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Variables and Expressions

Explore variables, expressions, order of operations, and simplify algebraic expressions in pre-algebra lessons. Practice converting units and evaluating equations with examples.

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Variables and Expressions

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  1. 6 7 51 weeks Variables and Expressions PRE-ALGEBRA LESSON 1-1 How many weeks in 363 days? 1-1

  2. Variables and Expressions PRE-ALGEBRA LESSON 1-1 Complete each equation. 1. 1 week = days 2. 1 foot = inches 3. 1 nickel = cents 4. 1 gallon = quarts 5. 1 yard = feet Check Skills You’ll Need 1-1

  3. Variables and Expressions PRE-ALGEBRA LESSON 1-1 Solutions 1. 1 week = days 2. 1 foot = inches 3. 1 nickel = cents 4. 1 gallon = quarts 5. 1 yard = feet 7 12 5 4 3 1-1

  4. Variables and Expressions PRE-ALGEBRA LESSON 1-1 Identify each expression as a numerical expression or a variable expression. For a variable expression, name the variable. a. 7  3 numerical expression b.  4t variable expression; t is the variable. Quick Check 1-1

  5. Words number of pens 29¢ times p Let = number of pens. p Expression 29 • Variables and Expressions PRE-ALGEBRA LESSON 1-1 Write a variable expression for the cost of p pens priced at 29¢ each. The variable expression 29 • p, or 29p, describes the cost of p pens. Quick Check 1-1

  6. Variables and Expressions PRE-ALGEBRA LESSON 1-1 Write a variable expression for each word phrase. 1. the total of h and 56 2. three less than d 3.p decreased by three 4.a divided by 7 h + 56 d – 3 p – 3 a ÷ 7 1-1

  7. The Order of Operations PRE-ALGEBRA LESSON 1-2 Divide. Round answers to the nearest tenth. a. 75 ÷ 8 b. 740 ÷ 7 105.7 9.4 1-2

  8. The Order of Operations PRE-ALGEBRA LESSON 1-2 (For help, go to Skills Handbook p. 760.) Find each quotient. 1. 164 ÷ 2 2. 344 ÷ 8 3. 284 ÷ 4 4. 133 ÷ 7 5. 182 ÷ 13 6. 650 ÷ 25 Check Skills You’ll Need 1-2

  9. Solutions 1. Estimate: 2. Estimate: 3. Estimate: 160 ÷ 2 = 80 360 ÷ 8 = 45 280 ÷ 4 = 70 82 43 71 2 164 8 344 4 284 –16 –32 –28 4 24 4 – 4 – 24 – 4 0 0 0 4. Estimate: 5. Estimate: 6. Estimate: 140 ÷ 7 = 20 195 ÷ 13 = 15 625 ÷ 25 = 25 19 14 26 7 133 13 182 25 650 –7 –13 –50 63 52 150 –63 – 52 –150 0 0 0 The Order of Operations PRE-ALGEBRA LESSON 1-2 1-2

  10. First multiply. 8 – 4 4 Then subtract. The Order of Operations PRE-ALGEBRA LESSON 1-2 Simplify 8 – 2 • 2. 8 – 2 • 2  Quick Check 1-2

  11. 4 – 2 + 1 Multiply and divide from left to right. Add and subtract from left to right. 2 + 1 3 Add. The Order of Operations PRE-ALGEBRA LESSON 1-2 Simplify 12 ÷ 3 – 1 • 2 + 1. 12 ÷ 3 – 1 • 2 + 1 Quick Check 1-2

  12. 20 – 3[ 7 – 1] Add within parentheses. Subtract within brackets. 20 – 3 [6] Multiply from left to right. 20 – 18 2 Subtract. The Order of Operations PRE-ALGEBRA LESSON 1-2 Simplify 20 – 3[(5 + 2) – 1]. 20 – 3[(5 + 2) – 1] Quick Check 1-2

  13. The Order of Operations PRE-ALGEBRA LESSON 1-2 Simplify each expression. 1. 7(3) – 2 • 4 2. 6 ÷ 2 + 1 • 5 3. 10 ÷ (4 + 1) 4. 3[9 • 2 ÷ (10 – 4)] 13 8 2 9 1-2

