Mathematics Study Guide for the TABE Test

General Information

The TABE test levels are L, E, M, D, and A, with each letter representing an increase in difficulty. As you progress through the levels, you will be expected to have a more developed understanding of the concepts and address questions at an increased level of competency. All skills from lower test levels may be assessed at the higher test levels.

The TABE Testing Procedure

Before everything else, you will be given a locator test in mathematics that will determine the level of your first TABE Mathematics test. This short test is given in two 10-minute parts.

No matter which TABE level you take after the locator test, you will have 60 minutes to complete the entire level E, M, D, or A test. The math tests are given in two parts: a 45-minute section followed by a 15-minute section. These combine to equal 60 minutes of testing time.

Using Our TABE Mathematics Study Guide

This edition of the TABE Mathematics test divides math concepts into only four categories, or broad areas of math, with these approximate percentages of coverage:

• Numbers and Operations—27.5%
• Algebraic Concepts—27.5%
• Geometry—20%
• Measurement, Data, and Probability—25%

Note: You will find that some concepts, such as graphing, are present in more than one category of math on this test.

The level L math test is read out loud to students. We do not provide preparation for that level here, as it covers only the most basic elementary-level mathematics skills.

• In this study guide, concepts are discussed across the range of difficulty from level E to level A. The topics within each concept area are mostly arranged in order of difficulty and in the order they would have been taught in school.

  • Work through the easiest parts first, then move on to the more complex ones as you gain skills and as you advance through the testing levels.

  • As you use our practice questions, be sure to come back to this study guide for more information about any questions you struggled with.

Use of a Calculator

There are two parts of the TABE Mathematics test: calculations and applied math. You will have access to the DESMOS calculator during the applied math part. This calculator is frequently used in classrooms in the US. The use of this calculator is explained in this reference sheet.

Formula Sheets

When you take the TABE locator test and any TABE Mathematics test, you will be given a list of formulas to use. The number and types of formulas differ for each test level. Look at this list of TABE math formulas so you’ll know what to expect. (There is one page for each test level.)

Numbers and Operations: Base Ten—Number Values

TABE 13&14 has combined several previous topic areas into one called “Numbers and Operations.” We will present these topics across separate sections. These are the concept areas currently covered in this broad Numbers and Operations topic:

• Number and Operations in Base Ten
• Numbers and Operations: Fractions
• Ratios and Proportional Relationships
• The Number System

To begin, you should know that the very foundation of math is the base ten system, which is a way of representing numbers using the following \(10\) digits:

\[0, \, 1, \, 2, \, 3, \, 4, \, 5, \, 6, \, 7, \, 8, \, 9\]

In this system, the value of a digit depends not only on the digit itself but also on its position, or place, within a number. Each place represents a power of \(10\), meaning that as you move from right to left, each place value becomes \(10\) times larger. As such, the number \(90\) is \(10\) times larger than \(9\), and \(900\) is \(10\) times larger than \(90\).

This structure allows us to write and work with both very large and very small numbers in an organized and efficient way. The base ten system is used in everyday life for counting, measuring, and calculating, making it an essential concept to understand for success on this test and beyond.

Counting

Counting is one of the first processes a person learns. The ability to count is fundamental to everything from shopping to exercise to computer programming. You will find that the ability to do basic counting from \(1\) to \(1\text{,}000\) is essential for the TABE.

You should also be able to count by factors of \(5\), \(10\), and \(100\):

• \(\bf{5}\)s: \(5, 10, 15, 20, 25, 30, 35, 40, 45, 50…\)
• \(\bf{10}\)s: \(10, 20, 30, 40, 50, 60…\)
• \(\bf{100}\)s: \(100, 200, 300, 400, 500, 600…\)

Place Value

Another key concept is place value, which tells you the relative value of a single digit within a number. The chart below shows the place values of numbers, including the first few for each decimal place (values between \(0\) and \(1\)):

The first row shows the place values of each digit in the number \(5\text{,}820\). The table explains that the digits in this number stand for five thousands, eight hundreds, two tens, and no ones. As you can see, \(5\text{,}820\) is a whole number with no decimal values.

The second row shows that the number \(26.227\) means two tens, six ones, two tenths, two hundredths, and seven thousandths. Notice the “ths” on the end of the place values on the right side of the decimal. That lets you know those values are fractional amounts (a concept we’ll cover in greater detail later in this guide).

The third row shows that the number \(352\text{,}000\) means three hundred thousands, five ten thousands, two thousands, and no hundreds, tens, or ones. Again, this is a whole number.

Using a Place Value Chart

In the place value chart above, as you go to the left, each place is \(10\) times the one to its right: \(100 = 10 \times 10\), \(1\text{,}000 = 10 \times 100\), \(10\text{,}000 = 10 \times 1\text{,}000\), and so forth. Likewise, as you go to the right, each place is one-tenth of the one to its left. That is how the base ten system works. Each place value is a multiple of \(10\) greater than the place value to its right.

Comparing Decimals

Place values to the right of the decimal point get smaller as you go farther to the right, which allows us to make comparisons between decimal numbers. For example, can you tell which is smaller, \(0.03\) or \(0.003\)? It must be \(0.003\), because that \(3\) is farther to the right of the decimal point.

What about \(0.0005\) and \(0.003\)? The smaller number is \(0.0005\). It doesn’t matter that \(5\) is bigger than \(3\), the \(5\) is farther to the right of the decimal point than the \(3\).

