Low Impact Development Training

Module 1.2: Math

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Sponsors

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

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Contributors

The Low Impact Development Center, Inc.

John Shorb Landscaping, Inc.Logo

Groundwork Anacostia River, D.C.

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Copyright

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.

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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

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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

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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

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Expected Outcomes

• Be able to estimate installation, maintenance, and repair costs

• Be able to calculate area and volume of landscape features

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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

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Estimating maintenance costs

• Materials costs– Plant materials– Hard goods

• Mulch• Stone

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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

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Other costs

• Equipment costs• Direct job overhead

– Permitting fees– Dumpster rental– Disposal fees for transfer station

• General overhead• Profit

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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/

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Basic math operations

• Purpose: to be able to set up and perform calculations correctly

• Estimates are no good if you get the wrong answer!

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Order of Operations

PEMDAS

1. Parentheses

2. Exponents

3. Multiplication and Division (left to right)

4. Addition and Subtraction (left to right)

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1. Parentheses

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2. Exponents

• Powers and square roots

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3. Multiplication and Division

• Proceed from Left to Right

CORRECT

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4. Addition and Subtraction

• Proceed left to right

CORRECT

INCORRECT

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Put it all together

1. Parenthesis

2. Exponents

3. Multiplication and Division L -> R

4. Addition and Subtraction L->R

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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

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Solving for x

Subtract 2 from both sides

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Solving for x

Divide both sides by 4

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Solving for x

Multiply both sides by 4

Add 5 to both sides

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Ratios and proportions

Ratio – the relative size of two quantities expressed as one divided by the other

Written as

or

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Proportion

• A proportion is a statement that two ratios are equal

• Examples

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Cross multiplication

If , then

For example,

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Solving problems

Use cross-multiplication to solve for x:

Divide both sides by 2

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Calculating quantities

According to the as-built plans, the bioretention media for this installation shall be composed of 1 part topsoil to 3 parts sand.

How much sand do you need to add to 9 cubic yards of sand?

You’ll need 3 cubic yards of topsoil

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Conversions

Proportions can be used to convert values from one number system to another

Example: convert 0.25 acres to sq ft

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Convert ratio to percentage

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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!

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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

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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

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• 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

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Rules for significant digits

1. Digits 1-9 are always significant

2. 0 is sometimes significant1. 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)

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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

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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

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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

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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)

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Area of a rectangle

Length (l)

Width (w)

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Area of a parallelogram

Base (b)

Height (h)

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Area of a trapezoid

Base1 (b1)

Height (h)

Base2 (b2)

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Area of a triangle

Base (b)

Height (h)

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Right triangle

• One angle is exactly 90º

• When you have a right triangle, calculating area is easy

b

h𝐴𝑟𝑒𝑎=

𝑏×h2

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Pythagorean Theorem

• For right triangles,

• Useful if the length of one side can’t be measured, or to check that an angle is square

a

b

c

𝑎2+𝑏2=𝑐2

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Area of a circle

rWhere r = radius

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Circumference of a circle

r

c

where r = radius

c

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Diameter of a circle

rwhere r = radiusd

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Area of an ellipse

rmajor

rminor

𝐴=𝜋 (𝑟𝑚𝑎𝑗𝑜𝑟×𝑟𝑚𝑖𝑛𝑜𝑟)

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Area of irregular shapes

• Geometric method• Offset method• Modified offset method

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Geometric method

• Used to calculate the area of spaces that are composed of simple geometric shapes

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Composite geometric forms

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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

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Offset method

• Step 1: establish a line along the longest axis

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Offset method

• Step 2: establish equally spaced offset lines perpendicular to the first line

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Offset method

• Step 3: measure each line from end to end

10 ft

100 ft

20 ft

24 ft

24 ft 21 ft 20 ft

20 ft

23 ft

24 ft

22 ft

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Offset method

• Step 4: Sum the lengths of all the offset lines

10 ft

100 ft

20 ft

24 ft

24 ft 21 ft 20 ft

20 ft

23 ft

24 ft

22 ft

20+24+24+21+20+20+23+24+22=198 𝑓𝑡

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Offset method

• Step 5: Multiply the sum by the distance between the offset lines

10 ft

100 ft

20 ft

24 ft

24 ft 21 ft 20 ft

20 ft

23 ft

24 ft

22 ft

198 𝑓𝑡×10 𝑓𝑡=1,980 𝑓𝑡2

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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

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Modified offset method

• Then, the area of the feature = the total area of the rectangle minus the area measured using the offset method

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Modified offset method

• Step 1: create a rectangle around the area to be measured

l

w

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Modified offset method

• Step 2: establish equally spaced offset lines

10 ft

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Modified offset method

• Step 3a: measure the lengths of each of the offset line segments

1 1 2 3 5 2

1 1 2 4 6 4

6

4

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Modified offset method

• Step 3b: Add up each pair of offset measurements

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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

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Modified offset method

• Step 5: Sum the widths of the figure calculated in Step 4

4 𝑓𝑡+12 𝑓𝑡+10 𝑓𝑡+10 𝑓𝑡+7 𝑓𝑡+3 𝑓𝑡+8 𝑓𝑡=54 𝑓𝑡

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Modified offset method

• Step 6: Multiply the summed value from Step 5 by the distance between offsets

(54 𝑓𝑡 )×(10 𝑓𝑡 )=540 𝑓𝑡2

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How would you estimate this area?

Photo Courtesy of The Low Impact Development Center, Inc.

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Calculating volume

• Used to estimate volumes of media, gravel, soil amendments, and topdressing

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Volume of shapes with parallel bases and equal cross-sections

𝑉𝑜𝑙𝑢𝑚𝑒=𝐴𝑟𝑒𝑎𝑜𝑓 𝑏𝑎𝑠𝑒×h h𝑒𝑖𝑔 𝑡

B

h

B

h

𝐵=𝑙×𝑤𝐵=𝜋𝑟2

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Volume of figures with tapered sides

𝑉=h×[ (𝐵𝑡𝑜𝑝+𝐵𝑏𝑜𝑡𝑡𝑜𝑚)÷2]

Btop

Bbottom

h

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Soil modification

• Calculate the volume of an amendment to be incorporated into the soil in an area

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Soil modification

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

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Soil modification

• Step 2: convert depth of soil to be modified to feet

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Soil modification

• Step 3: multiply area by depth to determine volume of soil to be modified

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Soil modification

• Step 4: determine the volume of each soil component that is needed

• Example: As-built specifies 75% sand, 20% topsoil, 5% compost by volume

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Soil modification

• Step 5: convert volumes from cubic feet to cubic yards

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Topdressing

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

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Topdressing

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

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Topdressing

• Step 2: convert depth of topdressing to feet

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Topdressing

• Step 3: multiply area by depth to determine volume of topdressing needed

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Topdressing

• Step 4: convert volume from cubic feet to cubic yards

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Water use

• Irrigation water is measured in acre-feet• One acre-foot is the amount of water

needed to cover one acre with water one foot deep

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Calculating water use

• How much water is needed to apply 1-inch of water to ½ acre?

• Step 1: convert to ac-ft

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Calculating water use

• Step 2: convert ac-ft to gallons

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Calculating slope

• Expressed as a ratio or a percentage

• E.g. 1:20 or

rise

run

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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