September 6, 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.

We use math and physics constantly, but it’s so natural to us that we don’t realize it. That’s because math and physics are just a languages that uses numbers instead of letters. OK, physics uses formulas, which include letters as placeholders for numbers that you put into the the formula. For example, you use physics, in the form of vectors and velocity, every time you merge onto a highway - you don’t calculate the numbers, but you are sorting out your speed and angle of entry from the ramp vs. the speed of traffic already on the highway. Who did do the calculations so that you can safely merge into oncoming traffic was an engineer (or two or three).

Studies have shown that most adults make an average of 3 to 5 calculations every day, with 85% of those being mental math. Most of these have to do with time and money (no surprise there). When you cook, you use math. When you plan a day, you use math to figure out how much time you can allocate to each task. When you go shopping, you use math to estimate how much your cart will cost vs. how much money is in your bank account.

Math is a challenge. Math is seen as a tough subject, because there are rules that need to be followed, and if you don’t know the rules, you can’t make math work. The good news is there are not as many rules in math as there are in English spelling!

Anyone involved in the construction trades needs math skills. Some need more complex math skills than others. For example HVAC techs, and electricians need to know a lot more formulas than carpenters or labourers.

The everyday math skills that construction trades need can be grouped into these five areas:

• Quantity and number

• Space and shape

• Data handling and chance

• Problem solving

• Patterns and relationships

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

We’ve also created simple evaluation forms for each category so you can rate yourself. 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). Here’s the one for Arithmetic, in PDF format for downloading. You don’t have to share this with anyone else, so rate yourself honestly!

Once you’ve rated yourself, give yourself a pat on the back for the areas where you are strong, and roll up your sleeves to improve your game in the other areas.

If you’ve got this competency down, tune in next week for geometry.

## Arithmetic

Arithmetic is the basic building block of all the rest of your math skills. Getting the basics right means all the rest of it becomes much easier.  It's important for accuracy, efficiency, and safety to make sure the work is done correctly.

People in the construction industry use math every day. You have to be able to know the dimensions of the materials you're working with, and make sure everything is of the right amount or size. You have to be able to take precise measurements and convert the units (within and between measurement systems!) using equations.

Arithmetic is simply doing calculations with numbers: addition, subtraction, multiplication, and division. Arithmetic also includes comparisons, using these symbols:

The equals sign (=) is the basic one that most people recognize.

The opposite of the equals sign is 'not equal to', an equals sign with a slash through it (≠).

Other symbols used for comparison are:

• Greater than (>): When one number is greater than another

• Greater than or equal to (≥): When one number is greater than or equal to another

• Less than (<): When one number is less than another

• Less than or equal to (≤): When one number is less than or equal to another

You might see these symbols when manufacturers are referring to tolerances, or relationships between chemicals, in standards that you use when specifying products, or when energy targets are required.

### Whole numbers: messing with whole things

Whole numbers, also called integers, are the numbers used for counting, including zero. The integers 0 through 9 are called digits. Integers can be made up of any number of digits and can be positive or negative numbers.

Some examples of integers are 7, 33, 429, and 10,488.

Negative numbers represent values that are less than 0. In construction, they don't apply very often to measurements of physical objects, but temperatures go below 0. You might need to calculate the difference between an outside temperature of -10F and an inside temperature of 65F, or -30C outside and 20C inside.

### Fractions and decimals: messing with parts of things

Fractions and decimals are numbers that are, or at least include, a value that is less than 1, or a part of a whole. Decimal numbers can be expressed as fractions, and vice versa.

In a fraction like 7/16, the top (or first) number, the numerator, tells us how many parts there are - in this case, 7. The bottom  (or second) number, the denominator, tells us how many parts the whole is divided into - in this case, 16. This is familiar ground: the North American construction industry works in feet and inches, and in fractions of an inch, where an inch is divided  into halves, quarters, then eighths, and then sixteenths.

The most important thing to remember with fractions is that the denominators , the bottom of all the numbers in the problem, must be the same before you can add or subtract them.

A measurement smaller than an inch is always a fraction, unless it's converted to a decimal.

Decimals are expressed in groups of 10. You have 10ths, 100ths, 1000ths, and so on. The decimal place moves to the right for each zero.

While a foot is divided into 12 parts, we don't often write it as 1/12th of an foot. But you could see an inch written as 0.083 feet, or eighty-three thousandths  of a foot.

### Rounding rules

When we're working in real-world measurements, but using a calculator, we end up with numbers that are more exact than what was actually measured in real life. The calculated measurement cannot be more precise than any of the initial measurements you took. The important thing to remember is that you are limited by the measurement with the least amount of precision (that's the number with the fewest digits after the decimal).

For example, Adding 7.89" + 8.5" gives us 16.39", but since 8.5" is only measured to a tenth of inch, the correct answer is 16.4". If you were adding 32.01m + 5.325m + 12m, you'll get a sum of 49.339m, but the answer should be 49 m, because 12m is the least accurate number in the calculation.

If you have 9.27 meters of conduit and you cut off 5.4 meters , you end up with 3.87 meters, but that gets rounded to the 10th decimal point instead of the 100th: 3.9 meters.