  14. Writing andEvaluating Expressions PRE-ALGEBRA LESSON 1-3 Evaluate each expression. a. 8 + 6 – 2 b. 16 – 8 + 2 c. 12 + 9 • 2 30 10 12 1-3

  15. Writing andEvaluating Expressions PRE-ALGEBRA LESSON 1-3 (For help, go to Lesson 1-2.) Simplify each expression. 1. 6(9 + 1) 2. 17 – 2 + 3 3. 9 + 8 • 2 + 4 4. [3(5) + 1] • 2 Check Skills You’ll Need 1-3

  16. Writing andEvaluating Expressions PRE-ALGEBRA LESSON 1-3 Solutions 1. 6(9 + 1) 2. 17 – 2 + 3 6(10) 15 + 3 60 18 3. 9 + 8 • 2 + 4 4. [3(5) + 1] • 2 9 + 16 + 4 [15 + 1] • 2 29 16 • 2 32 1-3

  17. Writing andEvaluating Expressions PRE-ALGEBRA LESSON 1-3 Evaluate 18 + 2g for g = 3. 18 + 2g= 18 + 2(3) Replace g with 3. = 18 + 6 Multiply. = 24 Add. Quick Check 1-3

  18. 2ab– = 2 • 3 • 4 – Replace the variables. c 3 c 3 9 3 Writing andEvaluating Expressions PRE-ALGEBRA LESSON 1-3 Evaluate 2ab – for a = 3, b = 4, and c = 9. = 2 • 3 • 4 – 3 Work within grouping symbols. = 6 • 4 – 3 Multiply from left to right. = 24 – 3 Multiply. = 21 Subtract. Quick Check 1-3

  19. x 36 x 36 180 36 = Evaluate for x = 180. Writing andEvaluating Expressions PRE-ALGEBRA LESSON 1-3 The Omelet Café buys cartons of 36 eggs. a. Write a variable expression for the number of cartons the Café should buy for x eggs. b. Evaluate the expression for 180 eggs. a.x eggs b. 180 eggs = 5 Divide. The Omelet Café should buy 5 cartons to get 180 eggs. Quick Check 1-3

  20. Words $16 for each pizza plus $2 delivery charge p Let = number of pizzas. Expression • p + 2 16 Writing andEvaluating Expressions PRE-ALGEBRA LESSON 1-3 The One Pizza restaurant makes only one kind of pizza, which costs $16. The delivery charge is $2. Write a variable expression for the cost of having pizzas delivered. Evaluate the expression to find the cost of having two pizzas delivered. Evaluate the expression for p = 2. 16 • p + 2 = 16 • 2 + 2 = 32 + 2 = 34 Quick Check It costs $34 to have two pizzas delivered. 1-3

  21. Writing andEvaluating Expressions PRE-ALGEBRA LESSON 1-3 Evaluate. 1. 7(b) – 4 for b = 3 2.h ÷ 2 + 1 for h = 12 3. 3c + 4 ÷ d for c = 8, d = 2 4.fg – g for f = 5, g = 7 17 7 26 28 1-3

  22. 1 8 – 1, – 0.2, , 0.5, 2 3 1 8 2 3 Integers and Absolute Value PRE-ALGEBRA LESSON 1-4 Order from least to greatest: , –1, 0.5, – 0.2, 1-4

  23. Integers and Absolute Value PRE-ALGEBRA LESSON 1-4 (For help, go to Skills Handbook p. 775.) Write an integer for each situation. 1. lose $7 2. find $9 3. 8 steps forward 4. 3 yards gained 5. 5 floors down Check Skills You’ll Need 1-4

  24. Integers and Absolute Value PRE-ALGEBRA LESSON 1-4 Solutions 1. lose $7 2. find $9 –7 9 3. 8 steps forward 4. 3 yards gained 8 3 5. 5 floors down –5 1-4

  25. Integers and Absolute Value PRE-ALGEBRA LESSON 1-4 Write a number to represent the temperature shown by the thermometer. The thermometer shows 2 degrees Celsius below zero, or –2°C. Quick Check 1-4

