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

Multiplication

Multiplication is a very important concept in mathematics. It’s a way of adding the same number together many times.

Definitions


Louis loves apples and eats exactly 2 apples every day. He never misses a day because he knows how healthy and tasty apples are.
If Louis eats 2 apples every day, how many apples will he eat in one week (7 days)?

If we want to know how many apples Louis eats in a week (7 days), we add 2 apples for each day:$$\textcolor{colorprop}{2}+\textcolor{colorprop}{2}+\textcolor{colorprop}{2}+\textcolor{colorprop}{2}+\textcolor{colorprop}{2}+\textcolor{colorprop}{2}+\textcolor{colorprop}{2}$$We find \(14\) apples. In this chapter, we will introduce multiplication to make it quicker and easier. When we say \(\textcolor{colordef}{7}\) groups of \(\textcolor{colorprop}{2}\) apples, we can write it as \(\textcolor{colordef}{7}\times \textcolor{colorprop}{2}\). The symbol \(\times\) means multiplied by or times.$$\textcolor{colordef}{7}\times \textcolor{colorprop}{2}=\textcolor{colorprop}{2}+\textcolor{colorprop}{2}+\textcolor{colorprop}{2}+\textcolor{colorprop}{2}+\textcolor{colorprop}{2}+\textcolor{colorprop}{2}+\textcolor{colorprop}{2}$$


Definition Multiplication
Multiplication is the process of repeated addition. When we multiply, we calculate the total by adding a number to itself a specified number of times.
Multiplication can be represented in several ways:
  • Numbers: $$\textcolor{colordef}{4}\times \textcolor{colorprop}{3}=\textcolor{olive}{12}$$
  • Groups: $$\textcolor{colordef}{4}\text{ groups of }\textcolor{colorprop}{3}=\textcolor{olive}{12}$$
  • Repeated addition: $$\textcolor{colorprop}{3}+\textcolor{colorprop}{3}+\textcolor{colorprop}{3}+\textcolor{colorprop}{3}=\textcolor{olive}{12}$$
  • Words:
    four times three equals twelve
  • Items:
  • Part-whole model:
Example
Write the repeated addition \(\textcolor{colorprop}{5}+\textcolor{colorprop}{5}+\textcolor{colorprop}{5}\) as a multiplication.

\( \textcolor{colorprop}{5}+\textcolor{colorprop}{5}+\textcolor{colorprop}{5}=\textcolor{colordef}{3}\times \textcolor{colorprop}{5}\)

In 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 ones 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 in number line
To evaluate \(4\times 3\), we start from 0 and we move \(\textcolor{colorprop}{3}\) ones to the right \(\textcolor{colordef}{4}\) times.
We end up at \(\textcolor{olive}{12}\), which is the result of the multiplication \(\textcolor{colordef}{4} \times \textcolor{colorprop}{3}\).

Representation of Multiplication in Word Problems

Method Groups of items
When we multiply, we often think about groups and the number of items in each group.$$\begin{aligned}[t]\textcolor{colordef}{\text{number of groups}} &\times \textcolor{colorprop}{\text{number of items in each group}} &=&\textcolor{olive}{\text{total}} \\\end{aligned}$$For example, there are \(\textcolor{colordef}{3}\) bags, and each bag contains \(\textcolor{colorprop}{2}\) apples. The total number of apples is:$$\begin{aligned}[t]\textcolor{colordef}{3} \times \textcolor{colorprop}{2} &= \textcolor{colorprop}{2} + \textcolor{colorprop}{2} + \textcolor{colorprop}{2}\\ &= \textcolor{olive}{6}\end{aligned}$$

Commutative


Two brothers, Hugo and Louis, want to count the number of cubes:
  • Louis calculates \(\textcolor{colordef}{2}\times \textcolor{colorprop}{3}\) to find the number of cubes.
  • Hugo calculates \(\textcolor{colordef}{3}\times \textcolor{colorprop}{2}\) to find the number of cubes.
Who is correct?
  • Only Hugo
  • Only Louis
  • Both Hugo and Louis

Both are correct:
  • Louis counts \(\textcolor{colordef}{2}\) groups of \(\textcolor{colorprop}{3}\):
    So, the total is \(\textcolor{colordef}{2}\times \textcolor{colorprop}{3}\) cubes.
  • Hugo counts \(\textcolor{colordef}{3}\) groups of \(\textcolor{colorprop}{2}\):
    So, the total is \(\textcolor{colordef}{3}\times \textcolor{colorprop}{2}\) cubes.
This shows that $$\textcolor{colordef}{2}\times \textcolor{colorprop}{3}=\textcolor{colordef}{3}\times \textcolor{colorprop}{2}$$


Proposition Commutative