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Understanding Proportions: A Comprehensive Guide for Middle School Students

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charles-crawford
28 diapositives · 420 Views · Nov 08, 2014

This PowerPoint presentation by Mr. Michael Braverman from Haverford Middle School covers the concept of proportions in mathematics. It explains definitions, solving techniques, and several examples, including cross-multiplication and simplification. Students will learn how to set up and solve proportions through practical exercises and problem-solving strategies. The presentation aims to provide a clear understanding of how to manipulate ratios and applies them to real-world contexts. Ideal for middle school math students seeking to strengthen their grasp of proportions.

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  1. Proportions PowerPoint Presentation By Mr. Michael Braverman Haverford Middle School School District of Haverford Township Havertown, PA 19083 Click mouse or press space bar to continue

  2. Proportions • Definition • Solving proportions • Setting up proportions • Extra practice problems

  3. Proportions Definition: • Two (or more) equivalent ratios make a proportion. • If a true proportion exists, we say that the variables are “in proportion.” a c b d

  4. Proportions Solving proportions: Example: a c d b a c b d In a proportion, the cross-products are equal. = ad bc

  5. Proportions Solving proportions: Example: 15 9 6 10 9 15 6 10 In a proportion, the cross-products are equal. 9*10 = 6*15 …therefore the original proportion is true 90 =90

  6. Proportions To solve a proportion: Example: x 15 6 10 x 15 6 10 1. Cross-multiply x*10 = 6 *15 2. Divide both sides by the co-efficient of the variable The variable is the “unknown quantity”in a problem – usually represented by a letter. In this case, “x” is the variable.

  7. Proportions To solve a proportion: Example: x 15 6 10 x 15 6 10 1. Cross-multiply x*10 = 6*15 2. Divide both sides by the co-efficient of the variable The coefficientis the number that is being multiplied by the variable. In this case, the coefficientis 10

  8. Proportions To solve a proportion: Example: x 15 6 10 x 15 6 10 1. Cross-multiply x*10 = 6*15 2. Divide both sides by the co-efficient of the variable 10 10 3. Simplify

  9. Proportions 3. Simplify Cancel the co-efficient. (You will ALWAYS be able to do this!) x*10 x*10 x*10 = = = 6*15 6*15 6*15 10 10 10 10 10 10

  10. Proportions 3. Simplify Cancel the co-efficient. (You will ALWAYS be able to do this!) x*10 x*10 x = = = 6*15 6*15 6*15 10 10 10 10 1 10

  11. Proportions 3. Simplify Cancel the co-efficient. x*10 x*10 = = 6*15 6*15 x = 9 90 6*15 = = 10 10 10 10 1 10 10 1

  12. Proportions 3. Simplify x x = = 90 9 9 90 6*15 6*15 = = = = 1 10 1 10 1 10 10 1

  13. Proportions 3. Simplify x 15 6 10 x = 9 x = 90 9 6*15 …so this makes the proportion = = 1 10 10 1

  14. Proportions 3. Simplify x 15 6 10 9 15 6 10 x = 9 x = 90 9 6*15 …so this makes the proportion = = 1 10 10 1

  15. Proportions Example 2 9 6 9 * 16 = 6 x x 16 3. Simplify both sides 1. Cross-multiply 2. Divide by the co-efficient (6) x 144 = x 24 = 6 6 6

  16. Proportions H To set up a proportion, you can use the following table to help you organize your variables and numbers Quantities: “Things” you are counting or measuring Need Have Quantity 1 Quantity 2

  17. Proportions H To set up a proportion, you can use the following table to help you organize your variables and numbers In the “Have” column, write the set of NUMBERS where you have BOTHquantities Need Have Quantity 1 Quantity 2

  18. Proportions H To set up a proportion, you can use the following table to help you organize your variables and numbers In the “Need” column, write the set of NUMBERS where you have Only one number AND a variable Need Have Quantity 1 Quantity 2

  19. Proportions H Example: One rectangle has dimensions of 3 and 8. A similar rectangle has a long side of 18. How long is the short side? 3 8 We “Have” the long side and short side of the small rectangle. Need Have long side Quantity 1 short side Quantity 2

  20. Proportions H Example: One rectangle has dimensions of 3 and 8. A similar rectangle has a long side of 18. How long is the short side? 3 8 We “Have” the long side and short side of the small rectangle. Need Have long side short side Quantity 1 Quantity 2

  21. Proportions H Example: One rectangle has dimensions of 3 and 8. A similar rectangle has a long side of 18. How long is the short side? We “Have” the long side of the big rectangle (18), but NEED the short side of the big rectangle (let’s call this s). Need Have 18 3 short side 8 long side s

  22. Proportions H Example: One rectangle has dimensions of 3 and 8. A similar rectangle has a long side of 18. How long is the short side? We “Have” the long side of the big rectangle (18), but NEED the short side of the big rectangle (let’s call this s). Need Have 3 s short side 8 long side 18

  23. Proportions H Need Need Have Have 3 3 s s short side short side 8 8 long side long side 18 18 …which you can now solve like the ones we solved earlier. This will set up your equation as a correct proportion. 3 s 8 18

  24. Proportions H Do you remember the steps? 3 x 18 = 8 x s 1. Cross-multiply 2. Divide both sides by the co-efficient of the variable 3 3 x 18 = 8 x s s 8 18 8 8

  25. Proportions H 2. Divide both sides by the co-efficient of the variable 3 x 18 = 8 x s 8 x 54 = s 3 x 18 = s 3. Simplify 8 8 54 = s 8 8 8

  26. Proportions H 54 = s = 54 s 8 8 = 27 s 4 = 3 6 s 4

  27. Proportions H Need Have 3 s short side 8 long side 18 Example: One rectangle has dimensions of 3 and 8. A similar rectangle has a long side of 18. How long is the short side? 3 3 = 3 s 6 6 s 8 4 4 18

  28. Proportions Extra Practice Problems: http://www.education.com/study-help/article/proportion-word-problems_answer/ http://www.ixl.com/math/grade-7/solve-proportions-word-problems http://www.homeschoolmath.net/worksheets/proportions.php