In order to succeed at studying math, you will first need to know the relevant vocabulary. This section summarises the necessary vocabulary, while equipping you with the skills required to tackle future topics in this study guide.
Posted in 4th December, 2014 | Mathematics KnowledgeExam Preparation
Exam Preparation
In this section, we aim to equip you with the necessary vocabulary and understanding required to understand the questions in the next section. Just as in every other science, mathematics requires you to grasp certain words, as it’s these words that define what it is you’re supposed to be doing. We are all familiar with being presented with a particular math problem, only not to understand the meaning of the language involved. This section should add some degree of clarification on this matter, allowing you to grapple and tackle with questions in a more confident manner. It may take some time to remember this terminology, but this becomes easier once you begin practicing the sample questions.
This section will go through seven basic mathematical terms employed in all future sections of this guide. Accompanying each term is an example, as this is designed to make it much easier to recall in the future. You should have a pen and paper to hand at this point, as it would be useful to take notes as you go along, revising these notes at the end of your study session to keep these terms fresh in your mind. We begin at once – starting with the common term, ‘Integer’, which continually appears throughout the ASVAB Mathematical Knowledge exam.
Understanding Terminology
- Integer – An integer is any whole number which is positive, negative, or zero. So, take ‘5’ for example – this is a ‘whole number’ as it isn’t a decimalized number (5.4). As a result, this and every other whole number is described as an integer.
- Exponent and Base – An exponent is a number to the power of another number. Take three multiplied by two; this can be written mathematically as 32. The smaller number here, ‘2’, is known as the exponent. It describes something about the number ‘3’. Similarly, 14 x 14 x 14 x 14 can be written mathematically as 14^4 –where the number ‘4’ is the exponent. When the exponent is 2, as in the example above, we describe is as ‘squared’ – in this case, three squared. If the exponent is 3, as in the example 5^3, we describe it as five cubed, which is the same as 5 x 5 x 5. The integer, when raised to an exponent in this way, is known as the base.
- Prime and Composite Numbers – Any number which can only be divided by itself and 1 is known as a prime number. Take the number 7, for example, where the only numbers that can evenly be divided into it are 7 and 1 – if we used any other number we would arrive at a decimal answer (7 divided by 3 is 2.3, for example). A composite number, on the other hand, is a number that can be divided by multiple factors. Take the number 6, for example, where the numbers that can divide evenly into it are 1, 2, 3, and 6.
- Square Root – We discussed exponents above, where we determined that 52 is also called ‘five squared’. The answer to this question is, obviously, 25. The square root allows us to figure out the original number multiplied by itself. The square root of 64, for example, is 8 – because 8 multiplied by itself is 64. The square root is known by the symbol √.
- Factorial – This mathematical concept allows you to calculate combinations. If we have a competition between 7 people and asked what the probability is that one of those winners should win the competition – we would use the factorial symbol (!). So, 7! is the same as writing 7 x 6 x 5 x 4 x 3 x 2 x 1. When you see the exclamation mark, you should be aware you will have to expand the number back to one, as highlighted in the example above.
- Reciprocal – Let’s recall something fundamental, that an integer is the same as its number divided by 1. In other words 5 = 5⁄1 – these two forms are exactly the same. If you are asked to find the reciprocal of a number, you simply reverse the number. In this case, if you were asked to find the reciprocal of 5, the answer would be 1⁄5. If the number is a negative number, the reciprocal will retain the negative form.
- Rounding – This is the word we use to tidy up numbers. So, for example, if you were asked: round the following to the nearest whole number 4.37? The answer, in this case, is either 4 or 5 – we simply need to eliminate the .37. Given 3 is nearer to 0, we can round 4.37 down to 4. If, however, the number was 4.71, we would round the number up to 5. By rounding numbers to whole numbers, we make the number much more manageable. You might be asked, though, the following: round 4.59174 to two decimal places? This means we have to look at the third number after the decimal point – 1. Given 1 is nearer to zero, the answer would be 4.59.
These seven sections may appear daunting at first. Yet, you will acclimate to using these terms once you begin answering questions. It’s only by answering questions do you have any chance of becoming more and more familiar with these terms. The next part of our exam preparation involves knowing the BOMDAS rule. This rule allows you to determine the correct sequence of events when solving a particular mathematical problem. You cannot guess which part of an equation you want to solve first – it needs to follow a structured and orderly approach.
The BOMDAS Rule
What does the BOMDAS Rule stand for? Let’s take a look:
- Brackets
- Of
- Multiplication
- Division
- Addition
- Subtraction
This means that when faced with a math problem, you must solve the problem in this precise order. Are there any brackets in the question? No? Move on. Are there any ‘of’ or ‘multiplication’ in the question? No? Move on. So, you continue the sequence until you have solved the entire problem. Let’s glance at an example problem below to highlight how it should be used correctly. It’s quite easy to ignore this rule and solve a math problem without it, though you would be risking quite a lot unless you are familiar with how to work out that type of problem in the first place.
