1. Proportions, Measurement Conversions, Scale, and Percents by Lauren McCluskey

2. Credits “Prentice Hall Mathematics: Algebra I” “Prentice Hall Mathematics: Algebra I” “Changing Percents” by D. Fisher “Changing Percents” by D. Fisher “Percent I” by Monica and Bob Yuskaitis “Percent I” by Monica and Bob Yuskaitis “Percent II” by Monica and Bob Yuskaitis “Percent II” by Monica and Bob Yuskaitis “Percent Formula Word Problems” by Rush Strong “Percent Formula Word Problems” by Rush Strong “Math Flash Measurement I” by Monica and Bob Yuskaitis “Math Flash Measurement I” by Monica and Bob Yuskaitis

3. Ratios, Rates, and Proportions: “A ratio is a comparison of two numbers by division.” “A ratio is a comparison of two numbers by division.” A rate is a ratio which compares two different units, such as 20 pages /per 10 minutes. A rate is a ratio which compares two different units, such as 20 pages /per 10 minutes. “A unit rate is a rate with a denominator of 1.” An example of this is miles / per hour. “A unit rate is a rate with a denominator of 1.” An example of this is miles / per hour. from Prentice Hall Algebra I from Prentice Hall Algebra I

4. Try It! Find the unit rate: Find the unit rate: 1) $57 / 6 hr. 2) $2 / 5 lb. 5) A 10-ounce bottle of shampoo costs $2.40. What is the cost per ounce? from Prentice Hall, Algebra I from Prentice Hall, Algebra I

5. Proportions: “A proportion is an equation that states that two ratios are equal.” “A proportion is an equation that states that two ratios are equal.” “The products ad and bc are the cross products of the proportion a/b = c/d.” “The products ad and bc are the cross products of the proportion a/b = c/d.” Example: 3/12 = x / 24 from Prentice Hall Algebra I

6. Multi-step Proportions: X+ 3 7 4 8 4 8 Use cross products: 4 * 7 = 8(x + 3) 28 = 8x + 24 28 = 8x + 24 -24 -24 4= 8x 4= 8x 8 8 8 8 x = / x = 1 / 2 = from Prentice Hall Algebra I a – 6 5 = 7 12 Now you try it!

7. a – 6 5 = 7 12 7(a – 6) = 5 * 7 7a – 42 = 35 +42 +42 7a = 77 7 a= 11

8. Proportions can be used when: Solving Unit Rate problems Solving Unit Rate problems Converting Measurements Converting Measurements Indirect Measurements via Similar Figures Indirect Measurements via Similar Figures Converting between Scale and the actual object/ distance Converting between Scale and the actual object/ distance Solving Percent problems Solving Percent problems

9. Try It! (Unit Rates:) 30) A canary’s heart beats 200 times in 12 seconds. How many times does it beat in 1 hour? from Prentice Hall Algebra I from Prentice Hall Algebra I

10. Proportions: 31) “Suppose you traveled 66 km in 1.25 hours. Moving at the same speed, how many km would you cover in 2 hours?” from Prentice Hall Algebra I

11. Measurement Conversions: 52) “The peregrine falcon has a record diving speed of 168 miles per hour. Write this speed in feet per second.” from Prentice Hall Algebra I

12. How large is a millimeter? The width of a pin from “Math Flash Measurement I” by M. and B. Yuskaitis

13. How large is a centimeter? The width of the top of your finger from “Math Flash Measurement I” by M. and B. Yuskaitis

14. How large is a meter? About the width of one & 1/2 doors 1 meter from “Math Flash Measurement I” by M. and B. Yuskaitis

15. How large is a kilometer? A little over 1/2 of a mile 1 kilometer Whitmore Walter White from “Math Flash Measurement I” by M. and B. Yuskaitis

