# Arithmetic properties: Distributive property

### Arithmetic properties: Distributive property

#### Lessons

Notes:

• The distributive property is what happens when you multiply a number (called a multiplier or factor) with a sum of two or more numbers (addends inside of brackets).
• Ex. 2 × (9 + 5) =

• To distribute means to spread out or to hand around
• So, the distributive property makes you distribute the multiplier
• The multiplier/factor is distributed (given to) all the addends in brackets
• Ex. 2 × (9 + 5) = 2×9 + 2×5
• = 18 + 10 = 28
• In other words, multiplying a sum of two numbers is equal to the sum of each addend multiplied by the factor

• A common mistake that many students make with the distributive property is that they do not FULLY distribute the multiplier/factor:
• Ex. 2 × (9 + 5) the 2 should be multiplied with both addends = 2 × 9 + 2 × 5
• The common mistake is to only multiply the with the first addend:
• 2 × (9 + 5) ? 2 × 9 + 5
• 2 × 9 + 5 = 18 + 5 = 23
• The correct answer should have been 28; not distributing will give the incorrect answer of 23

• The distributive property can be demonstrated using area block models:
• Area is given by two dimensions (i.e. length × width or height × length)
• Ex. 2 × (9 + 5) means an area block with a height of 2, and a combined length of 9 and 5. The total number of area blocks is 28.

• The general formula for the distributive property (where $a$, $b$ and $c$ are variables that represent real numbers) is:

• The distributive property works for any type of real number as the multiplier and/or addends (such as integers, fractions, and/or decimals):
• Ex. -3 x $(\frac{1}{5} + \frac{3}{5}) = (-3 \,x \, \frac{1}{5}) + (-3 \, x \, \frac{3}{5}) = \frac{-3}{5} + \frac{-9}{5} = \frac{-12}{5}$
• Ex. 5 × (0.2+ 0.05) = (5×0.2) + (5×0.05) = 1.0 + 0.25 = 1.25
• Introduction
Introduction to the distributive property:
a)
Showing that a × (b + c) = (a × b) + (a × c)

b)
Common mistakes to avoid when using the distributive property

c)
Using area blocks to demonstrate the distributive property

d)
Why is it called the "distributive" property?

e)
The general formula for the distributive property

f)
Using the distributive property for other types of real numbers (decimals, fractions, integers)

• 1.
Distributive Property Equations
Use the distributive property to fill in the blanks.
a)
6 × (3 + 5 + 4) = 6 × ___ + 6 × ___ + 6 × ___
$\qquad \qquad \qquad \qquad$ = _______+ _______+ ______
$\qquad \qquad \qquad \qquad$ = _______
$\qquad \qquad \qquad \qquad$ 6 × (___) = _______
$\qquad \qquad \qquad \qquad$ _______ = _______

b)
12 × (10. 5 + 1.2) = 12 × ___ + 12 × ___
$\qquad \qquad \qquad \qquad$ = _______ + _______
$\qquad \qquad \qquad \qquad$ = _______
$\qquad \qquad \qquad \qquad$ 12 × (___) = _______
$\qquad \qquad \qquad \qquad$ _______ = _______

c)
3 × ($\frac{3}{5} + \frac{4}{5}$) = 3 × ___ + 3 × ___
$\qquad \qquad \qquad \qquad$ = _______ + _______
$\qquad \qquad \qquad \qquad$ = _______
$\qquad \qquad \qquad \qquad$ 3 × (___) = _______
$\qquad \qquad \qquad \qquad$ _______ = _______

d)
8 × (6 + __) = 8 × ___ + 8 × ___
$\qquad \qquad \qquad \qquad$ = _______ + _______
$\qquad \qquad \qquad \qquad$ = 104
$\qquad \qquad \qquad \qquad$ 8 × (___) = 104
$\qquad \qquad \qquad \qquad$ _______ = 104

• 2.
Distributive property as a shortcut
Use the distributive property to split up one of the factors. This can be a shortcut so that you can use your times tables knowledge. Solve and check the answer using long (normal) multiplication
a)
6 × 15

b)
44 × 12

c)
18 × 2.5

d)
60 × $\frac{5}{4}$

• 3.
Distributive property using area blocks
Use an area block to show each distributive property, then solve.
a)
5 × (3 + 8)

b)
4 × (11 + 2)

• 4.
Rewriting a sum as a distributive property
Rewrite each sum as a distributive property expression. Recall that the general formula for the distributive property is: a × (b + c) = a × b + a × c
• Hint: find the greatest common factor of each number first.

a)
28 + 32

b)
27 + 21

c)
66 + 48

• 5.
Distributive property word problem 1
Eight people go to a restaurant for a buffet dinner. The cost per person for the meal is $11.50 and there is a separate drink cost of$2.10. How much did they pay all together?

• 6.
Distributive property word problem 2
In a movie theatre, there are 3 sections you can sit in (left, middle, right). Each row has 6 seats in the left section, 14 in the middle section, and 8 in the right section.
• If there are 10 rows, how many seats are there in each section? And how many total seats are there in the theatre?