Distributive Property

Background

We've found that addition and multiplication are both commutative (a + b = b + a and the same for multiplication) and associative (a + (b + c) = (a + b) + c and the same for multiplication). Are there any more properties we can explore? Yup.

Question

Does 2 x (4 + 3) = 8 + 6?

Answer

Yes - they both equal 14. We can show that from the Distributive Property.

Analysis

A form of multiplication that happens commonly is to have a number, like the 2, multiplying across the terms in a bracket, just like we're seeing with the question 2 x (4 + 3).

We can work our question using our usual procedure of the Order of Operations, and when we do that, we do the bracket first:

2 x (4 + 3) = 2 x 7 = 14

Awesome! But can we work this a different way? And the answer is yes - if we look at the Factors and Factoring entry, where we "factored" out the 2 in order to get from 8 + 6 to 2 x (4 + 3), we can also go the other way:

2 x (4 + 3) = 8 + 6 = 14

This ability to perform the multiplication first is called the Distributive Property and it can be expressed this way:

a x (b + c) = a x b + a x c

Ok - so how does this work?

When we did 2 x (4 + 3) = 2 x 7 = 14, we were looking for the number of times we'd move 2 to the right on a number line:



When we did it the other way, 2 x (4 + 3) = 8 + 6 = 14, what we're in effect doing is making two smaller jumps to have it add up to the big jump:



There are other ways we can use this property and we'll explore those in future entries.

Vocabulary used:

For more information check out these links (comment to add your favourite link):

Where might you have come from?

Fact-orials Index

Operations:

• Multiplication
• Order of Operations (PEMDAS)

Associated Operations:

• Factors and Factoring

Properties:

• Associative Property

Where might we go?

FOIL

Associative Property

Background

We looked at the ordering of terms in the Commutative Property post, asking if a operator b was the same thing as b operator a. So what happens if we group things in different ways? That is, if we have (a operator b) same operator c, is it the same as a operator (b same operator c)?

Question

Evaluate the following to see if they are true:

• 3 + (2 + 4) = (3 + 2) + 4
• 3 - ( 2 - 4) = (3 - 2) - 4
• 3 x (2 x 4) = (3 x 2) x 4
• 3 ÷ (2 ÷ 4) = (3 ÷ 2) ÷ 4  

Answer

Addition and Multiplication are Associative and so the first and third questions above are true in that they equal each other. The subtraction and division questions are false - they don't equal each other.

Analysis

Let's work each question to test to see if they are true or not:

Addition

3 + (2 + 4) = (3 + 2) + 4

3 + 6 = 5 + 4

9 = 9 ✅

And in fact this is true regardless of the numbers we put in for a, b, and c.

Addition is associative.

Subtraction

3 - (2 - 4) = (3 - 2) - 4

3 - (-2) = 1 - 4

5 = -3 X

While we can choose certain values for a, b, and c to make the expression true, it's not always true (like we just saw here).

Subtraction is not associative.

Multiplication

3 x (2 x 4) = (3 x 2) x 4

3 x 8 = 6 x 4

24 = 24 ✅

And in fact this is true regardless of the numbers we put in for a, b, and c.

Multiplication is associative.

Division

3 ÷ (2 ÷ 4) = (3 ÷ 2) ÷ 4

I'll rearrange how we write these to reflect the groupings:







   X

While we can choose certain values for a, b, and c to make the expression true, it's not always true (like we just saw here).

Subtraction is not associative.

Vocabulary used:

• Commutative Property - the property that states that a + b = b + a, and a x b = b x a

For more information check out these links (comment to add your favourite link):

Where might you have come from?

Fact-orials Index

Operations:

• Addition
• Subtraction
• Multiplication
• Division
• Grouping Terms, Parentheses
• Order of Operations (PEMDAS)

Operations with different kinds of numbers:

• Addition, Subtraction, and Whole Numbers

Properties:

• Commutative Property

Where might we go?

Operations with different kinds of numbers:

• Addition, Subtraction, and Whole Numbers - Practice Problems

Properties:

• Distributive Property

Commutative Property

Background 

When we're doing arithmetic (addition, subtraction, multiplication, division), we have a number and an operator and another number. Does it matter which number comes first?

Question 

Does 3 + 2 = 2 + 3?

Does 3 - 2 = 2 - 3?

Does 3 x 2 = 2 x 3?

Does 3 ÷ 2  = 2  ÷ 3?

Which operations are Commutative? 

 Answer 

Of the four arithmetic operations, only Addition and Multiplication are Commutative.

Analysis  

We're talking about the importance of knowing which term goes first. Does it really matter? It's the Commutative Property that helps to tell us.

For an operation to be Commutative, it needs to satisfy the relation A operator B = B operator A.

We can work examples to show which operations are Commutative and which aren't.

To help with my examples, I'm going to say that A = 3 and B = 2.

Addition

For addition to be commutative, A + B has to be equal to B + A. Is it?

3 + 2 = 2 + 3

5 = 5 ✅

And we can see this on a number line (3 + 2 is blue and 2 + 3 is purple):



Addition is commutative.

Subtraction

For subtraction to be commutative, A - B has to be equal to B - A. Is it?

3 - 2 = 2 - 3

1 = -1 No.

And we can see this on a number line (3 - 2 is blue and 2 - 3 is purple):



Subtraction is not commutative.

Multiplication

For multiplication to be commutative, A X B has to be equal to B X A. Is it?

3 x 2 = 2 x 3

6 = 6 ✅

And we can see this using cubes.

This is 3 x 2:



This is 2 x 3:



Multiplication is commutative.

Division

For division to be commutative, A ÷ B has to be equal to B ÷ A. Is it?

3 ÷ 2 = 2 ÷ 3

   No.

We can do this with blocks with  above and  below (the result of the division is one section of each of the set of blocks):



Division is not commutative.

For more information check out these links (comment to add your favourite link): 

Where might you have come from? 

Fact-orials Index

Numbers:

• Negative Numbers

Operations:

• Addition
• Subtraction
• Multiplication
• Division
• Order of Operations (PEMDAS)

Where might we go?

Operations with different kinds of numbers:

• Multiplication Table
• Addition, Subtraction, and Whole Numbers - Practice Problems

Properties:

• Associative Property