The rule of thumb is that you always go to the fewest numbers after the decimal point.

When rounding to the nearest 10, or 1, or tenth, or hundredth place, there are four generally followed rules:

1. Figure out what "place" you're rounding to and look at the digit to the right of it

2. If that digit is less than 5, the "nearest place" digit stays the same and you can drop all the digits to the right.

3. If that digit is greater than 5, the "nearest place" digit increases by 1 and the digits to the right are dropped.

4. When it's exactly 5, some people choose to consistently round the place digit up (or down), OR you can decide each time based on the value of the place value digit AND the one to the left of it.

### Exponents

A number that is expressed as an exponent has two parts, the base and the exponent. The exponent is written as a small number to the right and above the base number.The exponent of a number says how many times to use a base number in a multiplication. The exponent tells you the 'power' of the multiplication. In this example, it is eight to the power of two. The exponent "2" says to use the 8 two times in a multiplication.

The most common exponents used in the construction industry are the power of two, which we commonly call 'squared', and the power of three, which we commonly call 'cubed'.

### PEDMAS: the orderly order of mathematical functions

Sometimes you have to do a series of calculations using different functions. These calculations need to be done in the correct order for your result to be correct.

If you see a calculation in brackets, or parentheses. Do this first. Then, calculate any exponents. After those two are out of the way, then take care of any of the multiplication or division calculations. Finally, carry out the addition or subtraction calculations.

Here's a handy way to remember the order. Use the acronym PEMDAS. It stands for Parentheses, Exponents, Multiplication/Division, Addition/Subtraction. A good way to remember PEMDAS is "Please Excuse My Dear Aunt Sally".

Try these basic arithmetic exercises:

Exercises Part 1

## Algebra: adding letters to the numbers

Algebra, solving problems using equations with one unknown quantity, is useful for things like determining where to place holes, calculating roof angles, and using Ohm's law to check voltages. Algebra is math that substitutes letters for numbers. The letters in algebra represent variables.

A variable is a quantity that can change or take on different values. This is represented by the letter or the symbol. So, you could say the temperature on any given day is a variable, and we could represent the value that day with a capital T. T is your variable.

Algebra is the math of working with variables, and it can be used for everyday problem solving when it comes to unknown numbers, quantities, lengths, areas, or volumes. In the construction industry, algebra is used regularly, whether or not you're aware that you're using it.

### Is it an expression or an equation?

The first thing to understand is the difference between an algebraic expression and an equation. One way to look at them is that an expression is like a phrase, while an equation is a full sentence. You solve an equation, but you can only simplify an expression.

The primary difference is an equals sign, or another relation symbol like greater than or less than. An expression doesn't have a relation symbol. It has just a variable, or a variable and a known number. If we use the example of temperature, and the capital letter T as our variable, T plus 1 means the temperature just went up one degree. T + 1 is an expression. An algebraic equation would have an equals sign that acts to separate the two sides of the equation, like T + 1 = 72 degrees.

Another way of looking at this is: an algebraic equation represents a scale, and the equals sign is the balance point. This means that what is done on one side of the scale is also done to the other side of the scale.

In all algebraic equations and expressions, the variable is the unknown, and the numbers act as the constants.

### Formulas are specific, giving a desired result

A formula is a mathematical expression used to calculate a desired result, like finding the volume of a cylinder, or calculating surface area. It's going to use numbers, variables and the scale of the equation, just like any other equation or expression, but it's very specific to your need at hand. Some formulas use Greek letters that indicate a specific number is a constant, like the letter pi, written as π, which is used in formulas that relate to circles and cylinders: π is equal to 3.14, which is the ratio of the circumference of a circle to its diameter. More on this in the Geometry section, don’t panic!

Here are some examples of formulas:

Perimeter of a square or rectangle: P = 2(length + height)

The area of a rectangle: A = width x length

The volume of a cylinder: V = π x radius2 x height

Remember function order? Please Excuse My Dear Aunt Sally or PEMDAS?

In the perimeter formula, when you have the numbers in place, you add the length and height together before multiplying by 2, because the addition is within the parentheses.

In the volume formula you multiply the radius by itself (the exponent) before multiplying by π and then by the height.

Try these algebra exercises:

Resource 1

## Fitting words into numbers: word problems

All math problems in the real world start out as words.  For example: There are fifteen people coming to dig over a community garden this weekend. I have three shovels in the shed. How many more shovels do I need?

The words get translated into math functions so we can figure out the answer to the problem using numbers.

I have fewer shovels than people, three is less than fifteen, so 15 minus 3 gives me the answer to my question. I'm off to buy 12 shovels.

### How to frame up a word problem

Use these guidelines to translate the words into math for algebraic expressions and equations.

The words "is", "am", and "are" mean the math function will be an equals sign.

The words "and", "sum", "greater than", "longer", and "heavier" mean the math function will be a plus sign.

The words "minus", "less", "difference", "less than", and "below" mean the math function will be a minus sign.

The words "multiply", "product", and "of" mean the math function will be a multiplication sign.

The words "divide", "shared", and "per" mean the math function will be a division sign.

Carpentry Math

Basic Math from Construction Knowledge

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