Mathematics Study Guide for the TABE Test

Numbers and Operations: Base Ten—Operations 1

Numbers are combined and separated in a variety of different ways. These combinations are called operations. The common symbols \(+, -, \times,\) and \(\div\) are called operators and tell us what to do with the two numbers.

In this section, we’re going to go over two of the essential math operations using whole numbers: addition and subtraction. Addition enables us to combine numbers, and subtraction is a basic method of separating numbers.

Addition

Basic addition is the operation that combines two or more numbers into one larger number called the sum or total. You should be able to use column addition to add numbers from one digit to four digits as shown in the sections below. In a word problem, watch for the key words sum, total, added to, increased by, and plus, which tell you that you’ll need to use addition.

Note: It is possible to add negative numbers, which will result in a smaller number, but you’ll learn about that later.

Addition in Columns

Adding whole numbers without the aid of a calculator is simplified by first writing the addends (the numbers to be added) vertically. Align digits according to their place value: ones in the first, rightmost column, followed by the tens to the left, then hundreds, and so on.

To add \(1\text{,}045\) and \(34\), write the addends vertically, carefully aligning the digits according to place value:

\[\begin{align} 1\text{,}045& \\ \underline{+\quad 34}& \\ \end{align}\]

Start adding from the ones column, then go to the left one place value at a time. Write the sum of each column below the horizontal line, aligned by place value. In this example, the ones column sum is \(9\), the tens column sum is \(7\), the hundreds is \(0\), and the thousands is \(1\). As such, the sum of \(1\text{,}045\) and \(34\) is \(1\text{,}079\).

Once you are comfortable doing addition with two numbers, you can follow the same process for addition involving three or more numbers.

Regrouping (Decomposing/Composing)

To effectively add and subtract numbers with more than one digit, it is sometimes necessary to regroup the components of a number without changing its value. You can use your knowledge of expanded numbers (discussed above) as you do this. For example, we can decompose a number like this:

• You know that \(42\) equals four tens plus two ones (\(40 + 2\)).
• You also know that one ten equals \(10\) ones.
• So, we can also say that \(42\) equals three tens plus \(12\) ones (\(30 + 12\)).

To compose a number is to reverse the process. Say you have one ten and five ones (\(10 + 5\) in expanded form). You know that one ten is also \(10\) ones, so you really have \(15\) ones, or just \(15\).

If the sum of any of the columns is more than \(9\), as in the following example, you will need to regroup the digits:

\[\begin{align} 1\text{,}057& \\ \underline{+\quad 3\text{,}145}& \\ \end{align}\]

To do this, add the first column, which gives us \(7 + 5 = 12\). You can’t have a double-digit number in a column, but you can think of \(12\) as its expanded form, \(10 + 2\) (or one ten and two ones), so write \(2\) below the horizontal line in the ones column, and write \(1\) on top of the tens column:

\[\begin{align} \phantom{1,0}1\phantom{7}&\\ 1\text{,}057& \\ \underline{+\quad 3\text{,}145}& \\ 2& \\ \end{align}\]

Then, proceed to add the tens column including the \(1\) on top: \(1 + 5 + 4 = 10\) (which is one hundred and zero tens). So, we write \(0\) in the tens column and \(1\) on top of the hundreds column:

\[\begin{align} \phantom{1,}11\phantom{7}&\\ 1\text{,}057& \\ \underline{+\quad3\text{,}145}& \\ 02& \\ \end{align}\]

Now, add the hundreds column (\(1 + 0 + 1 = 2\)), followed by the thousands column (\(1 + 3 = 4\)). Therefore, the sum of \(1\text{,}057\) and \(3\text{,}145\) is \(4\text{,}202\):

\[\begin{align} \phantom{1,}11\phantom{7}&\\ 1\text{,}057& \\ \underline{+\quad3\text{,}145}& \\ 4\text{,}202& \\ \end{align}\]

Memorizing Addition Facts

For fluency in future math endeavors, you should memorize the basic math facts for adding single-digit numbers. This means knowing the sums for every addition operation from \(0 + 0\) to \(12 + 12\). You should be able to think of the answer instantly without making notes or counting on your fingers. This is a simple task, but very important. Later, when you have to do longer and more involved calculations, you’ll avoid having to stop to get a calculator or write the problem down, either of which can cause you to lose your place in the current problem. Knowing these addition facts will also enable you to answer simple subtraction problems quickly.

You can create a list of the basic addition facts or you can use a table like the one below. Simply pick a number in the top row and a number along the left side. Determine where the rows intersect and you’ll find the answer.

Extra Addition Techniques

You will also need to be able to add up to three whole numbers that have a sum not exceeding \(20\) in your head. If you’re struggling with this, there are a few techniques you can try that may help you learn the process.

One such technique is using something physical such as an object or drawing to see the logic of addition.