Sometimes it might not be immediately obvious, so one simple trick is circling the farthest left digit that’s not a zero in each number. The one you circled that is farther to the left belongs to the largest number. For instance, which of these two is larger?

\[\require{enclose} 0.0\enclose{circle}2 5 \text{ or } 0.\enclose{circle}1 04\]

The \(1\) is farther left than the \(2\), so \(0.104\) is the larger number.

If both digits are in the same place, then the larger digit wins. But what if both digits are the same and in the same place value? Then you look at the place to the right and see which one of those digits is larger. That number is larger. You continue doing this until you find two that digits are different.

It can help visually in these questions to write the numbers with one above the other, lining up the decimal points. When you do it this way, you can often glance at them and tell right away which one is larger:

\[\begin{array}{l} 0.003\\ 0.0005\\ \end {array}\]

Reading and Writing Numbers

On the TABE Mathematics test, you will see numbers presented in various forms. You will need to be familiar with all of these forms and be comfortable reading and writing them.

Numerals

Numbers are most commonly written using the numerals that you have been using for years, and you should be familiar with reading and writing them up to at least \(1\text{,}000\). Don’t forget that when you hit \(1\text{,}000\), you should put commas before every three places to help with reading larger numbers. After all, it’s much easier to recognize the value of \(2\text{,}067\text{,}104\) than it is \(2067104\).

Number Words

We looked at place value earlier, and that will help with writing numbers in words. We will just consider numbers up to three digits:

• One-digit numbers are easy to read correctly; just say the numeral: “one, two, three,” and so on.

• Two-digit numbers less than \(20\) (twenty) aren’t written in a consistent way like numbers greater than \(20\) are. So, for instance, \(10\) is written as “ten,” \(11\) is “eleven,” and \(12\) is “twelve.” After \(12\), the numbers end with teen, but whereas most of the numbers simply add teen to the ones numeral (\(14\) is “fourteen,” for instance), there are three exceptions: \(13\) is “thirteen,” \(15\) is “fifteen,” and \(18\) is “eighteen” (no double t).

• Two-digit numbers from twenty on up use words that end in ty (e.g., “twenty, thirty, forty”) plus the ones numeral. So, \(32\) is “thirty-two” and \(98\) is “ninety-eight.” Notice that hyphens are included.

• Three-digit numbers start with some number from one to nine plus the word hundred, then the two-digit number that follows. The number \(415\) is read as “four hundred fifteen,” while \(802\) is read as “eight hundred two.” Notice that the word and isn’t used after the hundred. Sometimes people say it, but it’s technically incorrect.

• Four-digit numbers start with some number from one to nine plus the word thousand, then the three-digit number that follows. The number \(2\text{,}138\) is read as “two thousand, one hundred thirty-eight.” Notice that there is no and after thousand or hundred.

Expanded Form

The expanded form of a number expresses that number as the sum of its place values. You should be able to write numbers up to \(1\text{,}000\) in an expanded form. Here are two examples:

• The number \(489\) can be written as \(400 + 80 + 9\).
• The number \(1\text{,}010\) can be written as \(1\text{,}000 + 0 + 10 + 0\).

Decimals

Using place values will also let you write decimal numbers in word form. If we saw the number \(4.56\), most of us would just say “four point fifty-six.” It’s hard to argue with that, but you should be able to word it this way: “four and fifty-six hundredths.” Likewise, \(12.022\) would be “twelve and twenty-two thousandths.” Here, the word “and” is used (in the place of the decimal).

Here are the steps to follow:

1. Read the whole number to the left of the decimal point.
2. Say “and.”
3. Read everything to the right of the decimal point as a whole number.
4. End by saying the place value of the last digit to the right.

Take the number \(19.198\), for example. This is how you read it aloud:

1. Say “nineteen.”
2. Say “and.”
3. Say “one hundred ninety-eight.”
4. Say “thousandths.”

Using Symbols to Compare Numbers

One of the most important basic skills you need in math is the ability to compare numbers and determine which is larger or smaller. There are three common symbols that help illustrate these relationships between numbers:

• \(<\) means “less than”
• \(=\) means “equal to”
• \(>\) means “greater than”

Sometimes the comparison will be obvious. For instance, \(10\text{,}000\) is clearly greater than \(10\) (which we can write as \(10\text{,}000>10\)).

In a situation where the comparison is not as obvious, you can use the trick you learned above and start by circling the farthest left digit that’s not a zero in each number. The one you circled that is farther to the left belongs to the largest number. If you were asked to compare \(135\) and \(97\), you could do the following:

\[\require{enclose} \enclose{circle}1 35 \text{ or } \enclose{circle}97\]

The \(1\) is farther left than the \(9\), so \(135\) is the larger of the two numbers:

\[135>97\]

Note: Remember that with a whole number, there is always an implied decimal point at the end. You are counting left from that point.

When the farthest left digits are in the same place in both numbers, the larger of the two numbers is the one with the larger digit. That’s why \(50\) is bigger than \(40\). If those two digits are the same, as in \(54\) and \(50\), you move to the right and compare those digits (\(54>50\)). It doesn’t matter how many digits the number has, the process is the same.

You’ve already seen how this process works with decimal numbers. So, for instance, if you are given the numbers \(0.0062\) or \(0.01\), you can circle the relevant digits:

\[\require{enclose} 0.00\enclose{circle}6 2 \text{ or } 0.0\enclose{circle}1\]

You can see that \(0.0062\) is less than \(0.01\), which we represent as:

\[0.0062<0.01\]

There are times when a value might be either equal to another value or less than or greater than that value. In such circumstances, you use these symbols:

• \(\le\) means “less than or equal to”
• \(\ge\) means “greater than or equal to”

Note: We’ll explore these concepts in greater depth later in this document.