1. Low Impact Development Training Module 1.2: Math

2. Sponsors 2 District Department of Transportation U.S. Department of Transportation Federal Highway Administration The Low Impact Development Center, Inc. University of the District of Columbia Funding for this project was provided through a grant from the Federal Highway Administration, U.S. Department of Transportation

3. Contributors 3 The Low Impact Development Center, Inc. John Shorb Landscaping, Inc. Logo Groundwork Anacostia River, D.C.

4. Copyright 4 Unless otherwise noted, Low Impact Development Training, funded by DDOT & DDOE, is licensed under a Creative Commons Attribution-NonCommerical- ShareAlike 3.0 Unported License. Content provided by cited entities remains the property of those entities and may not be used without their explicit permission.

5. Overview Some basic math skills are needed to perform bioretention maintenance activities Calculate quantities, lengths, and volumes of materials needed Estimate material and labor costs 5

6. Overview Basic math skills Conversions Geometry Calculating the area of landscaped features Calculating volumes for soil modification and topdressing Estimating water use Estimating maintenance costs 6

7. Supplemental Information Note that slides with orange backgrounds contain supplemental details that are provided for informational purposes, but which are not required content for this course 7

8. Expected Outcomes Be able to estimate installation, maintenance, and repair costs Be able to calculate area and volume of landscape features 8

9. Estimating costs Generating accurate estimates of material and labor needs allows you to provide better estimates to clients, and to avoid over or under purchasing from suppliers 9

10. Estimating maintenance costs Materials costs –Plant materials –Hard goods Mulch Stone 10

11. Estimating maintenance costs Labor costs –Wages –Benefits –Labor burden Worker’s compensation State and federal payroll taxes Unemployment taxes –Labor overhead Production rate –Published values for most tasks –Adjust for difficult site conditions 11

12. Other costs Equipment costs Direct job overhead –Permitting fees –Dumpster rental –Disposal fees for transfer station General overhead Profit 12

13. Published construction estimate guides RS Means publishes data that can be used to generate accurate cost estimates A volume dedicated to Site Work and Landscape Cost Data can be purchased http://rsmeans.reedconstructiondata.com/ 13

14. Basic math operations Purpose: to be able to set up and perform calculations correctly Estimates are no good if you get the wrong answer! 14

15. Order of Operations PEMDAS 1.Parentheses 2.Exponents 3.Multiplication and Division (left to right) 4.Addition and Subtraction (left to right) 15

16. 1. Parentheses 16

17. 2. Exponents 17

18. 3. Multiplication and Division 18

19. 4. Addition and Subtraction 19

20. Put it all together 20

21. Solving for x An unknown value in an equation is represented by a letter, usually x Determining which value x represents is done by isolating x on one side of the equation This is done by manipulating the equation to isolate x on one side of the equals sign X is isolated by making the same change to both sides of the equation 21

22. Solving for x 22

23. Solving for x 23

24. Solving for x 24

25. Ratios and proportions 25

26. Proportion 26

27. Cross multiplication 27

28. Solving problems 28

29. Calculating quantities 29

30. Conversions 30

31. Convert ratio to percentage 31

32. Accuracy and Significant Digits Measurements are only as accurate as the measuring device you’re using Calculations are only as accurate as the measurements they are based on You can’t always trust your calculator! 32

33. Accuracy Accuracy: how close a measured value is to the “true” value Expressed as ± For example, a scale may have an accuracy of ± 0.1 g Weights measured with this scale should be written as: Measured weight ± 0.1 g 33

34. Significant digits Significant digits are all of the digits in a number that can be measured by an instrument Examples: A scale can measure to a tenth of an ounce 20.1 ounces has 3 significant digits 34

35. When reading a ruler or tape measure, record one more digit than can be read on the scale Between 6 1/8” (6 4/32”) and 6 3/16” (6 6/32”), so record as 6 5/32”, or 6.16” Rulers 35

36. Rules for significant digits 1.Digits 1-9 are always significant 2.0 is sometimes significant 1.When found between two non-zero digits (e.g. 2045) 2.To the right of a number with a decimal point (e.g. 30.10) 3.NOT to the right of a number without a decimal point (e.g. 2,000) 4.NOT to the left (e.g. 0.0001) 36

37. Rounding When performing calculations, a calculator will give you values with as many digits as the display allows BUT, the calculated value is only as accurate as the least accurate measurement used in the calculation Calculated values need to be rounded to the number of significant digits of the least accurate measurement 37

38. Rules for rounding 1.If the number to the right of the last significant digit is less than 5, drop this digit and all the digits to the right of it Reduce 5.04329 to 3 significant digits 5.04329 5.04 38