Suppose Angela owns three bracelets, Claire owns six, and Maria owns four. How many bracelets do they own all together? You could model these numbers by using disks, as shown in the image below:

With that model before you, you could then just count the disks. Let’s start from the left:

\[3 \text{ disks } \ \ \ \ \ \ 6 \text{ disks } \ \ \ \ \ \ 4\text{ disks }\] \[1, 2, 3 \ \ \ \ 4, 5, 6, 7, 8, 9 \ \ \ \ 10, 11, 12, 13\]

The sum is:

\[3+6+4=13\]

Another useful technique is known as making tens. Here’s how it works.

Suppose you had the problem \(9 + 8 + 2\). If you add the \(8\) and \(2\) first, you will get \(9 + 10\), which is pretty easy to add and get \(19\). That’s a bit easier to do than adding \(9+8\) first.

Look for numbers that will add up to \(10\), such as \(1\) and \(9\), \(2\) and \(8\), \(3\) and \(7\), \(4\) and \(6\), and \(5\) and \(5\). This is a good trick, but you won’t be able to use it in every problem.

Another mental trick doesn’t have an official name, but you might think of it as making twins. Adding two of the same number is easier for many of us than adding two different numbers. Make use of that. If you have \(8+9\), mentally break up the \(9\) into \(8\) and \(1\). Now you have \(8 + 8 + 1\). Now, you might know that \(8 + 8 = 16\), so now just add the \(1\) and get \(17\).

Subtraction

Subtraction is the operation that finds the difference between two numbers. It is the opposite of addition. You can think of it as finding out how much larger one number is than another. In a word problem, some of the key words that tell you to subtract are difference, less or less than, more or more than, minus, and decrease.

Models of Subtraction

To help visualize a simple subtraction problem, we generally begin with concrete models. Here are a couple of ideas to use for the problem \(7 - 3\):

• Physically gather seven objects, then remove three. You’ll find that there are four objects remaining. This is your answer.

• Sketch seven objects of any shape, as we’ve done with circles below, and cross out the number to be subtracted:

Subtraction in Columns

As with other arithmetic operations, subtraction is much easier to perform if done vertically. Start subtracting from the ones column, then move to the left (the larger place values). In \(76 - 25\), \(76\) is the minuend and \(25\) is the subtrahend. When setting up a vertical subtraction problem, write the minuend on top and the subtrahend below, carefully aligning the digits that correspond to the same place value:

\[\begin{align} 76& \\ \underline{-\quad 25}& \\ \end{align}\]

Write the difference of the digits in each column under the horizontal line. The difference for the ones column is \(6 – 5 = 1\), and the difference for the tens column is \(7 – 2 = 5\). The result of subtracting \(25\) from \(76\) is, therefore, \(51\):

\[\begin{align} 76& \\ \underline{-\quad 25}& \\ 51& \\ \end{align}\]

This calculation can be verified by adding the resulting answer, \(51\), to the portion subtracted, \(25\), to arrive back at the original whole number:

\[51+25=76\]

Regrouping (Composing/Decomposing)

As with addition, when any digit of the subtrahend (bottom) is larger than the minuend (top), there is a need to perform regrouping, as in this example:

\[\begin{align} 2\text{,}061& \\ \underline{-\quad 543}& \\ \end{align}\]

To make the process easier to understand, we’ll expand each number to show the value of each digit:

\[\begin{align} {2\text{,}000 + 0 + 60 + 1}& \\ \underline{-\quad 500 + 40 + 3}& \\ \end{align}\]

Since we cannot subtract a number from another number that is smaller within a column (such as \(3\) from \(1\) and \(500\) from \(0\)), we “borrow” one unit from the number on the left column. We regroup the minuend by borrowing one unit from the tens column and adding that to the number in the ones column and by borrowing one unit from the thousands column and adding that to the number in the hundreds column:

\[\begin{align} {1\text{,}000 + 1\text{,}000 + 50 + 11}& \\ \underline{-\quad (500 + 40 + 3)}& \\ {1\text{,}000 + 500 + 10 + 8}& \\ \end{align}\]

Thus, our answer is \(1\text{,}518\).

Using a Number Line

A number line is a line with evenly spaced marks that we label with numbers that get bigger as they go to the right. Number lines give us a picture of what we are doing when we use addition and subtraction.

Representing Numbers

To represent a whole number on a number line, we draw a line from \(0\) to that whole number on the line. For example, the second figure below represents the number \(5\), and the third figure represents the number \(2\):

Counting On and Counting Back

We can show addition by drawing an arrow for each number as shown below. We write these above the line to help make the arrows clear. When adding, the arrows point to the right. This figure below shows \(2 + 4 = 6\):

This is known as “counting on” because you start with a number (your first addend) and count past it (your second addend) to arrive at your sum.

To show subtraction, such as \(4 - 2 = 2\), we draw the larger number going to the right and the subtracted number going left from the first number:

Subtracting on a number line is referred to as “counting back” because you start by moving up to your minuend and counting back from it the value of your subtrahend to arrive at the difference.