As an expression involving the subtraction sign, Johnny’s net worth is now .
Activities for this section:
Let’s look at subtraction with integers using the same three main ideas we used with addition: checks and bills, number lines, and investigating patterns. Remember that we are going to build from what we already know about subtraction, so you may want to re-read these sections in our text. Also, if you need to refresh your memory on checks and bills, chips, or number lines, you can go back to the section about models.
Checks and Bills
Let’s begin with one of our most basic subtraction examples.
Johnny has apples, and Suzy takes away of his apples.
How many apples does Johnny have now? Write an expression using the subtraction sign which solves this problem.
If we change this example to a story about checks and bills, we might use the following instead.
Johnny opens his
business for today with a net worth of $, and then he sends a check for $. What is Johnny’s net worth now?
Johnny
begins with a net worth of $. From our checks and bills model, we know that sending something means we should add the value to subtract the value from
Johnny’s total net worth. Since the object he receives is a check, the value should be positive negative
. Therefore, the expression we would write which solves this problem is .
Notice that Johnny’s total net worth should decrease in this situation: if he sends a check to someone else, he should have less money overall!
Let’s begin to include some negative numbers in our stories.
Johnny opens his business for today with a net worth
of $, and then he sends a check for $. What is Johnny’s net worth now?
Again, after sending a check for $, Johnny’s net worth should be less than it was at the beginning of the
day.
Johnny opens his business for today with a net worth of $, and then he sends a bill for $. What is Johnny’s net
worth now?
In this question, notice that Johnny is now sending a bill. Since we assume this bill will be paid by the person to
whom it was sent, Johnny’s total net worth should be greater than it was at the beginning of the
day.
As an expression involving the subtraction sign, Johnny’s net worth is now .
You may have learned some rules about subtraction in the past. In particular, you might be familiar with what happens when we subtract a negative number. How can we make sense of what we already know, but in terms of checks and bills?
When we subtract a negative number, our checks and bills method says we are receiving sending
a check bill
. So, when writing such a story, we don’t care what Johnny’s net worth is at the beginning of the day. Let’s call it $.
Let’s also say our bill is worth $. We want to compute . After the person to whom Johnny sent this bill pays, the
result to Johnny will be that he has $ more than $. So, we combine Johnny’s original net worth $ with the
additional amount $ to see that \[ A - (-B) = \answer [given]{A + B}. \]
We have a second way of thinking about subtraction problems which also lends itself nicely to stories about checks and bills. Again, we begin with an example about apples, and then transition this example to one about checks and bills.
Johnny had apples, and then Suzy gave Johnny some apples. Johnny now has apples. How many apples did
Suzy give Johnny?
As an expression using the subtraction sign, Johnny got apples from Suzy.
In this example, we might have been thinking of the related addition problem \[ 12 + ? = 19 \] which is equivalent to the subtraction problem \[ 19 - 12. \]
Now, let’s use our checks and bills context!
Johnny opens his business for today with a net worth of $ and then
received something in the mail. After this, his net worth is now $. What was the value of what Johnny received in
the mail?
As an expression using the subtraction sign, Johnny got in the mail.
Johnny opens his business for today with a net worth of $ and then the mail arrives. Afterwards, Johnny’s net
worth is now $. What was the value of what Johnny received in the mail?
As an expression using the subtraction sign, Johnny got in the mail.
Again, notice how the checks and bills story helps us understand why subtracting a negative number is equivalent to adding the value of the bill. Don’t hesitate to ask any questions you have about this concept!
With subtraction problems, notice that we are sometimes sending something in the mail, but sometimes receiving something as well! Be careful as you phrase your questions that you have the appropriate operation!
Along these lines, we should again be very careful when asking questions in our story problems. Here are some common pitfalls.
Johnny starts the day with a net worth of $, and then sends a bill for $. How much did Johnny’s net worth change?
This story looks very similar to some of our other stories, but if we read the question carefully, we see an important
difference. We are asked how much Johnny’s net worth changes, so we do not need the information about his
beginning net worth. Our bill was for $, and he sent this bill to someone else, so his net worth changes by . This is
not a story problem for .
Johnny starts the day with a net worth of $, and then receives a bill in the mail. If Johnny now has $, how much is
the bill Johnny received?
Any time you write a story problem, it’s an excellent practice to go back and try to answer the question from an objective perspective. Is the question really asking what you intend? Is there any other way the question could be interpreted? Especially in class or on a homework assignment, ask someone else for their opinion!
Number Lines
Next, let’s use a number line to solve some subtraction problems with integers. You can write a story problem to go along with each of these expressions for extra practice.
Imagine using a number line like the one below to solve the subtraction problem .
We begin by standing on the number line at the tick marked with . Since we are subtracting, we will move forward backward . We will face right left since is positive and then move spaces. Where on the number line are we now?
We are located at the tick labeled .
Imagine using a number line like the one below to solve the subtraction problem .
We begin by standing on the number line at the tick marked with . Since we are subtracting, we will move forward backward . We will face right left since is positive, and move spaces. Where on the number line are we now?
We are located at the tick labeled .
Imagine using a number line like the one below to solve the subtraction problem .
We begin by standing on the number line at the tick marked with . Since we are subtracting, we will move forward backward . We will face right left since is negative and then move spaces. Where on the number line are we now?
We are located at the tick labeled .
Why does subtracting a negative number give us the same result as adding that number? Using number lines, we can see that if we are subtracting, we are facing left while moving backward. The net result is the same as if we were facing right while moving forward. Try this out with some friends if you are skeptical.
Patterns
Finally, we investigate subtraction of negative numbers via patterns.
Consider the sequence of subtraction problems.
\begin{align*} 5 - 4 &= \answer [given]{1} \\ 5 - 3 &= \answer [given]{2} \\ 5 - 2 &= \answer [given]{3} \\ 5 - 1 &= \answer [given]{4} \\ 5 - 0 &= \answer [given]{5} \end{align*}
Try your hand at recognizing patterns with some other subtraction problems.
As we move down the chart, moving one row down results in the final answer increasing by . So, if the pattern continues to hold, we expect the answer to to be , since it is one less than . We can also notice that the answer to is the same as the answer to .
Finally, notice that no matter how we approach the problems in this section, we are getting consistent answers. Also, no matter how we look at subtraction of a negative number, we can see that it should be the same as addition. Subtraction as an operation remains the same, no matter how we model it.
2025-08-13 00:54:59