Mathematics Study Guide for the TABE Test

Advanced Number System Concepts

As you move forward in your math journey, you will begin to work with numbers in deeper and more meaningful ways. Up to this point, you have focused on basic operations and number types, but math extends far beyond that. In this section, we will explore more advanced ideas within the number system, including different types of numbers, how they behave, and how they relate to one another. You will also strengthen your skills in operations, factors, multiples, and patterns.

Rational and Irrational Numbers

Progressing through the number system, we now arrive at rational and irrational numbers. A rational number is any number that can be written as a fraction with whole numbers (integers). These numbers are all rational numbers:

$\frac{3}{8}$ is a rational number because $3$ and $8$ are whole numbers.
$15$ can be written as $\frac{15}{1}$.
$7\frac{3}{5}$ can be written as $\frac{38}{5}$.
$8.6$ can be written as $\frac{86}{10}$ (or even $\frac{43}{5}$).

So far, we’ve dealt almost exclusively with rational numbers, but there are plenty of examples of numbers that are not rational numbers because they cannot be written as a fraction with whole numbers. These are known as irrational numbers, and they include many roots, like $\sqrt{3}$ or $\sqrt[3]{21}$, as well as pi ($\pi$), a concept you’ll learn more about in the geometry section.

For example, the square root of $5$ is approximately $2.236$, but that’s not the exact value. It would be impossible to write an exact decimal number for $\sqrt{5}$ because the decimal numbers go on forever—it is a non-terminating decimal. And if you square $2.236$, you will get $4.999696$. That’s close, but it’s not $5$. The same is true of the square roots of all numbers that aren’t perfect squares.

Approximating Irrational Numbers

Irrational numbers are a bit weird when put in decimal form for two reasons: the decimals never have a repeating pattern and they never end. That means we can never write the complete exact value of an irrational number. Whatever we write down will be an approximation, though we can get a really good approximation by figuring out more decimal places.

You will need to be able to compare irrational numbers to rational numbers or other irrational numbers, and a calculator won’t always be available. That might seem impossible, but with approximations it’s actually quite simple.

Say you are asked to determine which is greater, $4$ or $\sqrt{18}$. Off the top of your head, you probably don’t know what the square root of $18$ is, but there are other common square roots that you should know:

\[\sqrt{4}=2 \ \ \ \ \sqrt{9}=3 \ \ \ \ \sqrt{16}=4 \ \ \ \ \sqrt{25} = 5\]

As you know, $18$ is between $16$ and $25$, so $\sqrt{18}$ must be between $4$ and $5$. That means that $\sqrt{18}$ is more than $4$ but less than $5$, and we have our answer: $\sqrt{18}$ is greater than $4$. A number line representation should make this clearer to see:

23 Irrational Numbers on Number Line.png

Extending Multiplication and Division Skills

Many of the skills you’ve already learned use simple rational numbers. The good thing about these types of skills is that once you learn them with one- or two-digit numbers, you can do them in exactly the same way with larger numbers. We’re going to dive a little further into multiplication and division of rational numbers.

Multiplying and Dividing Fluently

You’ve learned how to multiply and divide two-digit numbers. You’ve also learned how to do those operations with decimals. Once you feel comfortable with those skills, the next step involves adding digits. You should be able to multiply two three-digits numbers, like this:

\[\begin{align} 211& \\ \times \underline{\quad 415}& \\ 1055& \\ 211\phantom{8}& \\ \underline{844\phantom{8}\phantom{8}}& \\ 87565& \\ \end{align}\]

Notice how, with each new line, the digits moved over one space. You can put zeros in those spaces if it helps you keep track of the position.

Likewise, be able to use standard long division to divide numbers with several digits the way you see below:

\[\require{enclose} \begin {array}{r} 13.5\phantom{.} \\[-3pt] 24 \enclose{longdiv}{324.0}\\[-3pt] \underline{24}\phantom{4.0.}\\[-3pt] 84\phantom{.0.}\\[-3pt] \underline{72}\phantom{.0.}\\[-3pt] 120\phantom{0}\\[-3pt] \underline{120}\phantom{0}\\[-3pt] 0\phantom{0}\\[-3pt] \end{array}\]

Factors and Multiples

When it comes to multiplying and dividing numbers, factors and multiples are two key concepts that help us understand how smaller numbers relate to larger numbers.

Factors

If you multiply two numbers together to get a third number, the two numbers you multiplied are factors of the third number. So, for instance, you know that $3$ times $4$ equals $12$. As such, we can say that $3$ and $4$ are factors of $12$. They are a factor pair of $12$.

