As an expression involving the multiplication sign, Johnny has apples.
1 Activities for this section:
Now that we’ve conquered addition and subtraction, let’s move on to multiplication. The first thing we should notice when comparing multiplication to addition and subtraction is that with multiplication, the wholes for each of our quantities are different, while with addition and subtraction they are all the same. For instance, we might have something like the following.
(apples) + (apples) = (apples)
(baskets of apples) (apples per basket) = (apples)
We need to continue to be careful in how we phrase our problems!
1.1 Checks and Bills
As we have done previously, let’s begin by looking at a multiplication problem with whole numbers, and then slowly introduce integers. Remember to look back at our sections about multiplication in the text!
Johnny has lunch bags, and each lunch bag has apples in it. How many apples does Johnny have all
together?
We can think of this problem as making copies of the apples per bag. As a story about checks and bills, then, we could think of making copies of $.
Let’s start to write our story problem for . Johnny is our usual small business owner. Starting with the second
number, we know that these should be checks bills
, each worth $, since the is positive.
For the first number, we know that Johnny receives sends of these checks, because the is positive.
But what should be our question? We could start by saying that Johnny’s net worth is $, and ask what his net worth is now. The natural answer to this question would be but that extra zero can feel a little bit confusing.
We might instead ask, “How has Johnny’s net worth changed?” In this case, no matter what he had to start with, he now has more than he did previously. Let’s proceed with this second formulation, but cautiously. We will need to make sure all of this still makes sense as we introduce negative numbers.
For our first question, let’s change the second number to be negative.
Johnny receives bills, each for $. After all
bills are paid, how has Johnny’s net worth changed?
Since Johnny is paying bills in this situation, it makes sense for his net worth to decrease, or for the change to be
negative.
As an expression involving the multiplication sign, Johnny’s net worth has changed by $.
What if our first number is negative? In other words, what would it mean to have negative groups? It may feel a little bit forced, but we will use the convention that a negative group means we are sending that many copies of our check or bill.
Johnny opens his business for today with a net worth of $, and then sends checks for $ each. What is Johnny’s net
worth now?
While beginning with a net worth of $ helps us to understand why negative groups might be represented by sending, this convention may still feel artificial. Instead, we could ask the following.
Johnny sends three checks, each for
eight dollars. How much has Johnny’s net worth changed?
You might object that what we are doing here is actually calculating , and reasoning that the overall answer
should be negative. That’s okay! Remember that this concept is difficult, and we are doing the best we
can.
Johnny’s net worth has changed by $.
Finally, let’s look at what happens when we multiply two negatives together.
Johnny sends bills, each for $. After
all bills are paid, how much has Johnny’s net worth changed?
In terms of the situation with checks and bills, should this answer be positive or negative? If Johnny is sending bills
to other people, and then these other people pay their bills, this money comes back to Johnny. So, his net worth
overall should increase, meaning the change should be positive. Notice that we didn’t need to memorize any rules
about negative numbers in order to come to this conclusion!
Johnny’s net worth has changed by $.
1.2 Number Lines
Next, let’s use a number line to solve some multiplication 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 multiplication problem .
Remember that for whole numbers, we can think of multiplication as repeated addition. We can make sense out of this for the negatives of whole numbers, too. We want to add five times, or make five copies of . As with our story problems, the question or starting point requires care. With our stories, our starting net worth was , so with a number line we begin by standing at the tick marked with . Since we are adding, we will move forward backwards . Now, the amount we are adding each time is , which is negative, so we will face right left . Finally, we can think of the steps as our groups, meaning the amount we move with each step should be spaces. After taking such steps, 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 multiplication problem .
We begin by standing on the number line at the tick marked with . Since our groups are negative, we will move forward backward , and take total steps. We will face right left , and each step will move spaces, since is negative. Where on the number line are we now?
We are located at the tick labeled .
Notice again that if we multiply two negative numbers, we face backwards while moving backwards, for a net result of moving in the positive direction along the number line. Try this out with some friends if you are skeptical.
1.3 Patterns
Finally, we investigate subtraction of negative numbers via patterns.
