The Distributive Property for Seventh Graders

If we look at the following examples, we can see that we have broken down the larger number into several smaller numbers that are more manageable. The value is the same as before, but now we can distribute a complex operation into several easy operations.

The distributive property can be described as:

$Z \times (X + Y) = ZX + ZY$

$Z \times (X - Y) = ZX - ZY$

Sometimes, an expression will require us to perform both addition and subtraction within our parentheses. Not to worry! The distributive property can simplify these expressions too.

Let's see some examples:

• $(X + 2) \times (X + 3) =$
  $X² + 3X + 2X + 6 = X² + 5X + 6$
  
• $(X - 4) \times (X - 3) =$
  $X² - 3X - 4X + 12 = X² - 7X + 12$

How would you go about using the distributive property in an equation with two sets of parentheses?

First, we multiply the first term of the first parenthesis by the first and second terms of the second parenthesis..

Next, we multiply the second term of the first parenthesis and multiply it by the first and second terms of the second parenthesis.

Remember to place the addition and subtraction signs in the correct places.

Another way to describe the distributive property:

$(Z + T) \times (X + Y) = ZX + ZY + TX + TY$

$(Z - T) \times (X - Y) = ZX - ZY - TX + TY$

When first learning about the distributive property, we practice only with known whole numbers (without variables or fractions) in order to understand the idea of breaking down a larger number into smaller numbers. In this first stage, we use the distributive property mainly to simplify calculation, especially with large numbers.

For example:

$3\times 102=3\times(100+2)=300+6=306$

$7\times 96=7\times(100-4)=700-28=672$

By this point, most students have already mastered long addition and subtraction, but they might have less experience multiplying large numbers. The distributive property helps them to solve these problems by reducing into simpler multiplications equations.

In middle school, the distributive property gets more interesting. Now, we will start to use not just whole numbers, but variables and exponents too!

For example:

• $(X + 5) \times (X + 6) =$
  $X² + 6X + 5X+ 30 = X² + 11X + 30$
  
• $(X- 7) \times (X- 8) =$
  $X² - 8X - 7X + 56 = X² - 15X + 56$

As we mentioned earlier, there are other rules and properties out there that help us to simplify algebraic expressions. In this section we will briefly look at two of them: the associative property and the commutative property.

The associative property allows us to group several terms of an equation together without changing the final results, by moving the parentheses. However, we can only use this property to solve addition or multiplication exercises.

For example:

• $(10 + 2) + 8 =$
  $10 + (2 + 8) = 10 + 2 + 8 = 20$
  
• $2 \times (3 \times 6) =$
  $(2 \times 3) \times 6 = 2 \times 3 \times 6 = 36$

To learn more about the associative property, you can read the following: The Associative Property.

The commutative property allows us to change the order of the terms in an equation without changing the outcome of the equation. Like the associative property, the commutative property can only be used for addition and multiplication.

Let us look at some examples:

• $2 + 6 = 6 + 2 = 8$
• $3 \times 4 = 4 \times 3 = 12$

To learn more about the commutative property, you can read the following: The Commutative Property.

Task:

Using the distributive property, solve the following:

• $294:3=$
• $105\times 4=$
• $505:5=$
• $207\times 5=$
• $168:8=$

Solutions:

• $294:3=(300-6):3=300:3-6:3=100-2=98$
• $105\times 4=(100+5)\times 4=100\times 4+5\times 4=400+20=420$
• $505:5=(500+5):5=500:5+5:5=100+1=101$
• $207\times 5=(200+7)\times 5=200\times 5+7\times 5=1000+35=1035$
• $168:8=(160+8):8=160:8+8:8=20+1=21$

Task:

Three hundred fifty-one students from a high school were divided into nine equal groups.

How many students are in each group?

Solve the problem using the distributive property.

Solution:

We begin by expressing the problem numerically:

$351:9=(360-9):9=360:9-9:9=40-1=39$

Answer:

There are 39 students in each group.

Task:

Dani bought 15 boxes. In each box there were 9 pieces of candy.

How many pieces of candy did Dani buy in total?

Use the distributive property to solve the problem.

Solution:

We begin by expressing the problem numerically:

$15\times9=(10+5)\times9=10\times9+5\times9=90+45=135$

Answer:

Dani bought 135 pieces of candy.

