2. 1-1 Variables and Expressions Variable: a letter that stands for a number Variable Expression: a mathematical phrase that uses variables, numerals, and operation symbols - Examples: c + 5 8 - x

3. Writing Variable Expressions You can translate word phrases into variable expressions. - Examples: 1. Nine more than a number y y+9 2. 4 less than a number n n-4 3. A number z times three 3z

4. Examples 1. A number c divided by 12 2. 5 times the quantity 4 plus a number d 3. The value in cents of 10 dimes 4. The number of gallons in 7 quarts 5. Four times the quantity 8 minus p

5. Examples - Answers 1. A number c divided by 12 c/12 2. 5 times the quantity 4 plus a number d 5(4 + d) 3.The value in cents of 10 dimes.10(10) 4.The number of gallons in 7 quarts 7/4 5. Four times the quantity 8 minus p 4(8 - p)

6. 1-2 Order of Operations The order in which you perform operations can affect the value of an expression. P - Parenthesis ( ) [ ] E - Exponents x 2 M/D - Multiply/Divide x ÷ A/S - Add/Subtract + - Please Excuse My Dear Aunt Sally OR PEMDAS

7. Simplifying Expression 1. 4 + 15 x 3 =4 + 45 =49 2. 3 x 5 - 8 ÷ 4 + 6 = 15 - 2 + 6 = 13 + 6 = 19

8. Examples 1. 4 - 1 x 2 + 6 ÷ 3 2. 5 + 6 x 4 ÷ 3 - 1 3. 21 + 15 3 + 6 4. [2 + (6 x 8)] - 1

9. Examples - Answers 1. 4 - 1 x 2 + 6 ÷ 3 = 4 2. 5 + 6 x 4 ÷ 3 - 1 = 12 3. 21 + 15 = 4 3 + 6 4. [2 + (6 x 8)] - 1 = 49

10. Using Grouping Symbols Insert grouping symbols to make each sentence true. 1. 7 + 4 x 6 = 66 2. 7 x 8 - 6 + 3 = 17 3. 3 + 8 - 2 x 5 = 45

11. Using Grouping Symbols - Answers Insert grouping symbols to make each sentence true. 1. (7 + 4) x 6 = 66 2. [7 x (8 - 6)] + 3 = 17 3.[3 + (8 - 2)] x 5 = 45

12. Using, or = 15 x 3 - 2 15 x (3 - 2) 12 ÷ 3 + 9 x 4 12 ÷ (3 + 9) x 4 (19 - 15) ÷ (3 + 1) 19 - 15 ÷ 3 + 1

13. Using, or = (Answers) 15 x 3 - 2 15 x (3 - 2) > 12 ÷ 3 + 9 x 4 12 ÷ (3 + 9) x 4 > (19 - 15) ÷ (3 + 1) 19 - 15 ÷ 3 + 1 <

14. To EVALUATE a variable expression, you first replace each variable with a number. Then use order of operations to simplify. Example:Evaluate 4y - 15 for y = 9. 4y - 15 = 4(9) - 15 = 36 - 15 = 21 1-3 Writing and Evaluating Expressions

15. Evaluate: 1. 4(t+3) + 1 for t = 8. 2. 63 - 5x for x = 7 3. 3x + 2 for x = 8 Examples

16. Evaluate: 1. 4(t+3) + 1 for t = 8 45 2. 63 - 5x for x = 7 28 3. 3x + 2 for x = 8 26 Examples - Answers

17. Replacing More Than One Variable Example: Evaluate 3ab + c ⁄ 2 for a = 2, b = 5, and c = 10. 3ab + c ⁄ 2 = 3(2)(5) + 10 ⁄ 2 = 6 * 5 + 5 = 30 + 5 = 35

18. Evaluate: 1. (9 +y) ⁄ x for x = 2 and y = 3. 2. 2xy - z for x = 4, y = 3, z = 1 3. r + s for r = 10, s = 9 2 Examples

19. Evaluate: 1. (9 +y) ⁄ x for x = 2 and y = 3. 6 2. 2xy - z for x = 4, y = 3, z = 1 23 3. r + s for r = 10, s = 9 9.5 2 Examples - Answers

20. 1-4 Integers and Absolute Value Integers are whole numbers and their opposites. Zero separates positive and negative numbers on a number line. Zero is neither positive nor negative and is its own opposite. Absolute value is a number’s distance from zero.

21. Integers 32 degrees above zero = +32°F or 32°F 32 degrees below zero = -32 °F A number line helps you compare positive and negative numbers and arrange them in order.

