Table of Contents
- What is Multiplication?
- How do we draw an Area Model to Multiply Two Numbers?
- How can we apply Distributive Property to Multiply Three Numbers?
- Commutative Property of Multiplication
- Associative Property of Multiplication
- How can we apply Associative Property to Multiply Three Numbers?
- Real Life Examples
- Frequently Asked Questions
What is Multiplication?
Multiplication is the process of calculating the result when a number “a” is multiplied by “b” in simple algebra. The product of a and b is the result of an operation called multiplication in mathematics, and each of the numbers a and b is a factor of the product ab. The letters a.b, a × b, (a)(b), or simply ab are used to represent multiplication. The symbol “×” is known as the multiplication sign.
How do we draw an Area Model to Multiply Two Numbers?
The area model method is based on a simple equation for calculating a rectangle’s area: length times width equals total area (Length × Width = Area).
The steps for drawing an area model are:
Step 1: Write the numbers in their expanded form.
Step 2: Now, draw the area model to represent the expanded form.
Step 3: Write the product of each number’s multiplication in its own block.
Step 4: Add all the numbers that are in the block and write the result that will give the area of the whole block.
Here we are looking at an example of multiplying 12 by 11 using the area model of multiplication.
And as you can observe, the product using the area model is 132, which will also be the same when we just multiply 12 by 11, instead of using the area model.
How can we apply Distributive Property to Multiply Three Numbers?
The distributive property for multiplication states that when we multiply the sum or the difference of two or more numbers, it produces the same result. This indicates that we can multiply each addend or subtrahend individually by a number and then add or subtract the products in the subsequent step.
Suppose the given numbers are 1, 2, and 3. So, the distributive property for addition can be stated as:
1 x ( 2 + 3 ) = 1 x 2 + 1 x 3
1 x 5 = 2 + 3
5 = 5
Here’s an example of the distributive property of multiplication.
Example: What are the two ways to break apart 5 x 7 using the distributive property.
The two ways to break apart 5 x 7 using the distributive property are:
Case 1: Let’s break 7 as 9 – 2
So, 5(9 – 2)
= (5 x 9) – (5 x 2) [Distributive Property]
= 45-10 [Simplify]
= 35
Case 2: Let us break apart 5 as 3 + 2
So, 7 x (3 + 2)
= (7 x 3) + (7 x 2) [Distributive Property]
= 21 + 14
= 35
Commutative Property of Multiplication
The commutative property is a mathematical principle that states that the order in which numbers are multiplied has no effect on the final product.
We can observe the commutative property of multiplication with the following example.
Let’s consider two real numbers 6 and 3,
6 x 3 = 3 x 6
18 = 18
The commutative property is also referred to as the “order property of multiplication.”
Associative Property of Multiplication
According to the associative property of multiplication, when three or more numbers are multiplied with each other, the result is the same regardless of how the three numbers are grouped. This property can be written in the general form as (a x b) x c = a x (b × c).
We can substitute numbers in this expression. We can take a simple example of the numbers 2,3 and 4.
The left-hand expression shows that 2 and 3 are grouped together:
L.H.S = (2 × 3) × 4
The right-hand expression shows that 2 and 4 are grouped together:
R.H.S = 3 × (2 × 4)
However, when we multiply all of the numbers together, the end result is the same.
(2 × 3) × 4 = 3 × (2 × 4) Using the associative property of multiplication
6 × 4 = 3 × 8
24 = 24
How can we apply Associative Property to Multiply Three Numbers?
The associative property is also referred to as the “grouping property of multiplication.”
Example: Assume three different numbers and write two different equations by using the associative property.
Let us take the three factors as 2, 3, and 6. Using the associative property, we can write the multiplication of the given factors in different ways.
Case 1: Let us assume the order as given above.
So, ( 2 × 3) × 6 = 2 × ( 3 × 6 ) [Associative Property]
6 × 6 = 2 × 18 [Simplify]
36 = 36
Thus, the above factors satisfy the condition of the associative property of multiplication.
Case 2: Let us change the position of factors to their successive positions.