Those are not the only factors of $12$, though. After all, multiplying $2$ by $6$ also equals $12$, so those are two more factors, as are $1$ and $12$. In total, there are six factors of $12$ and three factor pairs.

Note: A number will always have at least two factors: the number itself and $1$.

Given a number from $1$ to $100$, you should be able to list all of its factor pairs. Can you list the factor pairs of $48$?

You already know that $1$ and $48$ are factors, so that’s the first factor pair. We’re working with an even number, so we know $48$ is divisible by $2$ and another number, in this case $24$. That’s the second pair. From there, work up the numbers in order:

$3 \times 18 = 48$
$4 \times 12 = 48$
$6 \times 8 = 48$

We can stop here, because $7$ is not a factor of $48$, and we already have $8$ in one of the factor pairs. So, in total, there are five factor pairs of $48$.

Multiples:

A multiple of a specific number is any value that is reached by multiplying that number by any other number. So, for instance, just as $3$ and $4$ are factors of $12$, $12$ is a multiple of both $3$ and $4$. It’s also a multiple of $2$ and $6$ because they are factors of $12$.

Here are some factors of $30$:

$2 \times 15 = 30$
$3 \times 10 = 30$
$5 \times 6 = 30$

Since the factors here are $2, \,3,\, 5, \,6, \,10,$ and $15$, we can say that $30$ is a multiple of each one of those numbers. Even though they aren’t in the table, don’t forget that $1$ and $30$ are also factors of $30$. In total, $30$ is a multiple of eight numbers: $1, \,2, \,3, \,5,\, 6, \,10, \,15,$ and $30$.

At the same time, there are an infinite number of multiples of any of those numbers, because you can multiply a number by any number in existence.

Using Greatest Common Factors and Least Common Multiples

In our discussion of fractions earlier, we introduced the concept of the least common multiple (LCM), in which you find the smallest number of which two numbers are both factors. There is a reciprocal concept known as the greatest common factor (GCF), which means finding the largest number that is a factor of two separate numbers. Being familiar with both of these concepts is essential. Furthermore, you should be comfortable finding the LCM of any two numbers up to $12$ and the GCF of any two numbers up to $100$.

Let’s find the least common multiple of $4$ and $5$:

$\boldsymbol{4}\text{: } 4, \,8, \,12, \,16, \,20, \,24, \,28,\, 32, \,36, \,40, \,44…$
$\boldsymbol{5}\text{: } 5, \,10,\, 15, \,20, \,25,\, 30,\, 35,\, 40, \,45…$

There are an infinite number of multiples for each number, but you can stop after the first few. Now, pick out the multiples that are common (the same) to both numbers: $20, 40$. Which is the lowest one? It is $20$, so that’s the LCM of $4$ and $5$.

Now, let’s find the greatest common factor of $30$ and $48$.

The first step is listing all factors of both numbers:

$\boldsymbol{30}\text{: } 1, \,2, \,3, \,5,\, 6, \,10,\, 15,\, 30$
$\boldsymbol{48}\text{: } 1, \,2,\, 3, \,4,\, 6, \,8, \,12, \,16,\, 24, \,48$

Now, pick out the factors that are common to both numbers: $1, \,2,\, 3,\, 6$. Which is the greatest common factor? It is $6$, so that is the GCF of $30$ and $48$.

Once you feel comfortable with both concepts, you will ideally be able to figure out the multiples and factors in your head without needing to write them down.

Working with Positive and Negative Numbers

If you reach into your pocket to count your money but find nothing there, you could say you have exactly zero cents. But can you have less than zero cents? Well, in a way, yes. What if your pockets are empty but you also owe your mom the $20$ dollars she loaned you for lunch last week. You’re worse than broke. You would have to earn $20$ dollars just to get up to broke. This kind of situation is why negative numbers were invented. On an expense report, a debt of $20$ dollars can be written as $-\$20$.

When representing negative and positive numbers, we often refer to signed numbers. That is, numbers that have either the positive ($+$) or negative ($-$) sign in front of them. If there is no sign in front of the number, it is assumed to be positive.

Number Lines and Absolute Value

You learned how to represent numbers on a number line earlier. We can expand that concept to include negative numbers. Many times, instead of a number line starting with $0$ and increasing as it moves to the right, a number line will have a $0$ in the center and expand in both directions, decreasing into negative numbers on the left and increasing into positive numbers on the right:

The absolute value of a number refers to its distance from $0$. This value is always positive, even when talking about negative numbers. It is represented by two vertical lines on each side of the number:

\[\vert 12 \vert = 12 \,\text{ and }\, \vert -12 \vert =12\]

The number line is very helpful for visualizing absolute value. Just as $4$ is four spaces away from $0$, $-4$ is four spaces away from $0$. Their absolute values are the same.

