September 27, 2018

‘Numeracy’ is being able to read numbers, and apply math to real-life situations. It’s similar to ‘literacy’, which is being able to read and understand words. Numeracy has mathematics as its core.

For the Energy Advisor Foundation training, you are required to understand mainly quantity and number as well as space and shape. Problem solving is a good skill to have, as well as understanding patterns and relationships. We have boiled down the numeracy competency requirements to three categories:

  • Arithmetic

  • Geometry

  • Conversions

This week’s article is about Conversions, how to translate between the Imperial or US system and the metric system.

Here’s the evaluation form for Conversions, so you can rate yourself on your existing understanding. The rating is from 0 (I know nothing about this topic) to 4 (I’m an expert in this topic and can handle complex tasks on the daily). You don’t have to share this with anyone else, so rate yourself honestly!

If you find this part of Geometry a challenge, read on. There are links to lots of free exercises and workbooks in this article.  

If you’ve got this part of the competency down, tune in next week for the first part of Plans Reading.

Conversions

The construction industry wasn’t just created in one place by a single group of people. Over time, our knowledge and skills had to change and grow, especially when international trade gave rise to the need for a standardized system of measurement. Almost everyone agreed to use the metric system. The United States uses a system of measurement called the US Customary System, which is an adaptation of the British Imperial system. The Canadian construction industry uses a mixture of British Imperial and metric, depending on where you are and who you're working with.

 

Here’s what you need to be able to do:

  • Convert units within and between the metric and imperial systems in length, area, volume, and weight

  • Convert temperature from Celsius to Fahrenheit and back again

  • Convert between units of energy

  • Convert between ratios and percentages

Introduction

The metric system, also known as the International System of Units (SI), is the most common system of weights and measures used throughout the world. The metric system was designed to have properties that make it easy to use and widely applicable, including units based on the natural world, like ten fingers. There are no complicated conversion factors between units.

The Imperial system of weights and measures originally developed in England. The US System is similar but some unit sizes vary. The Imperial System has been replaced by the Metric System in most countries (including England).

The metric system records measurements with a base unit and a prefix, in powers of 10. There is one base unit for each type of measurement.

For example, length is measured in meters, no matter how long or short the distance is that you're measuring, and prefixes are added for very small or very large lengths, like centimetre or kilometre. Area is recorded in square metres, or meters squared.

The base unit for volume is the litre, or cubic meters, and for mass, or weight, it’s the gram.

Here’s a video on how the metric system works: Math Antics

The Imperial System, or the US system, has a standard set of units for length but their relation to each other isn’t as neat as a factor of ten. Mostly, we measure length in feet, inches, and fractions of an inch. There are 12 inches in a foot and an inch is split up into quarters and then eighths. Area is recorded in square feet, square inches, and square fractions of an inch. The yard and the mile are used for longer lengths, and acres for large areas. A yard is 3 feet and a mile is 1760 yards.

Volume in the Imperial or US system is commonly measured in cubic inches, cubic feet, and cubic yards. One cubic foot is 1728 cubic inches, and one cubic yard is 27 cubic feet.

Here are two worksheets (with answers on the second page) to see how you do in converting between metric and imperial units:
Worksheet 1
Worksheet 2

 

Length

In most systems of measurement, the unit of length is a base unit, from which other units, like area and volume, are derived. Length is a measure of one dimension.

In the metric system, the three most commonly used prefixes to add to the metre when measuring length are milli-, centi-, and kilo-. One metre can be split up into 100 centimetres or 1000 millimetres, and one thousand metres make up one kilometre.

In the US Customary system, length is measured in yards, feet, inches, and fractions of an inch - a half, quarters, or eighths of an inch - or in decimal feet, like "4.33 ft". Notating a measurement in decimal feet makes it easier to perform operations on it.

To convert measurements between the metric and US systems, you need to know how the different units relate to each other. This comes down to the conversion factor.

Memorize these critical metric to US or imperial relationships for length:

1 inch = 2.54 centimeters or 25.4 millimeters

1 foot = 30.48 centimeters

1 meter = 39.4 inches

With these relationships memorized, you can figure out every conversion for length unless you’re working with miles and kilometers. For most of the homebuilding and renovation industry, these are adequate.

Here are two videos on converting between metric and imperial for length: Corbett Maths  and StudyForce

Here is a great interactive website for converting rates and measurements: IXL

Area

Measuring area is required to calculate the amount of material needed to complete a job. Once you've found out how much material you'll need to cover a floor or a wall, you might need to convert your measurement from metric to imperial units, or the reverse.

Area is always measured in square units, the unit for length with an exponent of 2. In the metric system, area units are m2, or square meters, and the meter's related units, square centimeters, square millimeters, and square kilometers.