Simplify: 4 – 6 x 8 ÷ 2(9 + 8x)
Where do you start? Take a look at the BOMDAS Rule above and see which steps you would take, and in which order you would take these steps. The answer to this problem is examined in a step-by-step process below. Again, keep the rule in mind as we continue to work through the answer:
- 4 – 6 x 8 ÷ 2(8 + 8x) [Brackets]
- 4 – 6 x 8 ÷ 16 + 16x [Multiplication]
- 4 – 48 ÷ 16 + 16x [Division]
- 4 – 3 + 16x [Subtraction]
- 1 + 16x
Take note of how the question is phrased – simplify. This word means reduce the equation to its smallest possible state. When it comes to (1 + 16x), we cannot simplify it any more, as you cannot add one term with an x and one term without an x. Also note in this example there is no active addition step – which means we can skip it and move onto the last step – subtraction. Don’t be afraid to skip a step in the BOMDAS Rule if there is no such step involved in solving the math problem.
Solving Simple Equations
In this final segment of exam preparation, we will take a look at solving simple equations. There are certain fundamental rules you need to follow regardless of the algebraic equation involved. You must commit these rules to memory, as they will appear time and time again. One of the more rudimentary examples is the idea of dealing with both sides of an equation, and to isolate the relevant factor. This will all become clear as we go through some examples below. Whatever your existing level of algebra, it’s worth revising these principles as they are endemic in many areas of the ASVAB Mathematics Knowledge exam.
Take a look at the following question:
Simplify: x^2 + 4x + 8 = 9x + 2x^2 – 3
- Only ‘like terms’ can be added or subtracted. In other words, you cannot add an integer (such as 3) to an x value. You can, of course, add x values together and add integers together – but they must remain separate at all times.
- When you move a variable from one side of the equation to the other side, the sign of that variable will change to its opposite. In the example above, if I move (+8) to the right-hand side of the equation, it will change to a value of (-8).
- You cannot add x values if they have different exponents. So, you cannot add x and x^3, for example.
With these rules in mind, let’s go about solving the problem in a step-by-step way:
- x^2 + 4x + 8 = 9x + 2x^2 – 3
- x^2 + 4x + 8 – 9x – 2x^2 + 3 = 0
- x^2 – 2x^2 + 4x -9x + 8 + 3 = 0
- -1x^2 -5x + 11 = 0
- x^2 + 5x – 11 = 0
In this example, we brought everything over from the right-hand side of the equation, and added it to the left-hand side of the equation. This meant we had to change the sign of each value, all the while taking each value from its respective other – in other words, we grouped integers, x values, and x^2 values, and dealt with them separately. This is in full accord with the rule above. Also take note of the final step at the end, where we eliminated the negative x^2 value by multiplying everything by -1. This isn’t a necessary step, per se, but it is considered the norm to ‘clean up’ the final equation, which involves eliminating a negative value in the first variable.
Take a look at this question:
Solve for x: (2x+6)/3 = 4x + 9
This is a slightly different example, as it concerns the use of a fraction. The point we wish to raise here is that we must eliminate the fraction to have any real chance of solving the problem. When faced with a situation such as this, all you need to do is multiply the denominator (the name given to the value of the bottom of a fraction) by the other side of the equation – in this case it involves multiplying 2 by (4x + 8). With this in mind, let’s take a look at how to solve the question in a step-by-step manner:
- (16x + 6) / 2 = 4x + 8
- 16x + 6 = 2(4x + 8)
- 16x + 6 = 8x + 16
- 16x – 8x = 16 – 6
- 8x = 10
- x = 10⁄8
- x = 5/4
- x = 1 1/4
Note how we immediately sought to eliminate the fraction by multiplying the other side of the equation by the denominator. After that, we eliminated the brackets by multiplying each term in the brackets by 2. Given this question asked us to “Solve for x”, we need to isolate x on one side of the equation, and place the integers on the other side of the equation. After resolving this issue, we needed to eliminate the 8 from 8x, as we were asked to solve for x. This means we had to divide the remaining term (10), by 8. We then simplified the fraction by dividing the top and the bottom by 2, and thereafter simplified further by realizing 5/4 is the same as 4/4 + ¼. 4/4 is the same as 1. It might seem like a long and difficult example, but its underlying principles apply across the algebraic world.
This section has aimed to build your arsenal of basic mathematical principles. This involved taking a quick look at the common range of vocabulary that infuses the subject of math. Thereafter, we emphasized the importance of the BOMDAS Rule, as recalling this simple acronym will massively assist you when it comes to figuring out algebraic problems. Toward the end of this section, we looked at basic principles of algebra, more specifically the necessary operations you need to remember for your forthcoming study in this area. With these tools at hand, we can now advance onto study strategies, before going on to look at subject matter topics themselves.
The next section begins out analysis of mathematics study strategies – meaning you will be taught how to study math in the most effective way.
Share