  26. Integers and Absolute Value PRE-ALGEBRA LESSON 1-4 Graph 2, –2, and –3 on a number line. Order the numbers from least to greatest. The numbers from least to greatest are –3, –2, and 2. Quick Check 1-4

  27. |–5| = 5 |5| = 5 Integers and Absolute Value PRE-ALGEBRA LESSON 1-4 Use a number line to find |–5| and |5|. Quick Check 1-4

  28. Integers and Absolute Value PRE-ALGEBRA LESSON 1-4 Write an integer to represent each situation. 1. A debt of $50 2. A dive of 23 feet below the surface Simplify. 3. | –12 | 4. | 8 | –50 –23 12 8 1-4

  29. 9 10 , , , , 1 3 5 6 3 5 1 6 2 3 2 7 1 4 3 5 1 5 1 3 1 4 1 5 2 7 2 3 1 6 5 6 2 7 Adding Integers PRE-ALGEBRA LESSON 1-5 List each set of fractions in order from greatest to least. a. , , b. , , c. , , 9 10 , , 1-5

  30. Adding Integers PRE-ALGEBRA LESSON 1-5 (For help, go to Lesson 1-4.) Compare. Use >, <, or = to complete each statement. 1. –6 –3 2. 2 –15 3. –5 | 5 | 4. | 10 | | –10 | 5. | 9 | | –2 | 6. | –8 | | 0 | Check Skills You’ll Need 1-5

  31. Adding Integers PRE-ALGEBRA LESSON 1-5 Solutions 1. –6 < –3 2. 2 > –15 3. –5 < | 5 | 4. | 10 | = | –10 | 5. | 9 | > | –2 | 6. | –8 | > | 0 | 1-5

  32. (–7) + 3 Model the sum. – 4 Group and remove zero pairs. There are four negative tiles left. Adding Integers PRE-ALGEBRA LESSON 1-5 Use tiles to find (–7) + 3. (–7) + 3 = – 4 Quick Check 1-5

  33. Start at 0. To represent –20, move left 20 units. To add positive 4, move right 4 units to –16. Adding Integers PRE-ALGEBRA LESSON 1-5 From the surface, a diver goes down 20 feet and then comes back up 4 feet. Find –20 + 4 to find where the diver is. –20 + 4 = –16 The diver is 16 feet below the surface. Quick Check 1-5

  34. Since both integers are negative, the sum is negative. –20 + (–15) = –35 Find the difference of the absolute values. |–17| – |13| = 17 – 13 Simplify. = 4 Since –17 has the greater absolute value, the sum is negative. 13 + (–17) = – 4 Adding Integers PRE-ALGEBRA LESSON 1-5 Find each sum. a. –20 + (–15) b. 13 + (–17) Quick Check 1-5

  35. 22 + (–30) Write an expression. |–30| – |22| = 30 – 22 Find the difference of the absolute values. Simplify. = 8 22 + (–30) = – 8 Since –30 has the greater absolute value, the sum is negative. Adding Integers PRE-ALGEBRA LESSON 1-5 A player scores 22 points. He then gets a penalty of 30 points. What is the player’s score after the penalty? The player’s score is – 8. Quick Check 1-5

  36. –7 + (– 4) + 13 + (–5)  Add from left to right. –11 + 13 + (–5) The sum of the two negative integers is negative. 2 + (–5) |13| – |11| = 2. Since 13 has the greater absolute value, the sum is positive. |5| – |2| = 3. Since –5 has the greater absolute value, the sum is negative. –3 Adding Integers PRE-ALGEBRA LESSON 1-5 Find –7 + (– 4) + 13 + (–5). –7 + (– 4) + 13 + (–5) = –3 Quick Check 1-5

  37. Adding Integers PRE-ALGEBRA LESSON 1-5 Find each sum. 1. –37 + (–5) 2. 14 + (–4) 3. –100 + 5 + (–3) 4. Evaluate 33 + t for t = –11. –42 10 –98 22 1-5

  38. Subtracting Integers PRE-ALGEBRA LESSON 1-6 Find all possible number pairs for which the sum of two whole numbers is 63 and the difference between the numbers is less than 10. 31 and 32; 30 and 33; 29 and 34; 28 and 35; 27 and 36 1-6