In the US system, area is measured in square feet, square inches, and square yards.

There are many conversion factors but you don't have to memorize them all. Most often, in residential construction, we are dealing with either inches and feet in the US/Imperial system, centimeters and metres in metric. We can simplify this into two key conversions that you either multiply or divide by, depending on which way your conversion is going. When you are trying to decide which way your conversion is going, keep this in mind: a square inch is bigger than a square centimetre, but a square metre is bigger than a square foot.

1 square inch =  6.45 square centimetres
Multiply square inches by 6.45 to get square centimeters. Divide square centimetres by 6.45 to get square inches.

1 square metre = 10.76 square feet
Divide square meters by 10.76 to get square feet. Multiply square meters by 10.76 to get square feet.

Here is a video on converting metric units of area: Corbett Maths

 

Volume

Remember that volume is a three-dimensional measurement of the space something takes up, so the units will have an exponent of 3, which you would say is cubed.

You might need to figure out the volume of a hot water tank to see if it's the appropriate size for the house, or the volume of air in an entire house to use in air flow calculations. The US system uses cubic inches, cubic feet, and cubic yards to measure volume. The metric system uses millilters and liters for liquids, and cubic centimeters and cubic meters for solids.

When it comes to volume, the difference between metric and the US or Imperial systems is complex. For example, in the US/Imperial system, pints, quarts and gallons are used to measure liquids. but here is a difference between a US and an Imperial liquid measurements. For instance, there are three point seven nine litres in a US gallon, but there are 4.54 litres in an Imperial gallon. Make sure you convert from and to the correct units

1 cubic inch is a one by one by one inch cube.

1 cubic foot = 12 by 12 by 12 inches = 1728 cubic inches

1 cubic yard = 3 feet by three feet by three feet = 27 cubic feet

By contrast, in the metric system uses only litres and cubic meters in units that are larger or smaller by base ten factors.
1 millilitre = 1/1000th of a liter

1 kilometer = one thousand metres

Much easier to convert between smaller and larger units.

The wide variety of units in the US/Imperial system make it a challenge to memorize two or three all-purpose conversions. The good news is that, in the residential construction industry, there are only a few US/Imperial units of volume that are commonly used. Gallons, as in how much paint is needed for a job, or how big a hot water tank is. Cubic feet and cubic yards, as in how much fill or concrete is needed for a foundation.

Gallons are typically converted to litres (make sure you know whether the gallon is US or Imperial). In Canada, 4.54L is a gallon. In the US, a gallon is 3.79L.

If you're starting with gallons, multiply by 4.54 or 3.79 to get litres.
If you're starting with litres, divide by 4.54 or 3.79 to convert to gallons.

Here are two examples:
10 Imperial gallons times 4.54 equals 45.4 litres
10 litres divided by 3.79 equals 2.64 US gallons

When it comes to converting cubic measurements for solids like cubic feet or cubic metres, it's easier to memorize the metric to US/Imperial units.  

1 cubic metre = 35.3 cubic feet.

If you're starting with cubic meters, multiply to get cubic feet.

If you’re starting with cubic feet, divide to get cubic meters.

1 cubic metre = 1.31 cubic yards.
If you're starting with cubic meters, multiply to get cubic yards.
If you’re starting with cubic yards, divide to get cubic metres.

Here’s two examples:
10 cubic metres x 35.3 = 353 cubic feet
10 cubic yards/ 1.31 = 7.63 cubic meters

Here is a video on converting between metric and imperial units of volume: StudyForce

Weight and Mass

Weight and mass are not exactly the same thing, but as long as we're building houses on Earth, where weight and mass are almost exactly proportional to each other, they can be used to describe the same thing. Mass is a measurement of how much "matter" is in an object, while weight is a measurement of the force exerted on the object by gravity. "Directly proportional" means that an object one hundred times as massive as a pencil would also be one hundred times heavier.

The metric system measures weight in grams, kilograms, and metric tons, while the US/Imperial system uses pounds, ounces, and short and long tons.

In the metric system, all weight units are based on the gram.

The gram is a very small measurement. one gram is roughly equal to 1 small paper clip or a pen cap. Smaller units are milligrams, larger units are kilograms.

In the residential construction industry, we are more likely to work with kilograms of material.

The most common units of weight in the US/Imperial system are the ounce, the pound, and the ton. One pound is equal to 16 ounces, and a short ton is 2000 pounds. (Fun fact: a long ton is 2240 lbs, and a metric ton is roughly 2204 lbs).

Memorize these two conversions:

1 kilogram = 2.2 pounds.
Multiply kilograms by 2.2 to convert to pounds.
Divide pounds by 2.2 to convert to kilograms.

