1. Operations The verbs of mathematics.

2. Subtraction Same as: adding a negative number. 4 - 3 = 4 + (-3)

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 Multiplication by the inverse or reciprocal of a number. This definition of division is essential when working with fractions!!!

5. Your turn: 1.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) “Motorcycle my riding fun I have.” (Persian)

7. Order of Operations (PEMDAS) “Please Excuse My Dear Aunt Sally.” ParenthesesExponentsMultiplicationDivisionAdditionSubtraction

8. Your turn: 4.

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

11. Associative Property of Addition 2 + 3 + 4 Can you add 3 numbers at the same time? Pick 2 of the 3 numbers, add them together. Add the 3 rd number to the sum of the 1 st two. 2 + 3 = 5 5 + 4 = 9

12. Associative Property of Addition 2 + 3 + 4 We use PEMDAS (parentheses) to “associate” the first 2 numbers together. the first 2 numbers together. (2 + 3) + 4 = 5 + 4 = 9 2 + (3 + 4) = 2 + 7 = 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.

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

14. Your turn: 5.Simplify the following expression using the commutative (order) and associative (grouping) commutative (order) and associative (grouping) properties. properties.

15. Associative Property of Multiplication 2 x 3 x 4 We use PEMDAS (parentheses) to “associate” the first 2 numbers together. the first 2 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.

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

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

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

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

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

21. Inverse Property of Addition 23 + x = 0 “What number do you add so the sum equals zero x = ? -25 + x = 0 x = ? We will use this property to solve equations.

22. Inverse Property of Multiplication What number do you multiply by so the product is 1 (one)? 10 * x = 1 x = ? 10 times its “reciprocal” equals 1 10 divided by itself equals 1 We will use this property to solve equations.

23. Addition Property of Equality a = b a + 1 = b + 1 a + 1 = b + 1 Equivalent Equations

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

25. Subtraction Property of Equality a = b a - 1 = b - 1 a - 1 = b - 1 Equivalent Equations

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

27. Multiplication Property of Equality a = b a * 2 = b * 2 a * 2 = b * 2 Equivalent Equations

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

29. Division Property of Equality a = b a ÷ 2 = b ÷ 2 a ÷ 2 = b ÷ 2 Equivalent Equations

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

31. 11. 11. 2 = 3 + x Your turn: 12. 12. -27 = x - 3 13. 13. 12 = 3x 14. 14. = -2

32. Combinations “Un-doing” operations Use “reverse” PEMDAS. What do you do 1 st : subtraction or multiplication? - 1 * 2 x = 8 x = ?

33. 15. 15. 12 = 3 + 3x Your turn: 16. 16. -8 = - 5 17. 17. 24 - x = 3x 18. 18. - 4 = -8