algebraic Expression 6m2 – 12m – 7m2 = -m2 – 12m 3(x-2) – 5(x-8) = 3x – 6 – 5x +40 = -2x + 34