1 ounce = 28.35 grams.
Multiply ounces by 28.35 to convert to grams.
Divide grams by 28.35 to convert to ounces.

Here’s two examples:
10 kilograms times 2.2 equals 22 pounds.
10 grams divided by 28.35 equals point three five ounces.

A short ton is 2000 pounds, while a metric tonne is 2000 grams, or 2204 pounds.

Memorize this conversion:
One metric tonne equals 1.1 short ton.
Multiply short tons by 1.1 to get metric tonnes.
Divide metric tonnes by 1.1 to get short tons.

Here are two examples:

10 short tons multiplied by 1.1 equals 11 metric tonnes
10 metric tonnes divided by 1.1 equals nine point one short tons

Here is a video on how to convert metric and imperial mass (weight): Corbett Maths

Temperature

 

You can measure kinetic energy (energy of motion) with a thermometer. When you measure the air temperature in your backyard, you’re really measuring the average kinetic energy of the gas particles in your backyard. The faster those particles are moving, the higher the temperature is.

The Celcius and Farenheit scales are based on observable properties of water: the freezing point and the boiling point.

On the Celcius scale, 0 degrees C is the freezing point of water, and 100 degrees C is its boiling point.

On Fahrenheit scale the freezing point of water is 32 degrees F, and its boiling point is 212 degrees F.

The relationship between the two scales isn't directly proportional, or even linear. So knowing that 0 C = 32 F doesn't mean you can add 32 to any Celsius measurement to get Fahrenheit.

When you know the °F, and you need to solve for °C, the formula is:
°C = 5/9 (°F – 32)

Take the °F temperature and subtract 32. Multiply this number by 5.
Divide that number by 9 to obtain your answer in °C.

When you know the °F, and you need to solve for °C, the formula is:
   °F = 9/5 (°C) + 32
Multiply your Celsius temperature by 9/5ths and then add 32. NOTE: 9/5ths is equal to 1.8.

Here is a worksheet on converting temperatures.

Ratios

In general a ratio is a comparison of one number to another. Sometimes ratios look like fractions, like 2/3, and sometimes the two numbers are separated by a colon, like this 2:3.

The slope of a roof is a ratio, it is the vertical height of the roof compared to the horizontal distance.

A ratio is representative of a quantity, and can be used to determine the required quantity of materials, like the amount of water needed for a specific concrete mix, or the amount of resin and a hardener needed for a good bond with a two-part epoxy adhesive.

In other instances, ratios help to determine energy performance and other characteristics in houses. Two important ratios are the perimeter to area ratio, and its extrapolation, the volume to surface area ratio.

Here is a video from Math Antics on Ratios and Rates

Equivalent Fractions

It's important to understand how equivalent fractions work, especially when it comes to ratios or adding and subtracting fractions. Equivalent fractions have the same value, even though they look different. What do we know about the number 1? Any number divided by itself is 1. So, 2/2 is 1, 7/7 is 1, 137/137 is 1. We also know that a number multiplied by 1 is itself. So, (2/3) * (4/4) is like saying 2/3 times 1. THAT'S why equivalent fractions can have the same value. The equivalent fraction would be 8/12.

When you look at a ruler. It's clear that 1/2 inch is the same as 2 quarter-inches.

Think of a pizza. When there are no slices gone we could represent the pizza with 1/1. When we cut the pizza into 8 slices but no one\s eaten any yet it could be 8/8. Once 3 pieces are gone, it's 5/8 left. Once 4 pieces are gone it could be 4/8, or we could divide both the numerator and the denominator by 4 to reduce the fraction down to 1/2.

Here’s a video on Percents and Equivalent Fractions: Math Antics

Converting A Ratio to Percentage

A ratio is a comparison of any two numbers by division. A percent is a special ratio that compares any number to 100, with 100 representing one whole. Representing the unit whole by 100 makes comparisons simple. For example, comparing 15% and 40% is easier than comparing the ratios 3 to 20 and 2 to 5. Any fraction or decimal can be converted into a percentage.

To convert a ratio into a percentage, divide the ratio, then multiply by one hundred.

Here's how to convert the two examples above:
3 to 20 can also be written as 3/20.
Three divided by twenty equals 0.15. Multiply .015 by one hundred to get 15%.

2 to 5 can also be written as 2/5. Two divided by five is 0.4. Multiply 0.4 by one hundred to get 40%.

To convert in the opposite direction, you need to know one portion of the ratio. For example, if you are trying to find slope from forty percent grade, you have the bottom half of your ratio: 12, because slope is always noted as ‘x’ over 12.
Divide 40 percent by 100. This gives the decimal version of the ratio 0.4.
Multiply by twelve.
This gives you the top half of the slope ratio, 4.8.