22. Graphing on a number line Graph -1, 4, and -5 on a number line. Then order them from least to greatest. -5-1 04 Least to greatest -5, -1, 4

23. Finding Absolute Value Numbers that are the same distance from zero on a number line but in opposite directions are called opposites. Example: -3 and 3 |-3| and |3| are both 3; this means they are both 3 units from zero Examples: 1. - |-10| 2. |-23| 3. 200 ft below sea level 4. A profit of $300

24. Use, or = Examples: 1. -8 0 2. |-3| |50| 3. |-10| |10|

25. Use, or = (Answers) Examples: 1. -8 0 < 2. |-3| |50| > 3. |-10| |10| =

26. 1-5 Adding Integers Adding Integer Rules: Same Sign: Sum of 2 positive integers = positive Sum of 2 negative integers = negative (Add the numbers and keep the same sign) Different Signs: 1. Find difference of the 2 numbers 2. The sum has the sign of the integer with the greater absolute value (Subtract the numbers and take the sign of the larger number)

27. Adding Integers with the same sign Examples: -2 + (-2) -22 + (-16) (-25) + (-15)

28. Adding Integers with the same sign (Answers) Examples: -2 + (-2) = -4 -22 + (-16) = -38 (-25) + (-15) = -40

29. Adding Integers with different signs Examples: 1. 2 + (-6) 2. (-1) + 4 3. 7 + (-18) 4. 5 + (-3)

30. Adding Integers with different signs (Answers) Examples: 1. 2 + (-6) = -4 2. (-1) + 4 =3 3. 7 + (-18) =-11 4. 5 + (-3) =2

31. Using Order of Operations Examples 1. -12 + (-6) + 15 + (-2) 2. 1 + (-3) + 2 + (-10) 3. -250 + 200 + (-100) + 220

32. Using Order of Operations - Answers Examples 1. -12 + (-6) + 15 + (-2) = -5 2. 1 + (-3) + 2 + (-10) =-10 3. -250 + 200 + (-100) + 220 =70

33. 1-6 Subtracting Integers To subtract integers, add its opposite. (Leave the 1st number the same, change the subtraction sign to an addition sign, and change the next number to the opposite sign) Example: 4 - 8 = 4 + (-8) = -4

34. Examples 1. -7 - (-3) 2. -2 - 3 3. 87 - (-9) 4. 16 - (-8) 5. -90 - (-80) -20 6. An airplane takes off, climbs 5,000 ft, and then descends 700 ft. What is the airplane’s current height?

35. Examples - Answers 1. -7 - (-3) = -4 2. -2 - 3 =-5 3. 87 - (-9) =96 4. 16 - (-8) =24 5. -90 - (-80) -20 =-30 6. An airplane takes off, climbs 5,000 ft, and then descends 700 ft. What is the airplane’s current height? 4300 ft

36. 1-7 Inductive Reasoning 1-8 Look for a Pattern Inductive Reasoning is making conclusions based on pattern you observe. Conjecture is a conclusion you reach by inductive reasoning.

37. Writing a Rule for Patterns 30, 25, 20, 15… RULE: Start with 30 and subtract 5 repeatedly. 1, 3, 4, 12, 13… RULE: Start with 1. Alternate multiplying 3 and adding 1. Examples: 1. 3, 9, 27, 81… 2. 4, 9, 14, 19

38. Is each conjecture correct? If not, give a counterexample? 1. All birds can fly. 2. Every square is a rectangle. 3. The product of 2 numbers is never less than either of the numbers.

39. Is each conjecture correct? If not, give a counterexample? (Answers) 1. All birds can fly. No, ostriches can’t fly 2. Every square is a rectangle. Yes 3. The product of 2 numbers is never less than either of the numbers. No, 1/2 x 1/2 = 1/4

40. 1-9 Multiply/Divide Integers Rules: The product/quotient of 2 integers with the same sign is positive. The product/quotient of 2 integers with different signs is negative.

41. Examples 1. -17 x 3 2. -65 ÷ -5 3. -6(-3) ÷ 2 4. -15(-3) -9 5. -3 x 7(-2)

42. Examples - Answers 1. -17 x 3 =-51 2. -65 ÷ -5 =13 3. -6(-3) ÷ 2 = 9 4. -15(-3) =-5 -9 5. -3 x 7(-2) =42

43. Find the average 1. Temperature: 6°, -15°, -24°, 3°, -25° 2. Stock Price Changes: $52, -$7, $20, -$63, -$82

44. Find the average - Answers 1. Temperature: 6°, -15°, -24°, 3°, -25° -11 2. Stock Price Changes: $52, -$7, $20, -$63, -$82 -16

45. 1-10 The Coordinate Plane Coordinate plane: formed by the intersection of 2 number lines. X-axis: the horizontal number line Y-axis: the vertical number line The x-axis and y-axis divide the coordinate plane into 4 quadrants Origin: where the axes intersect Ordered pair: gives the coordinates and location of a point

46. Quadrant II Quadrant I Y-axis X-axis Origin Quadrant III Quadrant IV