Consider the sequence of addition problems. \begin{align*} 5 \times 4 &= \answer [given]{20} \\ 5 \times 3 &= \answer [given]{15} \\ 5 \times 2 &= \answer [given]{10} \\ 5 \times 1 &= \answer [given]{5} \\ 5 \times 0 &= \answer [given]{0} \end{align*}
If we are convinced by this pattern that a positive number times a negative number should be negative, we can
use the commutative property to convince ourselves that a negative number times a positive number
should also be negative. Then, we can use our patterns again to convince ourselves that a negative
number times a negative number should be positive. As we move down the chart, moving one row down results in the final answer decreasing by . So, if the pattern continues to hold, we expect the answer to to be , since it is five less than .
Consider the sequence of addition problems. \begin{align*} -5 \times 4 &= \answer [given]{-20} \\ -5 \times 3 &= \answer [given]{-15} \\ -5 \times 2 &= \answer [given]{-10} \\ -5 \times 1 &= \answer [given]{-5} \\ -5 \times 0 &= \answer [given]{0} \end{align*}
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 five more than .
1.4 Extra Examples
Since multiplication with negatives is a complicated topic, here are two extra ways to think about this concept. First, we will talk about algebra, and second we will use a more advanced physical model.
We mentioned very briefly when talking about patterns that the properties of addition and subtraction can help us to understand multiplication with negative numbers.
We can model multiplication problems like without much trouble. In fact, most of our trouble came from
attempting to understand what negative groups should mean. If we instead wanted to consider , we might quickly
realize that the commutative property of multiplication gives us which we could model with a story, a number line,
or chips. Notice, however, that we’re making use of the idea that a positive number multiplied by a
negative number gives us a negative answer overall – and this fact requires justification some other
way!
Earlier in this section, our justification for why a positive number multiplied by a negative number is a negative number came mostly from describing the multiplication in a way we could understand or represent with a physical situation. Even number lines could be thought of as a physical situation, since we can use our bodies to move along the lines. Patterns are the beginning of looking at this concept algebraically, and let’s now do some more advanced algebra.
We know that if we have a negative number, like , we can write this as . Algebraically, why should this be true? We want our negative numbers to represent the opposite of our positive numbers. If Sue has dollars, she needs to obtain more dollars to make it back to zero. If Sam is standing on the tick labeled , he needs to move spaces to the right to get back to zero. In other words, should satisfy the equation
What about ?
Let’s look at to see if this is really equal to . If so, when we add to , we should get back to . First,
we see that there is a in each of these terms, so let’s factor it out. Inside the parenthesis, we should now know what
to do.
This may seem a little silly, but understanding and developing good rules for using negative numbers with algebra was a concern of many mathematicians before negative numbers were fully accepted.
There is actually one extra fact we have used in the previous argument that you may have noticed. To really be sure that , we should really be sure that the only number we can add to to get zero is . Proving this fact is possible, but we won’t look at that proof in this course.
Now that we know that we can write a negative number as multiplied by the positive version of that number, we can use more of our properties.
Let’s now move to a different kind of model for multiplication.
You might recall the equation Let’s say that zero is our position at noon, and we have been traveling east on an
east-west highway at a steady rate of miles per hour. Where were we at am?
Notice that am was three hours ago, and so this question is equivalent to asking for the solution to . Knowing that the answer is , we should interpret this result to say that at am, we were miles east west of where we were at noon.
1.5 Division
Now that we’ve had so much practice, we will generally leave you to think about division as the inverse of multiplication. We have discussed many ways to understand the sign of a multiplication problem with integers. Once we understand whether a result should be positive or negative, the problem is reduced to computing with positive numbers. What we mean by thinking of division as the inverse of multiplication is to turn your division problem back into a multiplication problem, and then use what you know about multiplication to understand the sign of the answer.
What is ?
We can think of as a multiplication problem by using or While which problem we choose matters in our stories, the associative commutative distributive property of multiplication gives us the same answer in either case. We also know from our work with multiplication that the sign of the answer must be positive negative . Since , we now know that should equal .
As we have stated throughout, this is a complicated concept. You may need to work through the examples several times, as well as ask questions, before understanding completely. Don’t worry, though – many of history’s most notable mathematicians shared these struggles!
2025-11-14 22:16:17