As an expression involving the multiplication sign, Johnny’s net worth has changed by $.
1 Activities for this section:
Now that we’ve conquered addition and subtraction, let’s move on to multiplication. The biggest difference between what we did with addition and subtraction and what we will do with multiplication is the units on each of the numbers. Remember that with addition and subtraction, all of the units are the same. But with multiplication, the units are different. For instance, compare and contrast the following examples.
(apples) + (apples) = (apples)
(baskets of apples) (apples per basket) = (apples)
We will need a different strategy for multiplication!
2 Checks and Bills
We’re jumping right in with an example using the repeated addition model of multiplication. In other words, when we write a story problem for , we are thinking about this as copies of the quantity , or .
Let’s write a checks and bills story problem for . Johnny will still be our small business owner. Starting with the
second number, we know that these should be checks bills
, each worth $, since the is positive.
We will use the first number to tell us whether Johnny is sending or receiving these checks. Since the is positive, we will say that that Johnny receives sends checks.
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.
So, our story problem would be: Johnny receives checks each for dollars. How has his net worth changed? The answer would be that Johnny’s net worth goes up by $, or the answer is dollars.
Next, 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. Indeed, the change in his net worth is dollars.
What if the number of groups is negative? 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.
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 to model difficult ideas.
3 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 model the multiplication problem .
We are still thinking about multiplication as repeated addition here: should be five copies of . As with our story problems, the question or starting point requires care. Since we would like to investigate how our value changes after these jumps, it’s natural to start on zero. If we think about , we are adding, so 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 model the multiplication problem .
We begin by standing on the number line at the tick marked with . Since our groups are negative, we can think of this as subtracting our copies of , so we will move forward backward , and take total steps. We will face right left , because is negative, and each step will move spaces. 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.
We encourage you to try out using red and black chips to model some multiplication problems, and to investigate using patterns as we did with addition and subtraction. However, since our in-class work will focus mostly on addition and subtraction, we’d like to end this section with a few comments about division.
4 Division
Remember that we defined division as the inverse of multiplication, so we will use what we know about multiplication to think about division. We have used our models to explain why the following rules make sense.
- Multiplying two positive numbers gives a positive result.
- Multiplying one positive and one negative number gives a negative result.
- Multiplying two negative numbers gives a positive result.
Let’s use this to think about a division problem.
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 .
If we wanted to write a story problem using checks and bills for , we could do the following. Johnny sent identical items in the mail, and his net worth changed by $. What was the value of each item that was sent?
Notice that the answer to this question is that Johnny must have sent bills (so his net worth would go up from sending something), and these bills must have been for $ each, so that the total change is $.
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!
2026-07-17 02:43:10