  39. Subtracting Integers PRE-ALGEBRA LESSON 1-6 (For help, go to Lesson 1-5.) Find each sum. 1. 8 + (–9) 2. –11 + (–18) 3. –4 + (–6) 4. 14 + (–3) 5. 6 + (–6) 6. –13 + (–10) Check Skills You’ll Need 1-6

  40. Subtracting Integers PRE-ALGEBRA LESSON 1-6 Solutions 1. 8 + (–9) 4. 14 + (–3) | –9 | – | 8 | = 9 – 8 | 14 | – | –3 | = 14 – 3 = 1 = 11 8 + (–9) = –1 14 + (–3) = 11 2. –11 + (–18) 5. 6 + (–6) –11 + (–18) = –29 | –6 | – | 6 | = 6 – 6 = 0 6 + (–6) = 0 3. –4 + (–6) 6. –13 + (–10) –4 + (–6) = –10 –13 + (–10) = –23 1-6

  41. Start with 7 negative tiles. Take away 5 negative tiles. There are 2 negative tiles left. Subtracting Integers PRE-ALGEBRA LESSON 1-6 Find –7 – (–5). –7 – (–5) = –2 Quick Check 1-6

  42. Start with 2 positive tiles. There are not enough positive tiles to take away 8. Add 6 zero pairs. Take away 8 positive tiles. There are 6 negative tiles left. Subtracting Integers PRE-ALGEBRA LESSON 1-6 Find 2 – 8. 2 – 8 = –6 Quick Check 1-6

  43. Write an expression. 42 – 50 To subtract 50, add its opposite. 42 – 50 = 42 + (–50) = –8 Simplify. Subtracting Integers PRE-ALGEBRA LESSON 1-6 An airplane left Houston, Texas, where the temperature was 42°F. When the airplane landed in Anchorage, Alaska, the temperature was 50°F lower. What was the temperature in Anchorage? The temperature in Anchorage was –8°F. Quick Check 1-6

  44. Subtracting Integers PRE-ALGEBRA LESSON 1-6 Find each difference. 1. –24 – (–5) 2. 19 – (–4) 3. –33 – 11 4. 14 – 46 5. –200 – 50 – (–10) –19 23 –44 –32 –240 1-6

  45. 1 8 1 8 Inductive Reasoning PRE-ALGEBRA LESSON 1-7 3 12 Find the difference between and . 3 24 = 1-7

  46. Inductive Reasoning PRE-ALGEBRA LESSON 1-7 (For help, go to Skills Handbook.) Find each difference. 1. –3 – 4 2. –7 – 4 3. –11 – 4 4. –15 – 4 Check Skills You’ll Need 1-7

  47. Inductive Reasoning PRE-ALGEBRA LESSON 1-7 Solutions 1. –3 – 4 2. –7 – 4 –3 – 4 = –3 + (–4) –7 – 4 = –7 + (–4) = –7 = –11 3. –11 – 4 4. –15 – 4 –11 – 4 = –11 + (–4) –15 – 4 = –15 + (–4) = –15 = –19 1-7

  48. Inductive Reasoning PRE-ALGEBRA LESSON 1-7 Use inductive reasoning. Make a conjecture about the next figure in the pattern. Then draw the figure. Observation: The circles are rotating counterclockwise within the square. Conjecture: The next figure will have a shaded circle at the top right. Quick Check 1-7

  49. Inductive Reasoning PRE-ALGEBRA LESSON 1-7 Write a rule for each number pattern. a. 0, – 4, – 8, –12, . . . Start with 0 and subtract 4 repeatedly. Alternate 4 and its opposite. b. 4, – 4, 4, – 4, . . .   c. 1, 2, 4, 8, 10, . . .   Start with 1. Alternate multiplying by 2 and adding 2. Quick Check 1-7

  50. The first number is 110. 110, 100, 90, 80 The next numbers are found by subtracting 10.   – 10 – 10 – 10 Inductive Reasoning PRE-ALGEBRA LESSON 1-7 Write a rule for the number pattern 110, 100, 90, 80, . . . Find the next two numbers in the pattern. The rule is Start with 110 and subtract 10 repeatedly. The next two numbers in the pattern are 80 – 10 = 70 and 70 – 10 = 60. Quick Check 1-7

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