16. How large is a milliliter? About a drop of liquid from “Math Flash Measurement I” by M. and B. Yuskaitis

17. How large is a liter? Half of a large pop bottle 1 liter from “Math Flash Measurement I” by M. and B. Yuskaitis

18. How heavy is a gram? A paper clip weighs about 1 gram from “Math Flash Measurement I” by M. and B. Yuskaitis

19. How heavy is a kilogram? A kitten weighs about 1 kilogram from “Math Flash Measurement I” by M. and B. Yuskaitis

20. Measurement Conversions: 12m = _________km 12m = _________km 12m = _________mm 12m = _________mm 12m = _________cm 12m = _________cm 48 in. = _________ft. 48 in. = _________ft. 48 in. = _________ yd. 48 in. = _________ yd. 48 in. = _________ mile 48 in. = _________ mile

21. Similar Figures: “ Similar figures have the same shape but not necessarily the same size. “ Similar figures have the same shape but not necessarily the same size. In similar triangles, corresponding angels are congruent and corresponding sides are in proportion.” In similar triangles, corresponding angels are congruent and corresponding sides are in proportion.” from Prentice Hall Algebra I

22. What is the missing measure? x 15cm 20cm4cm 3cm 5cm 8m 4m 12m X

23. Scale: “A scale drawing is an enlarged or reduced drawing that is similar to an actual object or place. “A scale drawing is an enlarged or reduced drawing that is similar to an actual object or place. The ratio of a distance in the drawing to the corresponding actual distance is the scale of the drawing.” The ratio of a distance in the drawing to the corresponding actual distance is the scale of the drawing.” from Prentice Hall Algebra I

24. Scale 22) A blueprint scale is 1 in. : 9 ft. On the plan, the room measures 2.5 in. by 3 in. What are the actual dimensions of the room? from Prentice Hall Algebra I from Prentice Hall Algebra I

25. Proportions and Percent Equations: is % OR: part is % OR: part of 100 whole of 100 whole n 20 80 w 80 20 n 20 80 = w 80 20 100 25 100 25 100 z 100 25 100 25 100 z = = = Find the percent Find the partFind the whole

26. Understanding Percents: Percent can be defined as “of one hundred.” 100 of from “Percent I” by M. and B.Yuskaitis

27. “Cent” comes from the Latin and means 100. Many words have come from the root cent such as century, centimeter, centipede, & cent. from “Percent I” by M. and B.Yuskaitis

28. The letter C in Roman Numerals stands for 100 or “cent”. The letter C in Roman Numerals stands for 100 or “cent”. CCC means three hundred. from “Percent I” by M. and B.Yuskaitis

29. Percent and Money We write our change from a dollar in hundredths. We write our change from a dollar in hundredths. If you understand money, learning percents is a breeze. from “Percent I” by M. and B.Yuskaitis

30. Comparing Money & Percents $.25 is ¼ of a dollar $.25 is ¼ of a dollar 25% also means ¼ 25% also means ¼ = $1.00 25¢ from “Percent I” by M. and B.Yuskaitis

31. How to Find the Percent of a Whole Number The first thing to remember is “of” means multiply in mathematics. of = x from “Percent I” by M. and B.Yuskaitis

32. How to Find the Percent of a Whole Number Step 1 - When you see a percent problem you know when you read “of” in the problem you multiply. Step 1 - When you see a percent problem you know when you read “of” in the problem you multiply. 25% of 200 x from “Percent II” by M. and B.Yuskaitis

33. Step 2 – Change your percent to a decimal and then move it two places to the left. 25% x 200.. from “Percent I” by M. and B.Yuskaitis

34. Step 3 – Multiply just like a regular decimal multiplication problem. 200 x. 25 1000 +400 5000 from “Percent II” by M. and B.Yuskaitis

35. Step 4 – Place the decimal point 2 places to the left in your answer. 200 x. 25 1000 +400 5000. from “Percent II” by M. and B.Yuskaitis

36. Percent Problems: There are 3 types of percent problems: 1) What is ____% of ____? 2) What % of ____ is ____? 3) _____ is ___% of what #?