If you encounter an operation inside those symbols, complete the operation first, then determine the positive value of that result, as below:

\[\vert 5-8 \vert = \vert -3 \vert = 3\]

Operations with Signed Numbers

There are some differences in the way we complete operations when using signed numbers in comparison to the basics you learned earlier. Here are the essential concepts.

Addition and Subtraction

To add integers with like signs, use the absolute values of both integers, then affix the common sign to the sum. Consider adding $-23$ and $-44$:

\[\vert-23\vert + \vert-44\vert = 23 + 44 = 67\]

Then, since they had a like sign, we affix the common sign (in this case, $-$) to the sum to get $-67$.

For two positive numbers, the process is more clear since for any positive number $x$, $\vert x \vert = x$:

\[4 + 98 = 102\]

To add integers with unlike signs, find the difference of the integers, then affix the sign of the larger integer. Regardless of which number is negative and which is positive, do the subtraction with the absolute value of each number:

Consider adding $-56$ and $42$. First, we do subtraction with the absolute value of each number, subtracting the smaller one from the bigger one:

\[\vert-56\vert - \vert 42 \vert = 56 - 42 = 14\]

Then, we affix the sign of the larger number (in this case, $-56$) to get $-14$.

Similarly, if we want to add $-15$ and $36$, we first do subtraction with the absolute value of each number, subtracting the smaller one from the bigger one:

\[\vert 36 \vert - \vert -15 \vert = 36 - 15 = 21\]

Then, we affix the sign of the larger number (in this case, $36$) to get $21$.

Remember, if the larger number is negative, then the answer is negative, but if the larger number is positive, then the answer is positive.

Subtracting integers with like signs is simple if both signs are positive:

\[35 - 22 = 13\]

That is just the basic subtraction you’ve already done before. However, if both signs are negative, the minus sign and the subtrahend’s negative sign cancel each other out and become positive:

\[-10 - (-4) = -10 + 4 = -6\]

In a similar manner, when subtracting integers with unlike signs, we can convert the problem to addition by switching the sign of the subtrahend and adding the results, using the rules of addition found above:

\[100 - (-4) =100 + (+4) = 100 + 4 = 104\] \[-698 - (29) = -698 + (-29) = -727\]

Multiplication and Division

Multiplying and dividing integers with like signs is done like the usual process you already learned, with the answer always being positive:

\[-25 \times -20 = +500 = 500\] \[14 \times 29 = 406\] \[-75 \div -5 = +15 = 15\]

Multiplying and dividing integers with unlike signs again follows the same process you already know, but the answer will always be negative:

\[400\text{,}000 \times (-1) = -400\text{,}000\] \[-95 \times 4 = -380\] \[5 \div -2 = -2.5\]

Word Problems with Positive and Negative Numbers

Understanding positive and negative numbers becomes especially important when solving real-world problems. These situations often involve gains and losses, increases and decreases, or movement above and below a reference point. Let’s look at a couple of examples.

In the morning the temperature was $-3^\circ$C. By the afternoon, the temperature increased by $9$ degrees Celsius. What is the temperature in the afternoon?

Solution

First, we identify what operation is being described by the question. We are told the starting temperature is $-3$, and there is an increase of $+9$. That’s addition with unlike signs.

As you learned, in such a situation, you subtract the absolute value of the smaller number from the absolute value of the larger number, then add on the sign of the larger number:

\[\vert9\vert - \vert -3 \vert = 9 - 3 = 6\]

Since $9$ is the bigger number, that’s the sign we put on our answer, so:

\[-3 + 9 = 6\]

The temperature in the afternoon is $6^\circ$C.

*Note: If you can do the addition in your head, you don’t need to do this whole process. But it’s good to know as a way of checking your work, and it can be helpful when dealing with much bigger numbers.

Roger has $\$25$ in his bank account. He spends $\$40$ on school supplies. What is his new account balance?

Solution

Since Roger spent money, you can think of this as an addition problem with unlike signs, in which case the $\$40$ is a negative amount, or you can think of it as a subtraction problem with like signs. Let’s perform the calculation as though it were the addition problem:

\[25 + (-40) = 25 - 40 = -15\]

The final balance is $-\$15$, which means that Roger owes $\$15$ to the bank (he is in debt).

Note: As you can see, the addition operation turned into the subtraction operation we described.

These types of problems require you to carefully interpret whether values are positive or negative and then apply the correct operation. Always think about the real-world meaning behind the numbers to guide your solution.

