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\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
International Baccalaureate · PYP 2

Multiplication

⌚ ~138 min ▢ 15 blocks ✓ 73 exercises ➣ Prerequisites : Addition to 100 and beyond, Multiplication: Equal Groups
I What is Multiplication?

Louis loves apples and eats 2 apples every day: Figure.
How can we find how many apples he eats in one week (7 days)?
Figure

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 of the same number. We can show the idea of “\(\textcolor{colordef}{\text{four}}\) times \(\textcolor{colorprop}{\text{three}}\) equals \(\textcolor{olive}{\text{twelve}}\)” 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:
    Figure
  • With a Part-Whole Model:
    Figure
Example
Write the repeated addition \(\textcolor{colorprop}{5}+\textcolor{colorprop}{5}+\textcolor{colorprop}{5}\) as a multiplication.

The number 5 appears 3 times in the addition. So, the multiplication is: $$ \textcolor{colordef}{3}\times \textcolor{colorprop}{5}$$

Example
A packet holds \(20\) biscuits. Sam buys \(6\) packets. Write a multiplication and find how many biscuits Sam has.
Figure

Six packets of \(20\) is \(6\) times \(20\): $$6 \times 20 \text{ biscuits}$$ No need to add \(20+20+20+20+20+20\) one by one: \(20\) is \(2\) tens, and adding six \(2\)s is easy: \(2+2+2+2+2+2 = 12\). So \(6\) times \(2\) tens makes \(12\) tens, that is \(120\). $$6 \times 20 \text{ biscuits} = 120 \text{ biscuits}$$ The multiplication is much shorter to write than the repeated addition.

Skills to practice
  • Identifying Multiplications and Repeated Additions From Words
  • Identifying Multiplications and Repeated Additions From Groups
  • Finding the Number of Groups and the Number of Fruits in Each Group
  • Writing Multiplications from Groups and from Additions
  • Calculating Multiplications Using Cubes
  • Multiplying Whole Tens
II On the Number Line

Louis shows the multiplication \(\textcolor{colordef}{4} \times \textcolor{colorprop}{3}\) on a number line: starting from \(0\), he makes \(\textcolor{colordef}{4}\) jumps of \(\textcolor{colorprop}{3}\).
Figure
On which number does he land?

Each jump adds \(\textcolor{colorprop}{3}\): \(0 + \textcolor{colorprop}{3}\), \(3 + \textcolor{colorprop}{3}\), \(6 + \textcolor{colorprop}{3}\), \(9+\textcolor{colorprop}{3}\). After \(\textcolor{colordef}{4}\) jumps, Louis lands on \(\textcolor{olive}{12}\): $$\textcolor{colordef}{4} \times \textcolor{colorprop}{3} = \textcolor{colorprop}{3}+\textcolor{colorprop}{3}+\textcolor{colorprop}{3}+\textcolor{colorprop}{3} = \textcolor{olive}{12}$$


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 \(\textcolor{colorprop}{3}\).
Figure
Each jump represents adding 3. After 4 jumps, we land on \(\textcolor{olive}{12}\). So, \(\textcolor{colordef}{4} \times \textcolor{colorprop}{3} = \textcolor{olive}{12}\).
Skills to practice
  • Calculating Multiplications Using the Number Line
III 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: $$\textcolor{colordef}{3} \times \textcolor{colorprop}{2} = \textcolor{olive}{6}$$
Figure
Skills to practice
  • Solving Real-World Problems With Drawing
  • Solving Real-World Problems
IV Does the Order Matter?

Hugo and Louis are looking at this group of cubes.
Figure
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 correct? Can they both be correct?

They are both correct.
  • Louis sees \(\textcolor{colordef}{2}\) columns, with \(\textcolor{colorprop}{3}\) cubes in each column (2 groups of 3).
    Figure
    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).
    Figure
    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
Multiplication is commutative: changing the order of the numbers does not change the result. Cubes set out in rows and columns make an array: we can count it by columns or by rows.
Figure
Skills to practice
  • Finding Expressions from Arrays
  • Finding an Easier Way to Calculate Multiplication
  • Writing the Commutative Property
  • Using the Order of Multiplication
V Going further
Skills to practice
  • Completing Multiplications with Pictures
  • Solving Multiplication Problems
  • Quiz