Since, the factors are 6, 2, and 3
Hence, ( 6 × 2 ) × 3 = 6 × ( 2 × 3 ) [Associative Property]
12 × 3 = 6 × 6 [Simplify]
36 = 36
Thus, the above factors satisfy the condition of the associative property.
Real Life Examples
Example 1: Martha and Lucy, two of your friends, were born on the same day. On their birthday, you must give them the same pair of sandals and a shirt. What is the total cost of the gifts if the sandals are worth $25 and the shirt is worth $5?
Solution: There are two different methods to solve this problem:
Method 1:
Step 1: Find the total cost of each set.
= 25 + 5 = $30
Step 2: Find the final cost by multiplying the total cost by 2.
= 30 × 2 = $60
Method 2:
Step 1: As there are 2 friends, double the cost of the sandals and the shirt.
Sandals = 25 × 2 = $50
Shirt = 5 × 2 = $10
Step 2: Find the final cost by adding both quantities.
= 50 + 10 = $60
Example 2: Three friends have two nickels, three pennies, and ten dimes each. What is the total amount of money they have in dollars?
Solution: There are two different methods to solve this problem:
Method 1:
Step 1: Find the total cost of each type of coin.
Nickels:
2 × 5¢ = 10¢
Pennies:
3 × 1¢ = 3¢
Dimes:
10 × 10¢ = 100¢
Step 2: There are three friends, so multiply each type of coin by 3.
Nickels:
3 × 10¢ = 30¢
Pennies:
3 × 3¢ = 9¢
Dimes:
3 × 100¢ = 300¢
Step 3: Find the total amount of money.
30¢ + 9¢ + 300¢ = 339¢
Step 4: Convert to dollars.
\(\frac{339}{100}\) = $3.39
Method 2:
Step 1: Each person has two nickels, three pennies, and ten dimes.
2 × 5¢ + 3 × 1¢ + 10 × 10¢
Step 2: Total money each person has.
2 × 5¢ + 3 × 1¢ + 10 × 10¢ = 113¢
Step 3: Total money three people have.
113¢ + 113¢ + 113¢ = 339¢
Step 4: Convert to dollars.
\(\frac{339}{100}\) = $3.39
Example 3: Steve washes his car and then places a bowl of food for his pet dog.
If you observe, this is a commutative situation. It makes no difference what Steve does in no specific order. Steve has the option of washing his car first or placing a bowl of food for his dog first.
Example 4: Two trucks with five boxes each have arrived at a sports centre to deliver new basketballs. There are ten basketballs in each box. What is the total number of basketballs that have arrived at the sports centre?
If we first multiply the number of trucks by the number of boxes in each truck (2 x 5), we get the total number of boxes (10).
Then, we multiply by the number of basketballs in each box (10 x 10) and that gives us a total of 100 basketballs.
We’ll now solve the problem by grouping the variables in a new way.
This time, we’ll start by multiplying the number of boxes by the number of basketballs in each box (10 x 5), giving us 50, the number of basketballs in each truck.
Then we multiply by the number of trucks (50 x 2) to get a total of 100 basketballs.
Write down each factor of area in expanded form. After that, make your model. The area of each smaller rectangle is then calculated by multiplying the factors that are broken up. Finally, add all of the products together to get the total area.
When adding or multiplying, the associative property states that while grouping, the symbols can be rearranged without affecting the result. This is stated as (a + b) + c = a + (b + c)
An example of the associative property in addition is:
(1 + 2) + 3 = 1 + (2 + 3)
3 + 3 = 1 + 5
6 = 6
When we look at the application of the associative property in multiplication it looks like the example given:
(3 × 2) × 6 = 3 × (2 × 6)
6 × 6 = 6 × 6
18 = 18
The distributive property is a multiplication method that involves multiplying a number by all of the separate addends of another number. This is stated as a × (b + c) = a × b + a × c
Proof that the distributive property is true for 3, 7, and 9.
According to the distributive property, a × (b + c) = a × b + a × c
So, 3 × (7 + 8) = 3 × 7 + 3 × 8 Using the distributive property of multiplication
3 × 15 = 21 + 24
45 = 45