39. Rules for rounding 2.If the number to the right of the last significant digit is greater than or equal to five, increase the last significant digit by one and drop all the digits to the right of it Reduce 43.2379 to 4 significant digits 43.2379 43.24 39

40. Geometry How to calculate the area of different shapes Used to calculate the size of an area to be maintained Estimate plant quantities Estimate water needs Estimate materials (e.g. mulch) 40

41. Area of a rectangle 41 Length (l) Width (w)

42. Area of a parallelogram 42 Base (b) Height (h)

43. Area of a trapezoid 43 Base 1 (b 1 ) Height (h) Base 2 (b 2 )

44. Area of a triangle 44 Base (b) Height (h)

45. Right triangle One angle is exactly 90º When you have a right triangle, calculating area is easy 45 b h

46. Pythagorean Theorem For right triangles, Useful if the length of one side can’t be measured, or to check that an angle is square 46 a b c

47. Area of a circle 47 r

48. Circumference of a circle 48 r c

49. Diameter of a circle 49 r d

50. Area of an ellipse 50 r major r minor

51. Area of irregular shapes Geometric method Offset method Modified offset method 51

52. Geometric method Used to calculate the area of spaces that are composed of simple geometric shapes 52

53. Composite geometric forms 53

54. Offset method A method to measure the area of a feature that isn’t obviously composed of geometric shapes Basically, the feature is approximated by a series of rectangles of equal widths but different lengths 54

55. Offset method Step 1: establish a line along the longest axis 55

56. Offset method Step 2: establish equally spaced offset lines perpendicular to the first line 56

57. Offset method Step 3: measure each line from end to end 57 10 ft 100 ft 20 ft 24 ft 21 ft 20 ft 23 ft 24 ft22 ft

58. Offset method Step 4: Sum the lengths of all the offset lines 58 10 ft 100 ft 20 ft 24 ft 21 ft 20 ft 23 ft 24 ft22 ft

59. Offset method Step 5: Multiply the sum by the distance between the offset lines 59 10 ft 100 ft 20 ft 24 ft 21 ft 20 ft 23 ft 24 ft22 ft

60. Modified offset method Used when areas cannot easily be traversed to measure the offset lines This is the same kind of approximation as the offset method, but estimates by subtraction It works by drawing a rectangle around the outside of the feature, then using the offset method to measure the area within the rectangle that is NOT in the feature 60

61. Modified offset method Then, the area of the feature = the total area of the rectangle minus the area measured using the offset method 61

62. Modified offset method Step 1: create a rectangle around the area to be measured 62 l w

63. Modified offset method Step 2: establish equally spaced offset lines 63 10 ft

64. Modified offset method Step 3a: measure the lengths of each of the offset line segments 64 1 1 235 2 1 1 246 4 6 4

65. Modified offset method Step 3b: Add up each pair of offset measurements 65

66. Modified offset method Step 4: For each of the line segments, subtract each of the sums from the width of the rectangle. Each of the results = the actual width of the figure at the offset location 66

67. Modified offset method Step 5: Sum the widths of the figure calculated in Step 4 67

68. Modified offset method Step 6: Multiply the summed value from Step 5 by the distance between offsets 68

69. How would you estimate this area? 69 Photo Courtesy of The Low Impact Development Center, Inc.

70. Calculating volume Used to estimate volumes of media, gravel, soil amendments, and topdressing 70

71. Volume of shapes with parallel bases and equal cross-sections 71 B h B h

72. Volume of figures with tapered sides 72 B top B bottom h

73. Soil modification Calculate the volume of an amendment to be incorporated into the soil in an area 73

74. Soil modification Step 1: determine the surface area that needs to be modified using one of the methods discussed earlier –Geometric method –Offset method 74

75. Soil modification 75

76. Soil modification Step 3: multiply area by depth to determine volume of soil to be modified 76

77. Soil modification 77

78. Soil modification 78

79. Topdressing Calculate the volume of an amendment to be placed on top of existing soil surface (e.g. mulch) 79

80. Topdressing Step 1: determine the surface area that needs to be modified using one of the methods discussed earlier –Geometric method –Offset method 80

81. Topdressing 81

82. Topdressing Step 3: multiply area by depth to determine volume of topdressing needed 82

83. Topdressing 83

84. Water use 84

85. Calculating water use 85

86. Calculating water use 86

87. Calculating slope 87 rise run

88. Resources Mathematics for the Green Industry: Essential Calculations for Horticulture and Landscape Professionals by Michael L. Agnew, Nancy H. Agnew, Nick Christians and Ann Marie VanDerZanden. 2008. –Chapters 1-4, pp.174-190, 191-194, 211-221 88