The slope ratio is 4.8 over 12.

Energy

We want to build homes that are highly energy-efficient, because they save us money and put less pressure on the environment. How do we know how efficiently a house's systems are using energy? We need to measure it. How do we measure energy? What is energy, exactly?

Energy is the ability to do work. The complicated part is making sure you're talking about the right kind of energy, like thermal or electrical, and that you're using the right units.

This section will focus on thermal energy, or heat.

Heat is not the same as temperature. When you measure the temperature of something, you’re measuring the average kinetic energy of the individual particles. Heat, on the other hand, is a measure of the total amount of energy a substance possesses.

For example, a glass of water and a swimming pool may be the same temperature, but they contain vastly different amounts of heat to maintain a specific temperature. It takes much more energy to raise the temperature of a swimming pool 5°C than it does a glass of water, because there’s so much more water in the swimming pool.

Units of Energy

Energy is the amount of work being done by something. The fundamental unit of energy is the Joule.

One Joule is equal to the energy required to move an object that has 1 Newton of force acting on it over a 1 metre distance.

The Newton (N) is a standard unit of force, derived from the gravitational pull of the earth. 1 N = 1 kilogram times one metre per square second. A Newton is roughly equivalent to the force needed to lift an apple above your head.

A joule is a very small measure of energy.
1 Joule is about 0.2388 calories.
The kilocalorie (food calorie, or Calorie with a capital C) is equivalent to 1000 "little c" calories.
One joule is about 0.0009481 BTUs

When talking about energy and houses, megajoules (millions of joules) and gigajoules (billions of joules) are the most common metric units. Imperial units of energy use in houses are typically in British thermal units (Btus).

When talking about thermal electric energy, the most common term used is kilowatt hours (kWh).

Memorize these energy conversions:
1 kiloWatt = 3412 Btu

There are the same number of Btus in a kilowatt-hour.
Multiply kilowatts or kilowatt-hours by 3412 to get Btus.
Divide Btus by 3412 to get kilowatts or kilowatt-hours.

 

There are 3.6 megajoules in a kilowatt hour.
Multiply kilowatt-hours by 3.6 to get megajoules.
Divide by 3.6 to get kilowatt hours.

There are 1000 megajoules in a gigajoule.

Insulation Values

An insulating material's resistance to conductive heat flow is measured or rated in terms of its thermal resistance. In the US or imperial system, this is known as the R-value. In the metric system this is known as the RSI value. The higher the value per unit of thickness, the more resistance the material has to the movement of heat, and the more effective it is.

U-factor, or thermal transmittance is the reciprocal of the R-value, or RSI value. Thermal transmittance is the rate of transfer of heat through a structure, divided by the difference in temperature across that structure. Windows and doors are often rated by U-factor.

The lower the U-factor, the more energy efficient the system in question will be.

To convert RSI to R-value multiply by 5.678

To convert R-value to RSI divide by 5.678

For example, a fiberglass batt is rated at R-20. To convert it to RSI, divide by 5.678. The answer is 3.522. Because the precision of the original rating is a whole number, round the RSI conversion off to one decimal place for RSI 3.5

The U-factor is the reciprocal of R-value. That is, U equals 1 over R, and R equals 1 over U.

Here's an example, a material with an R-Value of 5 has a U-factor of 0.2 (1 divided by 5).

Air Movement

Air sealing a house makes it much more energy efficient, but you need to ensure the air inside the house is still moving around and that proper ventilation equipment is installed. Air flow can be measured in cubic feet per minute (CFM).

Knowing how to calculate air flows - and how to convert between cubic feet per minute and litres per second - is important when doing blower door testing, when designing or sizing forced air heating systems, air conditioning systems and ventilation systems.

The metric equivalent of cubic feet per minute is litres per second, litres being a measure of volume.

1 litre = 1 cubic decimetre (dm3)

1 litre per second = 2.12 cubic feet per minute.

 

To convert from cubic feet per minute to litres per second, divide the CFM by 2.12

Energy Intensity

Energy use intensity, EUI, shows a building’s energy use as a function of its floor space. It’s calculated by dividing the total energy consumed by the building in one year (measured in Btu or GJ) by the total gross floor area of the building. The EUI is expressed as energy per square foot per year or by Gigajoules per square meter per year.

As the performance improves, the EUI goes down. The EUI takes away the variable of house size, so homeowners, municipalities and lenders can compare and rate houses according to their energy use. Here's an example:

A house is 2700 square feet. The annual energy use for this house is 100,500,000 Btus. Divide the energy use by 2700 square feet. The EUI is 37,222 Btu per square foot.

 

Everybody can learn to do the math. 

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