Mathematics: Levels E, M, D, and A Study Guide for the TABE Test

Number and Operations in Base Ten: Part 1

Percentage of Test Level Specifically Assessing Number and Operations in Base Ten (— = Assumed)

L	E	M	D	A
40%	28%	15%	—	—

This section deals with basic numbers and mathematical operations with them.

Counting

Be able to do basic counting from 1 to 1,000, as well as counting by 5s, 10s, and 100s.

By 5s: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50…
By 10s: 10, 20, 30, 40, 50, 60…
By 100s: 100, 200, 300, 400, 500, 600…

Place Value

The table above shows values for each decimal place. The first row shows the number 5,820. The table shows you that the digits in this number stand for 5 thousands, 8 hundreds, 2 tens, and 0 ones.

The second row shows that the number 26.227 means 2 tens, 6 ones, 2 tenths, 2 hundredths, and 7 thousandths.

The third row shows that the number 352,000 means 3 hundred thousands, 5 ten thousands, 2 thousands, and 0 hundreds, tens, and ones.

Using a Place Value Chart

Something to notice in the place value table above is that as you go to the left, each place is ten times the one to its right: 100 = 10 x 10, 1,000 = 10 x 100, 10,000 = 10 x 1,000, and so forth. Likewise, as you go to the right, each place is 1/10 of the one to its left.

Comparing Decimals

Remember, place values of numbers to the right of the decimal point get smaller as you go farther to the right, so that should help you to make comparisons between decimal numbers. For example, 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.

Which is smaller, \(0.0005\) or \(0.003\)?

The answer 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 \(3\), so \(0.0005\) is smaller.

A simple way to do this is: Circle 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. 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 larger.

What if both digits are in the same place? The larger one wins.

What if both digits are the same number? Then you look at the place to the right and see which one of those is larger. That number is larger.

It may help visually in these questions to write the numbers with one above the other, lining up the decimal points. Often, you can glance at it and tell right away which one is larger.

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

Another thing to remember: the symbols < (less than), = (equal to), and > (greater than).

Picture answering a question like this:

True or False? \(0.0062 > 0.01\)

Answer: False because the statement says that \(0.0062\) is greater than \(0.01\), but the \(1\) is farther to the left than the \(6\).

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

Reading and Writing Numbers

In the sections below you will see a few different ways to write numbers.

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 1,000. Don’t forget that when you hit 1,000, you should start putting commas before every three places to help with reading big numbers.

Number Words

We looked at place value earlier and that will help with writing numbers in words. We will just look at numbers up to four digits.

• One-digit numbers are easy, 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 plus the ones numeral, so 32 is thirty-two and 98 is ninety-eight. Notice that hyphens are used.

• 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 would be read as four hundred fifteen and 802 would be read as eight hundred two. Notice that the word “and” isn’t used.

Expanded Form

Using what you know about place values, be able to write numbers up to 1,000 in an expanded form. Here are two examples:

• The number 489 can be written 4 x 100 + 8 x 10 + 9 x 1.

• The number 1,000 can be written 1 x 1,000 + 0 x 100 + 0 x 10 + 0 x 1.

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:

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

Using Symbols to Compare Numbers

In the section on comparing decimals, we first discussed using three comparison symbols in math:

< less than
= equal to
> greater than

The steps to compare whole numbers are the same as for comparing decimals. Here they are again:

Circle 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. Which of these two is larger?

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

The \(1\) is farther left than the \(9\), so \(135\) is larger.

What if both digits are in the same place? The larger one wins.

What if both digits are the same number? Then you look at the place to the right and see which one of those numerals is larger. That one is the larger number.