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Understanding Direct Variation in Mathematics

Learn to recognize direct variation by graphing tables of data and checking for constant ratios. Practice determining if data sets exhibit direct variation and find equations of direct variation. Discover how to identify direct variation relationships through practical examples and exercises. Adapt Math Unravel Strategies to solve direct variation problems effectively. Progress report due tomorrow.

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Understanding Direct Variation in Mathematics

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  1. February 13, 2008 Math 7/8 and Math 7 Warm-Up Direct Variation Class/HW: Practice 12-5 Practice 12E Progress Report signed and returned tomorrow Warm-Up One piece of material is nine inches plus half its length. How long would the piece of material and half of itself be? Use Math Unravel Strategies

  2. Direct Variation 12-5 Course 3 Learn to recognize direct variation by graphing tables of data and checking for constant ratios.

  3. Direct Variation 12-5 Course 3 Insert Lesson Title Here Vocabulary direct variation constant of proportionality

  4. Direct Variation 12-5 Course 3 IMPORTANT INFORMATION

  5. Direct Variation 12-5 Helpful Hint The graph of a direct-variation equation is always linear and always contains the point (0, 0). The variables x and y either increase together or decrease together. Course 3

  6. Direct Variation 12-5 Course 3 Additional Example 1A: Determining Whether a Data Set Varies Directly Determine whether the data set shows direct variation.

  7. Direct Variation 12-5 Course 3 Additional Example 1A Continued Make a graph that shows the relationship between Adam’s age and his length. The graph is not linear.

  8. Direct Variation 12-5 27 22 12 3 ? = Course 3 Additional Example 1A Continued You can also compare ratios to see if a direct variation occurs. 81 81 ≠ 264 The ratios are not proportional. 264 The relationship of the data is not a direct variation.

  9. Direct Variation 12-5 Course 3 Additional Example 1B: Determining Whether a Data Set Varies Directly Determine whether the data set shows direct variation.

  10. Direct Variation 12-5 Course 3 Additional Example 1B Continued Make a graph that shows the relationship between the number of minutes and the distance the train travels. Plot the points. The points lie in a straight line. (0, 0) is included.

  11. Direct Variation 12-5 25 10 75 100 50 30 40 20 Course 3 Additional Example 1B Continued You can also compare ratios to see if a direct variation occurs. Compare ratios. = = = The ratios are proportional. The relationship is a direct variation.

  12. Direct Variation 12-5 Course 3 Check It Out: Example 1A Determine whether the data set shows direct variation.

  13. Direct Variation 12-5 Course 3 Check It Out: Example 1A Continued Make a graph that shows the relationship between number of baskets and distance. The graph is not linear. 5 4 3 Number of Baskets 2 1 20 30 40 Distance (ft)

  14. Direct Variation 12-5 3 5 30 20 ? = Course 3 Check It Out: Example 1A Continued You can also compare ratios to see if a direct variation occurs. 60 150  60. The ratios are not proportional. 150 The relationship of the data is not a direct variation.

  15. Direct Variation 12-5 Course 3 Check It Out: Example 1B Determine whether the data set shows direct variation.

  16. Direct Variation 12-5 4 3 Number of Cups 2 1 32 8 16 24 Number of Ounces Course 3 Check It Out: Example 1B Continued Make a graph that shows the relationship between ounces and cups. Plot the points. The points lie in a straight line. (0, 0) is included.

  17. Direct Variation 12-5 1 = = = 8 3 4 2 24 32 16 Course 3 Check It Out: Example 1B Continued You can also compare ratios to see if a direct variation occurs. Compare ratios. The ratios are proportional. The relationship is a direct variation.

  18. Direct Variation 12-5 Course 3 Additional Example 2A: Finding Equations of Direct Variation Find each equation of direct variation, given that y varies directly with x. y is 54 when x is 6 y = kx y varies directly with x. 54 = k6 Substitute for x and y. 9 = k Solve for k. Substitute 9 for k in the original equation. y = 9x

  19. Direct Variation 12-5 = k y = x Substitute for k in the original equation. 5 5 5 4 4 4 Course 3 Additional Example 2B: Finding Equations of Direct Variation x is 12 when y is 15 y = kx y varies directly with x. 15 = k12 Substitute for x and y. Solve for k.

  20. Direct Variation 12-5 Course 3 Check It Out: Example 2A Find each equation of direct variation, given that y varies directly with x. y is 24 when x is 4 y = kx y varies directly with x. 24 = k4 Substitute for x and y. 6 = k Solve for k. Substitute 6 for k in the original equation. y = 6x

  21. Direct Variation 12-5 = k y = x Substitute for k in the original equation. 1 1 1 2 2 2 Course 3 Check It Out: Example 2B x is 28 when y is 14 y = kx y varies directly with x. 14 = k28 Substitute for x and y. Solve for k.

  22. Direct Variation 12-5 y = x y = x 1 6 9 5 Course 3 Insert Lesson Title Here Lesson Quiz: Part I Find each equation of direct variation, given that y varies directly with x. 1.y is 78 when x is 3. 2.x is 45 when y is 5. 3.y is 6 when x is 5. y = 26x

  23. Direct Variation 12-5 Course 3 Insert Lesson Title Here Lesson Quiz: Part II 4. The table shows the amount of money Bob makes for different amounts of time he works. Determine whether there is a direct variation between the two sets of data. If so, find the equation of direct variation. direct variation; y = 12x

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