Sec. Math 2 Lesson 1-2

Quiz #1-1 (Fractions) 1. What fraction represents the red portion of the circle? 2. Draw line segments from the center of the circle to each of the black dots. What is the new fraction? 3. Write a rule that explains how to divide fractions.

Mathematics is a language! What do languages consist of? Verbs: “action words” Math: what causes “action” in math? Vocabulary: “the meaning of a word” Math: also has vocabulary He hit me. operations: +, -, x, ÷

Mathematics is a language! What do languages consist of? Sentences: “a complete thought in words” What is a Math “sentence”? Phrase: example: “easy subject” is not a complete thought. complete thought. an expression (no equal sign) an equation What is a “phrase” in Math?

Mathematics is a language! What is an adjective? Adjective: “a word that describes a noun (person, place or thing)” What is an adjective in math?: 5 dogs, 3 cats, 2 triangles. Math: an expression (no equal sign) a number

Operations The verbs of mathematics.

Subtraction English: 5 “take away” 3 3 less than 5 Math: 5 – 3 3 less than “a number” x – 3

Subtraction Same as: adding a negative number Is the same as: 4 + (-3) Your turn: Is the same as: -7 + (-2) Convert the following to addition This definition is useful when trying to simplify the following:

Multiplication English: 5 “times” 3 One third of 30 Math: 5 * 3 Twice “a number” 2x

Multiplication “repeated addition.” = 3 x or 3 rows of 5 items.

Division English: 4 “fifths” Math:

Division Multiplication by the inverse or Multiplication by the inverse or reciprocal of a number. This definition is essential when working with fractions!!! Simplify the following using the definition of division: 3. Your turn:

4. Change this into addition: 4 – 1 5. Change this into multiplication: 6. Change this into multiplication:

“I have fun riding my motorcycle.” (English) “To ride a motorcycle fun I have.” (Persian) “Aez savar shodan-e mashin-e xodam, lezat miboram.” (Persian) lezat miboram.” (Persian) In each language there is a correct order of speaking words. speaking words. In math order of operations matters!!!!

Order of Operations (PEMDAS) “Please Excuse My Dear Aunt Sally.” Parentheses Exponents Multiplication Division Addition Subtraction

Properties The grammar of mathematics. 1. Associative property. 2. Commutative property. 3. Distributive property. 4. Identity property. 5. Inverse property. 6. Property of equality.

Associative Property of Addition We use PEMDAS (parentheses) to “associate” the 2 of the 3 numbers together so we know which to add together first. (2 + 3) + 4 = = (3 + 4) = = 9 The property says: when adding 3 or more numbers together, it doesn’t matter which two of numbers you together, it doesn’t matter which two of numbers you add together first (“associate”), you’ll always get the add together first (“associate”), you’ll always get the same answer. same answer.

Associative Property of Multiplication 2 x 3 x 4 We use PEMDAS (parentheses) to “associate” the 2 of the 3 numbers together. the 2 of the 3 numbers together. (2 x 3) x 4 = 6 x 4 = 24 2 x (3 x 4) = 2 x 12 = 24 The property says: when multiplying 3 or more numbers together, it doesn’t matter which two of numbers you together, it doesn’t matter which two of numbers you multiply together first (“associate”), you’ll always get the multiply together first (“associate”), you’ll always get the same answer. same answer.

Commutative Property of Addition Adding two numbers  doesn’t matter which number comes first. = 5

Commutative Property of Multiplication 2 x 3 = 3 x 2 multiplying two numbers  the order of multiplication does not matter.

Using the commutative and associative properties. 7 + x x = x + 2x Rearrange the order (commutative ) = (7 + 3) + x + 2x = (7 + 3) + x + 2x Group terms to add together (associative) (associative) = 10+ x + 2x What does “2x” mean? x + x

Showing work For now: to add or multiply two numbers together, they must be next to each other. 1.Rearrange the order to get 4 and 5 next to each other. Simplify the following: 2. Justify being able to “rearrange” the order with a property Commutative prop. of add. 3. Indicate you will add 4 and 5 together 4. Justify adding the 4 and 5 first with a property Associative prop. of add.

Showing work 5. Add 4 and 5 together. Is there a property for adding 4 and 5 together? addition 7. Indicate you will add x and 3x together 9. Add x and 3x together Associative prop. of add. 6. Justify the step 8. Justify the step 10. Justify the step addition

Another example: Simplify the following: Commutative property of addition. Associative property of addition. addition subtraction

Another example: Negative numbers (harder to justify) Simplify the following: Commutative property of addition. Associative property of add addition. addition Can I put parentheses around the -4 and -5 ? NO! Definition of subtraction Associative property of addition subtraction Definition of subtraction

Your turn: 7. Simplify the following expression using the commutative (order) and associative (grouping) commutative (order) and associative (grouping) properties. Justify each step. properties. Justify each step.

Your turn: 8. Simplify the following expression using the commutative (order) and associative (grouping) properties. Justify each step.

Distributive Property (of Addition over Multiplication) This property is important when variables are involved. 2(x + 4) = 2 x + (2 * 4) = 2x + 8 Simplify the following: 2(x + 4) Distributive property multiplication

Your turn: Simplify the following expressions using the distributive property (of addition over multiplication). Justify each step. distributive property (of addition over multiplication). Justify each step