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Warm Up

This lesson introduces students to proportional and equivalent ratios through various examples and practice problems. Students will learn to identify and write proportions, as well as find equivalent ratios by multiplying or dividing. The lesson also includes quizzes to assess understanding.

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Warm Up

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

  2. Warm Up Find the unit rate. 1. 18 miles in 3 hours 2. 6 apples for $3.30 3. 3 cans for $0.87 4. 5 CD’s for $43 6 mi/h $0.55 per apple $0.29 per can $8.60 per CD

  3. Problem of the Day Bob made a square table top with 100 white square tiles. He painted the tiles along the edge of the table red. How many tiles are red? 36

  4. Learn to find equivalent ratios and to identify proportions.

  5. Vocabulary equivalent ratios proportion

  6. Students in Mr. Howell’s math class are measuring the width w and the length lof their faces. The ratio of lto w is 6 inches to 4 inches for Jean and 21 centimeters to 14 centimeters for Pat.

  7. These ratios can be written as the fractions and . Since both simplify to , they are equivalent. Equivalent ratiosare ratios that name the same comparison. 6 4 21 14 3 2

  8. 21 14 6 4 = An equation stating that two ratios are equivalent is called a proportion. The equation, or proportion, below states that the ratios and are equivalent. 6 4 21 14 21 14 6 4 = Reading Math Read the proportion by saying “six is to four as twenty-one is to fourteen.”

  9. If two ratios are equivalent, they are said to be proportional to each other, or in proportion.

  10. 24 51 8 17 24 ÷ 3 51 ÷ 3 Simplify . = 72 ÷ 8 9 16 72 128 = Simplify . 128 ÷ 8 9 16 8 17 Since = , the ratios are not proportional. Additional Example 1A: Comparing Ratios in Simplest Forms Determine whether the ratios are proportional. 24 51 72 128 ,

  11. 150 105 10 7 150 ÷ 15 105 ÷ 15 Simplify . = 90 ÷ 9 10 7 90 63 = Simplify . 63 ÷ 9 10 7 10 7 Since = , the ratios are proportional. Additional Example 1B: Comparing Ratios in Simplest Forms Determine whether the ratios are proportional. 150 105 90 63 ,

  12. 54 63 6 7 54 ÷ 9 63 ÷ 9 Simplify . = 72 ÷ 72 1 2 72 144 = Simplify . 144 ÷ 72 1 2 6 7 Since = , the ratios are not proportional. Check It Out: Example 1A Determine whether the ratios are proportional. 54 63 72 144 ,

  13. 9 5 135 ÷ 15 75 ÷ 15 135 75 Simplify . = 9 4 9 4 is already in simplest form. 9 4 9 5 Since = , the ratios are not proportional. Check It Out: Example 1B Determine whether the ratios are proportional. 135 75 9 4 ,

  14. Servings of Rice Cups of Rice Cups of Water 12 3 6 40 10 19 Since = , the two ratios are not proportional. Additional Example 2: Comparing Ratios Using a Common Denominator Directions for making 12 servings of rice call for 3 cups of rice and 6 cups of water. For 40 servings, the directions call for 10 cups of rice and 19 cups of water. Determine whether the ratios of rice to water are proportional for both servings of rice. Write the ratios of rice to water for 12 servings and for 40 servings. 3 6 Ratio of rice to water, 12 servings: Write the ratio as a fraction. 10 19 Ratio of rice to water, 40 servings: Write the ratio as a fraction Write the ratios with a common denominator, such as 114. 3 6 10 19 60 114 3 · 19 6 · 19 57 114 10 · 6 19 · 6 = = = = 57 114 60 114

  15. Servings of Beans Cups of Beans Cups of Water 8 4 3 35 13 9 Since = , the two ratios are not proportional. Check It Out: Example 2 Use the data in the table to determine whether the ratios of beans to water are proportional for both servings of beans. Write the ratios of beans to water for 8 servings and for 35 servings. 4 3 Ratio of beans to water, 8 servings: Write the ratio as a fraction. 13 9 Ratio of beans to water, 35 servings: Write the ratio as a fraction Write the ratios with a common denominator, such as 27. 4 3 4 · 9 3 · 9 36 27 13 9 39 27 13 · 3 9 · 3 = = = = 39 27 36 27

  16. You can find an equivalent ratio by multiplying or dividing both terms of a ratio by the same number.

  17. Additional Example 3: Finding Equivalent Ratios and Writing Proportions Find a ratio equivalent to each ratio. Then use the ratios to find a proportion. Possible Answers: 3 5 A. 3 5 3 · 2 5 · 2 6 10 Multiply both the numerator and denominator by any number such as 2. = = 6 10 3 5 = Write a proportion. 28 16 B. Divide both the numerator and denominator by any number such as 4. 28 16 28 ÷ 4 16 ÷ 4 7 4 = = 7 4 28 16 = Write a proportion.

  18. Check It Out: Example 3 Find a ratio equivalent to each ratio. Then use the ratios to find a proportion. Possible Answers: 2 3 A. 2 3 2 · 3 3 · 3 6 9 Multiply both the numerator and denominator by any number such as 3. = = 6 9 2 3 = Write a proportion. 16 12 B. Divide both the numerator and denominator by any number such as 4. 16 ÷ 4 12 ÷ 4 4 3 16 12 = = 16 12 4 3 = Write a proportion.

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

  20. 3 10 4 7 3 10 2 3 , ; proportional , ; not proportional 3 8 3 7 15 40 9 21 15 40 9 21 = = ; ; Lesson Quiz: Part I Determine whether the ratios are proportional. 9 30 12 40 12 21 10 15 1. 2. , , Find a ratio equivalent to each ratio. Then use the ratios to write a proportion. 3 8 3 7 4. 3.

  21. 5 1 = Lesson Quiz: Part II 5. In preschool, there are 5 children for every one teacher. In another preschool there are 20 children for every 4 teachers. Determine whether the ratios of children to teachers are proportional in both preschools. 20 4

  22. Lesson Quiz for Student Response Systems 1. Determine whether the ratios are proportional. , A. proportional B. not proportional 12 20 18 30

  23. Lesson Quiz for Student Response Systems 2. Determine whether the ratios are proportional. , A. proportional B. not proportional 21 56 14 49

  24. Lesson Quiz for Student Response Systems 2 9 3. Identify a ratio equivalent to . Then use the ratios to write a proportion. A. ; = C. ; = B. ; = D. ; = 10 35 5 35 5 35 10 35 2 9 2 9 10 45 5 45 10 45 2 9 5 45 2 9

  25. Lesson Quiz for Student Response Systems 4 5 4. Identify a ratio equivalent to . Then use the ratios to write a proportion. A. ; = C. ; = B. ; = D. ; = 12 15 12 20 12 20 12 15 4 5 4 5 12 18 15 20 12 18 4 5 15 20 4 5

  26. Lesson Quiz for Student Response Systems 5. In one school, there are 12 students for every classroom. In another school, there are 36 students for every 4 classrooms. Determine whether the ratios of students to classrooms are proportional in both schools. A. = ; C. ≠ ; proportional not proportional B. = ; D. ≠ ; proportional not proportional 1 3 36 4 12 36 12 1 36 3 36 1 12 1 12 4

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