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Operations

Operations. The verbs of mathematics. Subtraction. Same as: adding a negative number. 4 + (-3). 4 - 3 =. Convert the following to addition: 5 - 2. Multiplication. Best understood as “ repeated addition .”. 3 x 5. = 5 + 5 + 5. or 3 rows of 5 items. Division.

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Operations

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  1. Operations The verbs of mathematics.

  2. Subtraction Same as: adding a negative number. 4 + (-3) 4 - 3 = Convert the following to addition: 5 - 2

  3. Multiplication Best understood as “repeated addition.” 3 x 5 = 5 + 5 + 5 or 3 rows of 5 items.

  4. Division Multiplication by the inverse or reciprocal of a number. This definition of division is essential when working with fractions!!! Convert the following to multiplication:

  5. Your turn: • Change this into addition: 4 – 1 2. Change this into multiplication: 3.

  6. Properties The grammar of mathematics. “I have fun riding my motorcycle.” (English) “To ride a motorcycle fun I have.” (Persian) “Aez savor shodan-e mashin-e xodaem, lezaet miboraem.” (Persian)

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

  8. Your turn: 4.

  9. Commutative Property of Addition 2 + 3 = 3 + 2 Adding two numbers  doesn’t matter which number comes first.

  10. Commutative Property of Multiplication 2 x 3 = 3 x 2 multiplying two numbers  doesn’t matter which number comes first.

  11. Associative Property of Addition 2 + 3 + 4 We use PEMDAS (parentheses) to “associate” the first 2 numbers together. 2 + (3 + 4) (2 + 3) + 4 = 2 + 7 = 5 + 4 = 9 = 9 The property says: when adding 3 or more numbers together, it doesn’t matter which two of numbers you add together first (“associate”), you’ll always get the same answer.

  12. Using the commutative and associative properties. 7 + x + 3 + 2x = ? = 7 + 3 + x + 2x Rearrange the order (commutative) = (7 + 3) + (x + 2x) Group terms to add together = 10+ 3x When doing problems, you don’t need to rewrite the equation to re-arrange the order, or to group terms together, you can do it in your head.

  13. Your turn: • Simplify the following expression using the • commutative (order) and associative (grouping) • properties.

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

  15. Your turn: 6. Simplify the following expression using the commutative (order) and associative (grouping) properties.

  16. Distributive Property of Addition over Multiplication 2(3 + 4) = (2 * 3) + (2 * 4) 2 ( 7 ) = 6 + 8 14 = 14 This property is important when variables are involved. 2(x + 4) = (2 x) + (2 * 4) = 2x + 8

  17. Your turn: 7. Simplify the following expression using the distributive property of “additional over mulitplication”.

  18. Your turn: Identify the property that allows the step indicated. 8. 9. 10.

  19. Equality Properties Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality

  20. Solving an Equation Addition Property of Equality: whatever we added to the left side of the ‘=‘ sign, we must add to the right side of the equation. . x – 1 = 5 + 1 + 1 x = 6 x = Inverse Property of Addition Identity Property of Addition

  21. Solving an Equation Subtraction Property of Equality: whatever we subtracted from the left side of the ‘=‘ sign, we must subtract from the right side of the equation. . x + 1 = 5 - 1 - 1 x = 4 x = Inverse Property of Addition Identity Property of Addition

  22. Multiplication Property of Equality: whatever we multiply the left side of the ‘=‘ sign by, we must multiply the right side of the equation. . Solving an Equation = 5 * 2 * 2 x = 10 Inverse Property of Multiplication Identity Property of Multiplication

  23. Division Property of Equality: whatever we divide the left side of the ‘=‘ sign by, we must divide the right side of the equation. . Solving an Equation 3x = 15 ÷ 3 ÷ 3 (Or: mult. by ⅓) (Or: mult. by ⅓) x = 5 Inverse Property of Multiplication Identity Property of Multiplication

  24. Your turn: 11. 2 = 3 + x 12. -27 = x - 3 13. 12 = 3x 14. = -2

  25. Combinations What do you do 1st: subtraction or multiplication? “Un-doing” operations - 1 - 1 Use “reverse” PEMDAS. Or: get the variable term all by itself. * 2 * 2 x = ? x = 8

  26. Your turn: 15. 12 = 3 + 3x 16. -8 = - 5 18. - 4= -8 17. 24 - x = 3x

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