Patterns

Math is built on patterns, and your ability to recognize and use those patterns will determine how well you perform in higher-level math. You will find patterns in all types of situations and forms. In this section, we’ll review the types of patterns you expect to see on the TABE.

Shape Patterns

Early in elementary school, you were introduced to patterns that mainly involved a series of shapes. You were then asked to continue the pattern. For example, imagine being given this pattern:

24 Shape Pattern.jpg

By studying the order of the shapes in the established pattern, you can determine that the shape needed to fill the blank is a square because that is the shape that always comes after the triangle.

For the TABE, you will need to be able to recognize two more advanced types of patterns: arithmetic sequences and geometric sequences.

You know that a pattern is an arithmetic sequence if the same value is added or subtracted to an element to get the next element. The series $5, \,8, \,11, \,14, \,17, \,20 …$ is an arithmetic sequence because $3$ is added to get the next element.

Note: The three dots, known as an ellipsis, means the sequence continues on forever. If there isn’t an ellipsis, that means you are seeing the complete series.

A geometric sequence is one in which the same value is multiplied or divided by an element to get to the next element. The series $48, \,24, \,12, \, 6, \,3$ is a geometric sequence because each element is divided by $2$ to produce the subsequent element.

There are more complicated patterns to explore, such as square number sequences, Fibonacci sequences, triangular sequences, etc. The patterns presented here, however, are the essentials.

Calculating Patterns

Some problems will ask you to find a specific element within a series or sequence. This element will often be represented as the $n$th element, with $n$ simply being a variable like $x$.

Calculating Arithmetic Sequences

Here is the formula to help you predict the $n$th element of an arithmetic sequence without having to do repetitive addition:

\[x_n = x_1 + d (n -1)\]

where $x_n$ is the $n$th element, $x_1$ is the first element, and $d$ is the common difference between elements.

Suppose we have this sequence:

\[5,\,8,\,11,\,14,\,17…\]

Quick addition reveals this is an arithmetic sequence that goes up by $3$ each time. Knowing that, we can solve for the $100$th element using the above formula:

\[x_{100} = 5 + 3(100-1) = 5 + 297 = 302\]

The Sum of Arithmetic Sequences

Sometimes, you may need to find the sum ($S_n$) of an arithmetic sequence up to a certain value. Again, this would be a lengthy process if you just added each value, but there is a formula that makes it much quicker:

\[S_n = \frac {n}{2} [(2 \cdot x_1) + d(n -1)]\]

We can now find the sum of the first six elements of the above sequence without adding them one by one:

\[S_6 = \frac {6}{2} [(2 \cdot 5) + 3(5)] = 3 (10 + 15) = 75\]

Calculating Geometric Sequences

There is also a formula for finding the $n$th element of a geometric sequence:

\[x_n = x_1 \cdot r^{(n-1)}\]

where $x_n$ is the $n$th element, $x_1$ is the first element, and $r$ is the common ratio (or common divisor).

Suppose we have this sequence:

\[3, \,6,\, 12, \,24, \,48, \,96…\]

For this given geometric sequence, in which each element is multiplied by $2$,, we can compute the $12$th element:

\[x_{12} = 3 \cdot 2^{11} = 6\text{,}144\]

The Sum of Geometric Sequences

We also have a formula for the sum of a geometric sequence:

\[S_n = x_1 \cdot \frac{(1 - r^n)}{(1 - r)}\]

To find the sum of the first six elements of the above sequence, we use the formula:

\[S_6 = 3 \cdot \frac{(1-2^6)}{(1-2)} = 3 \cdot \frac{(1-64)}{(1-2)} = 3 \cdot \frac{(-63)}{(-1)} = 189\]

Creating Patterns

You should also be able to make up a sequence rule and generate terms of the sequence. As we just saw above, an arithmetic sequence can start with any number and add any number over and over to get a sequence. So, pick a starting number, say $6$, and generate an arithmetic sequence by using $4$ to add to each term:

\[6, \,10, \,14,\, 18, \,22, \,26,\, 30…\]

Notice how each number is even? Well, what happens when we start with $7$ and use $4$ to add to each term:

\[7, \,11,\, 15, \,19, \,23,\, 27, \,31…\]

This time, each term is odd. This makes sense because our common difference is even. If we had used an odd common difference, the values would have switched back and forth between odd and even.

You can also create your own geometric sequence. Let’s start with $3$ and multiply each term by $-3$:

\[3,\, -9, \,27,\,- 81,\, 243 …\]

Notice how the numbers switch from negative to positive. Any time you multiply by a negative number, the sign changes. Since you are always multiplying by a $-3$ in this sequence, the signs always alternate.