37. Problem 1 Brittany Berrier became a famous skater. She won 85% of her meets. If she had 250 meets in 2000, how many did she win? from ”Percent Formula Word Problems” by R. Strong from ”Percent Formula Word Problems” by R. Strong

38. What is 85% of 250? 250 250 ● 0.85 1250 1250 20000 20000 21250 21250 So 212.50 is 85% of 250.

39. Problem 2 Matt Debord worked as a produce manager for Walmart. If 35 people bought green peppers and this was 28% of the total customers, how many customers did he have? from ”Percent Formula Word Problems” by R. Strong from ”Percent Formula Word Problems” by R. Strong

40. 35 = 28 So what is x? x 100 35 = 28 So what is x? x 100 Use cross products: Use cross products: 3500 = 28x 28 28 28 28 X = 125 customers

41. Problem 3 Brett Mull became a famous D.J. He played a total of 175 C.D’s in January. If he played 35 classical C.D.’s, what percent of CD’s were Classical? adapted from ”Percent Formula Word Problems” by R. Strong adapted from ”Percent Formula Word Problems” by R. Strong

42. 35 = x 175 100 Use cross products: 3500 = 175x 175 175 175 175 X = 20 % (or 1/5 of the CD’s played were classical)

43. Changing Fractions to Equivalent Percents: There are 3 ways to change a fraction to an equivalent percent:

44. 1) Divide the denominator into the numerator, then change the decimal to a percent. 6÷10 = 0.60 * 100= 0.60 60%

45. 2) Find an equivalent fraction with 100 as the denominator. 6 10 = ? 100 60

46. 3) Draw an illustration using a 100 grid. using a 100 grid. 1/10 6/10= 60%

47. Try it! 1) What is 20% as a fraction in simplest form? 2) What is 0.6 as a percent? 3) What is 3/5 as a decimal?

48. What is 20% as a fraction in simplest form? 20/100? 20/100 = ? 1/51/5 1/51/5 1/51/5 1/51/5 1/51/5 20 100 20 ÷ 20 100 ÷ 20 == 1 5 adapted from a slide by D. Fisher

49. Rewrite 0.6 as a percent. 6/10= ?% = 0.6 6÷ 10 adapted from a slide by D. Fisher

50. Remember: Multiply by 100 when changing a decimal to a percent because percent means “out of 100”. 0.6= 60% 0.6 * 100= 60 So…

51. What is 3/5 as a decimal? 1) Divide then multiply… 3 ÷ 5 = 0.6 3 ÷ 5 = 0.6 0.6 * 100 = 60 0.6 * 100 = 60 So 3/5 = 60% So 3/5 = 60% OR OR 2) Find an equivalent fraction 3/5 = ? /100 5 * 20 = 100 5 * 20 = 100 Do the same to the numerator: Do the same to the numerator: 3 * 20 = 60 3 * 20 = 60 So 3/5 = 60/100 which equals 60% So 3/5 = 60/100 which equals 60%

52. 3) Make an Illustration: 1/5 = 60 out of 100 or 60%

53. Percent of Change: “Percent of change is the ratio: “Percent of change is the ratio: amount of change original amount expressed as a percent. Try it! “Suppose you increase the strength in your elbow joint from 90 foot-pounds to 135 foot-pounds. Find the percent of increase to the nearest percent.” adapted from Prentice Hall Algebra I adapted from Prentice Hall Algebra I

54. 90 to 135: 135- 90 = 45 change 45 ÷ 135 = 0.33 or 33 1 / 3 % increase.

55. Maximum and Minimum Areas: “The greatest possible error in measurement is one half of that measuring unit.” “The greatest possible error in measurement is one half of that measuring unit.” Find the maximum and minimum areas for a room that is 13 ft. by 7ft. Find the maximum and minimum areas for a room that is 13 ft. by 7ft. from Prentice Hall Algebra I from Prentice Hall Algebra I

56. 13 ft could be 12.5 or 13.5 while 7 ft could be 6.5 or 7.5 12.5 * 6.5 = 81.25 ft 2 12.5 * 6.5 = 81.25 ft 2 (minimum) (minimum) 13.5 * 7.5 = 101.25 ft 2 (maximum) 13.5 * 7.5 = 101.25 ft 2 (maximum)