Task:

Isabel has packed 246 notebooks into 6 equal packages.

How many notebooks has Isabel put in each package?

Use the distributive property to solve the problem.

Solution:

We begin by expressing the problem numerically:

$246:6=(240+6):6=240:6+6:6=40+1=41$

Answer:

Isabel has put 41 notebooks in each packet.

Task:

A mother had 894 euros. She divided the money equally among her three children.

How much money did each child receive?

Use the distributive property to solve the problem.

Solution:

We begin by expressing the problem numerically:

$894:3=(900-6):3=900:3-6:3=300-2=298$

Answer:

Each child received 298 euros.

Task:

Solve the following:

$93:3=\text{?}$

Solution:

We simplify the number 93 into 4 smaller numers to make it easier for us to divide it by 3.

For example $30:3=10$

$93:3=\text{?}$

$(30+30+30+3):3=$

$+30:3=10$

$+30:3=10$

$+30:3=10$

$+3:3=1$

Then we will add the results and we will get:

$93:3=31$

Answer:

$31$

Task:

Solve the following:

$=90:5$

Solution:

We simplify the number 90 into 3 smaller numbers:

(50,20,20)

$90=5-+20+20$

Then we divide each of them by 5 and add the three results.

$+50:5=10$

$+20:5=4$

$+20:5=4$

Which gives us:

$90:5=10+4+4=18$

Answer:

$18$

Task:

Solve the following:

$=72:18$

Solution:

We simplify the number 72 into two smaller numbers $\left(36+36\right)$ and then divide each of them by 18.

$72:18=$

$(36+36):18=$

$+36:18=2$

$+36:18=2$

$72:18=4$

$4$

Answer:

$4$

Task:

Solve the following:

$\left(40+70+35−7\right)×9=$

Solution:

$\left(40+70+35−7\right)×9=$

$40\times9+70\times9+35\times9-7\times9=$

$=360+630+(30+5)9-63$

$=360+630+270+45-63=1242$

Answer:

$1242$

Task:

Solve the following:

$\left(35+4\right)×\left(10+5\right)=$

Solution:

$(35+4)(10+5)=35\times10+35\times5+4\times10+4\times5$

$=350+175+40+20=585$

Answer:

$585$

Task:

Solve the following:

$\left(7x+3\right)×\left(10+4\right)=$

Solution:

$(7X+3)(10+4)=7X\times10+7X\times4+3\times10+3\times4$

$=70X+28X+30+12=98X+42=238$,$-42$

$98X=196$,$:98$

$X=\frac{196}{98}=2$

Answer:

$2$

• $187\times (8-5)=$
• ${2\over3}\times (12+0-5)=$
• $5\times (2{1\over2}+1{1\over6}+{3\over4})=$
• $(10+5+18)\times 4=$
• $(5.5-0.8)\times 5=$
• $340:(12-7)=$
• $(29-4):5=$
• ${5\over1}:(6+1-5)=$
• $18:(5+7+4)=$
• $({1\over7}-{1\over3}):4=$
• $97\times 12=$
• $3\times 36=$
• $120:97=$
• $8:{1\over2}=$
• $151\times 23=$

What is the distributive property?

The distributive property is a method used to simplify expressions into smaller, more manageable pieces.

How is the distributive property used?

In an equation, we use the distributive property to break down a large number into two or more smaller numbers (using addition and subtraction), and then by distributing the multiplication.

Example

• $20 \times 8\times 7=20+8\times 7=20\times 7+8\times 7=140+56=196$

Can we use the distributive property in division?

Of course we can! In an expression with division, we break down the numerator into smaller numbers (using addition and subtraction), and then the division is distributed.

Example

• $150:6=120+30:6=120:6+30:6=20+5=25$

If you found this article helpful, you may also be interested in the following:

• Equivalent expressions / Equivalent algebraic expressions.
• The Associative Property
• The distributive property in the case of divisions
• The distributive property in the case of multiplication
• The distributive property: extension
• The commutative property of addition
• The commutative property of multiplication
• The associative property
• Associative property of addition
• Associative property of multiplication
• Distributive property

For a wide range of mathematics articles visit the Tutorelablog.