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Understanding Properties of Rational Numbers

Learn to utilize properties of rational numbers to create equivalent expressions and equations. Practice using the Associative, Distributive, and Identity properties to simplify calculations and solve real-world problems.

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Understanding Properties of Rational Numbers

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  1. Warm Up Problem of the Day Lesson Presentation Lesson Quizzes

  2. Warm Up Identify the property represented. 1.4 + (9 + 3) = (4 + 9) + 3 2.10(5 - 6) = 10 . 5 - 10 . 6 3. 17 . 1 = 17 Associative Property Distributive Property Identity Property

  3. Problem of the Day Fran made 18 three-point shots and 12 one-point shots. She had a total of 102 points for the basketball season. How many two-point shots did Fran make for the season? 18 two-point shots

  4. Learn to use properties of rational numbers to write equivalent expressions and equations.

  5. Orlando works part-time at a moving company to earn money for a car. He earns $12.75 per hour. You can use the Distributive Property to calculate Orlando’s total earnings two different ways. The Distributive Property can be used to help you perform calculations more easily by writing equivalent expressions.

  6. Remember The Distributive Property states: a(b + c) = ab + ac a(b - c) = ab - ac

  7. Additional Example 1: Writing Equivalent Expressions An art teacher pays $13.89 for one box of watercolor brushes. She buys 6 boxes in March and 5 boxes in April. Use the Distributive Property to write equivalent expressions showing two ways to calculate the total cost of the watercolor boxes. Write an expression to show how much the teacher pays for a box and how many boxes purchased. Then use the Distributive Property to write an equivalent expression

  8. Method 1 $13.89(6 + 5) $13.89(11) $152.79 Method 2 $13.89(6) + $13.89(5) $83.34 + $69.45 $152.79 Both methods result in a calculation of $152.79 for the amount of money spent of watercolor brushes.

  9. Check It Out : Example 1 Jamie earns $8.75 per hour. Last week she worked 15 hours and next week she will work 20 hours. Use the Distributive Property to write equivalent expressions showing two ways to calculate how much money she earned. Write an expression to show how much Jamie earns and the number of hours she works. Then use the Distributive Property to write an equivalent expression.

  10. Continued: Check It Out Example 1 Method 1 $8.75(15 + 20) $8.75(35) $306.25 Method 2 $8.75(15) + $8.75(20) $131.25 + $175 $306.25 Both methods result in a calculation of $306.25 for Jamie’s salary.

  11. 3 4 5 6 X + 7 = 3 4 5 6 x + 7 = 3 4 5 6 12 x+ 7 = 12 3 1 4 5 1 6 12 2 3 12 x + 12 (7) = Additional Example 2 : Writing Equivalent Expressions Write an equivalent equation for that does not contain fractions. Then solve the equation. The LCM of denominators is 12. Multiply both sides by 12. Simplify.

  12. -84 -84 9 9 2 9 x = -8 An equivalent equation is 9x + 84 = 10 and the solution is 2 9 x = -8 Additional Example 2 : Continued 9x + 84 = 10 is an equivalent expression 9x + 84 = 10 9x + 84 = 10 Subtract 84 from both sides. 9x = -74 Divide both side by 9

  13. 1 2 4 6 X + 9 = 1 2 4 6 X + 9 = 1 2 4 6 6 x+ 9 = 6 1 1 2 4 1 6 6 1 3 6 x + 6 (9) = Check It Out: Example 2 Write an equivalent equation for that does not contain fractions. Then solve the equation. The LCM of denominators is 6 Multiply both sides by 6. Simplify.

  14. 2 3 x = -16 An equivalent equation is 3x + 34 = 4 and the solution is 2 3 x = -16 Continued: Check It Out Example 2 3x + 54 = 4 3x + 54 = 4 is an equivalent expression 3x + 54 = 4 -54 -54 Subtract 54 from both sides. 3x = -50 Divide both side by 3 3 3

  15. Helpful Hint 75 100 0.75 can also be written as

  16. Additional Example 3: Construction Application The soccer team uses a 36.75-liter container to take water to games. The team manager fills 0.75 liter bottles from this. He has used 22.5 liters. How many more 0.75 liter bottles can he fill before he runs out of water? Write and solve an equivalent equation without decimals. Write an equation to represent the situation. 0.75x + 22.5 = 36.75

  17. Continued: Example 3 Write an equivalent equation without decimals. The equation has decimals to the hundredths, so multiply both sides by 100. 100(0.75x + 22.5) = (36.75)100 Use the Distributive Property 100(0.75x + 100(22.5) = (36.75)100 Simplify to get an equivalent equation without decimals 75x + 2,250 = 3,675

  18. Continued: Example 3 75x + 2,250 = 3,675 -2250 -2250 75x = 1,425 75 75 x = 19 The number of 0.75 liter bottles that he can fill before he runs out of water is 19.

  19. Check It Out: Example 3 …If the soccer team uses a 42.5-liter container, about how many 0.75 liter bottles can the manager fill before he runs out of water? Write an equation to represent the situation. 0.75x + 22.5 = 42.5 Write an equivalent equation without decimals. The equation has decimals to the hundredths, so multiply both sides by 100. 100(0.75x + 22.5) = (42.5)100

  20. Continued: Check It Out Example 3 Use the Distributive Property 100(0.75x + 100(22.5) = (42.5)100 Simplify to get an equivalent equation without decimals 75x + 2,250 = 4,250 75x + 2,250 = 4,250 -2250 -2250 75x = 2000 75 75

  21. Continued: Check It Out Example 3 x ≈ 26.6 The number of 0.75 liter bottles that he can fill before he runs out of water is 19.

  22. Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems

  23. 4 5 3 8 1 2 2. x + 4 = 8x + 40 = 5; x = -4 Lesson Quiz 1. Jai earns $9.75 per hour. Jai works 3 hours one day and then works 7 hours the next day. Use the Distributive Property to write equivalent expressions showing two ways to calculate Jai’s total earnings. 9.75(3) + 9.75(7); 9.75(3 + 7); $97.50 Write an equivalent equation that does not contain fractions. Then solve the equation.

  24. 2 3 3 8 1 4 3. x - 4 = 8x - 48 = 3; x = 6 Lesson Quiz 4. Joy has $67.85. She buys several pairs of earrings at $9.98 per pair and has $17.95 left. How many pairs of earrings did she buy? Write and solve an equivalent equation without decimals. 9.98x + 17.95 = 67.85; 998x + 1795 = 6785; x = 5; Joy bought 5 pairs of earrings.

  25. 1 3 4 6 x + 6 = Lesson Quiz for Student Response Systems 1. Write an equivalent equation that does not contain fractions. A. 2x + 6 = 4 B. 2x + 36 = 4 C. 3x + 6 = 4 D. 3x + 36 = 4

  26. 7 8 1 4 x + 2 = Lesson Quiz for Student Response Systems 2. Write an equivalent equation that does not contain fractions. A. 8 = 2x + 16 B. 8 = 2x + 2 C. 7 = 2x + 2 D. 7 = 2x + 16

  27. 6 8 1 4 x + 2 = Lesson Quiz for Student Response Systems 3. Solve the equation A. x = -4 B. x = -5 C. x = 3 D. x = 5

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