\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)

Multiplication

In math, we are always looking for faster ways to solve problems. Think about when you add the same number over and over again. This is called repeated addition. Multiplication is a powerful shortcut for repeated addition!

What is Multiplication?


Louis loves apples and eats 2 apples every day: .
How could we figure out how many apples he eats in one week (7 days)?

One way is to use repeated addition. We can add 2 apples for each of the 7 days:$$\textcolor{colorprop}{2}+\textcolor{colorprop}{2}+\textcolor{colorprop}{2}+\textcolor{colorprop}{2}+\textcolor{colorprop}{2}+\textcolor{colorprop}{2}+\textcolor{colorprop}{2} = 14$$This works, but it takes a long time to write. A much faster way is to use multiplication.
When we have \(\textcolor{colordef}{7}\) groups of \(\textcolor{colorprop}{2}\), we can write it as \(\textcolor{colordef}{7}\times \textcolor{colorprop}{2}\). The symbol \(\times\) means "times" or "groups of."$$\textcolor{colordef}{7}\times \textcolor{colorprop}{2} \quad \text{is the same as} \quad \textcolor{colorprop}{2}+\textcolor{colorprop}{2}+\textcolor{colorprop}{2}+\textcolor{colorprop}{2}+\textcolor{colorprop}{2}+\textcolor{colorprop}{2}+\textcolor{colorprop}{2}$$


Definition Multiplication
Multiplication is a fast way to show repeated addition. We can show the idea of "four times three equals twelve" in many different ways:
  • With Numbers: $$\textcolor{colordef}{4}\times \textcolor{colorprop}{3}=\textcolor{olive}{12}$$
  • In Groups: $$\textcolor{colordef}{4}\text{ groups of }\textcolor{colorprop}{3}=\textcolor{olive}{12}$$
  • As Repeated Addition: $$\textcolor{colorprop}{3}+\textcolor{colorprop}{3}+\textcolor{colorprop}{3}+\textcolor{colorprop}{3}=\textcolor{olive}{12}$$
  • With Cubes:
  • With a Part-Whole Model:
Example
Write the repeated addition \(\textcolor{colorprop}{5}+\textcolor{colorprop}{5}+\textcolor{colorprop}{5}\) as a multiplication.

We are adding the number 5, and we are adding it 3 times. So, the multiplication is:$$ \textcolor{colordef}{3}\times \textcolor{colorprop}{5}$$

On the Number Line


Let's consider the multiplication: \(\textcolor{colordef}{4} \times \textcolor{colorprop}{3}\) that is:$$\textcolor{colorprop}{3}+\textcolor{colorprop}{3}+\textcolor{colorprop}{3}+\textcolor{colorprop}{3}$$We can visualize this on a number line:
Starting from 0, we move 3 units to the right 4 times. Each move represents addition: \(0 + \textcolor{colorprop}{3}\), \(3 + \textcolor{colorprop}{3}\), \(6 + \textcolor{colorprop}{3}\), \(9+\textcolor{colorprop}{3}\). As you can see, we end up at \(\textcolor{olive}{12}\), which is the result of the multiplication \(\textcolor{colordef}{4} \times \textcolor{colorprop}{3}\).

Method Multiplication on the Number Line
We can show multiplication as "jumps" on a number line. To show \(\textcolor{colordef}{4} \times \textcolor{colorprop}{3}\), we can start at 0 and make \(\textcolor{colordef}{4}\) jumps of size \(\textcolor{colorprop}{3}\).
Each jump represents adding 3. After 4 jumps, we land on \(\textcolor{olive}{12}\). So, \(\textcolor{colordef}{4} \times \textcolor{colorprop}{3} = \textcolor{olive}{12}\).

Multiplication in Word Problems

Method Finding the Total with Groups
In word problems, we can find the total by multiplying the number of groups by the number of items in each group.$$\textcolor{colordef}{\text{Number of groups}} \times \textcolor{colorprop}{\text{Number in each group}} =\textcolor{olive}{\text{Total}}$$For example, if there are \(\textcolor{colordef}{3}\) bags and each bag has \(\textcolor{colorprop}{2}\) apples, the total number of apples is:$$\textcolor{colordef}{3} \times \textcolor{colorprop}{2} = \textcolor{olive}{6}$$

Does the Order Matter?


Hugo and Louis are looking at this group of cubes.
To find the total, Hugo thinks they should calculate \(\textcolor{colordef}{3}\times \textcolor{colorprop}{2}\), but Louis thinks it should be \(\textcolor{colordef}{2}\times \textcolor{colorprop}{3}\).Who is right? Can they both be right?

Let's look at their thinking. They are both correct!
  • Louis sees \(\textcolor{colordef}{2}\) columns, with \(\textcolor{colorprop}{3}\) cubes in each column (2 groups of 3).
    His total is \(\textcolor{colordef}{2}\times \textcolor{colorprop}{3} = 6\) cubes.
  • Hugo sees \(\textcolor{colordef}{3}\) rows, with \(\textcolor{colorprop}{2}\) cubes in each row (3 groups of 2).
    His total is \(\textcolor{colordef}{3}\times \textcolor{colorprop}{2} = 6\) cubes.
They both get the same answer! This shows that \(\textcolor{colordef}{2}\times \textcolor{colorprop}{3}=\textcolor{colordef}{3}\times \textcolor{colorprop}{2}\).


Proposition Commutative Property
In multiplication, changing the order of the numbers does not change the result.