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Teacher Guide: Properties of Numbers

Learn the fundamental rules that govern how numbers behave in arithmetic operations.

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Class quiz

10 questions on Basic Operations. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Identify and name the four main properties of numbers
  • Apply the commutative property to addition and multiplication
  • Apply the associative property to addition and multiplication
  • Use the identity property to identify when a number remains unchanged
  • Apply the distributive property to simplify calculations
  • Explain why the commutative and associative properties do not work for subtraction and division
Prerequisites
  • • Addition and subtraction of whole numbers
  • • Multiplication and division of whole numbers
  • • Understanding of parentheses and order of operations
Discussion Starters
  • 1. Can you think of a real-life example where changing the order doesn't matter?
  • 2. Why do you think mathematicians named these rules 'properties'?
  • 3. If you had to explain the distributive property to a younger student, what would you say?
  • 4. Which property do you think is most useful for mental math? Why?
Common Misconceptions

All properties work for all operations

Remediation: Provide counterexamples: Show that 8−3≠3−88 - 3 \neq 3 - 8 and 12÷4≠4÷1212 \div 4 \neq 4 \div 12. Have students verify with calculators.

The identity property means any number plus itself

Remediation: Clarify that identity means 'stays the same.' Zero is the additive identity because adding it changes nothing. Compare to looking in a mirror - your reflection is identical.

The distributive property works with subtraction inside: a(b−c)=ab+aca(b - c) = ab + ac

Remediation: Show that distribution preserves the sign: a(b−c)=ab−aca(b - c) = ab - ac. The minus sign distributes too!

Differentiation Ideas

For Struggling Students:

  • • Use physical manipulatives (blocks, counters) to demonstrate properties
  • • Focus on commutative property first with single-digit numbers
  • • Provide property reference cards with examples

For On-Level Students:

  • • Apply properties to two-digit mental math problems
  • • Identify which property is being used in given equations
  • • Create their own examples of each property

For Advanced Students:

  • • Prove why subtraction is not commutative using algebra
  • • Explore the distributive property with three addends: a(b+c+d)a(b + c + d)
  • • Apply properties to simplify algebraic expressions
Standards Alignment
  • 3.OA.B.5 (CCSS.MATH.CONTENT.3.OA.B.5)

    Apply properties of operations as strategies to multiply and divide

  • 4.NBT.B.5 (CCSS.MATH.CONTENT.4.NBT.B.5)

    Multiply using strategies based on place value and the properties of operations

  • 6.EE.A.3 (CCSS.MATH.CONTENT.6.EE.A.3)

    Apply the properties of operations to generate equivalent expressions

Lesson Resources
  • visualArea Model Explorer

    Show 6×(4+3)6 \times (4 + 3) as one rectangle split into two parts

  • activityProperty Sorting Game

    Match equations to their properties

  • worksheetMental Math Challenge

    Use properties to calculate faster

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

Properties of numbers are rules that describe how numbers behave when we add, subtract, multiply, or divide them. These properties help us understand patterns and simplify calculations.
The four main properties are:
  1. 1.Commutative Property: The order of numbers doesn't matter
a+b=b+aanda×b=b×aa + b = b + a \quad \text{and} \quad a \times b = b \times a
  1. 2.Associative Property: The grouping of numbers doesn't matter
(a+b)+c=a+(b+c)and(a×b)×c=a×(b×c)(a + b) + c = a + (b + c) \quad \text{and} \quad (a \times b) \times c = a \times (b \times c)
  1. 3.Identity Property: Special numbers that don't change a value
a+0=aanda×1=aa + 0 = a \quad \text{and} \quad a \times 1 = a
  1. 4.Distributive Property: Multiplication distributes over addition
a×(b+c)=a×b+a×ca \times (b + c) = a \times b + a \times c

Worked Examples

Show that 4+9=9+44 + 9 = 9 + 4
1

Calculate the left side

4+9=134 + 9 = 13
2

Calculate the right side

9+4=139 + 4 = 13
3

Compare both sides

13=1313 = 13 ✓ → Equal!

Common Mistakes

Thinking subtraction is commutative (5−3=3−55 - 3 = 3 - 5)

Why it's wrong: The commutative property only works for addition and multiplication, NOT for subtraction or division.

Correct: 5−3=25 - 3 = 2, but 3−5=−23 - 5 = -2. These are not equal!

Thinking division is associative: (8÷4)÷2=8÷(4÷2)(8 \div 4) \div 2 = 8 \div (4 \div 2)

Why it's wrong: The associative property only works for addition and multiplication, NOT for subtraction or division.

Correct: (8÷4)÷2=2÷2=1(8 \div 4) \div 2 = 2 \div 2 = 1, but 8÷(4÷2)=8÷2=48 \div (4 \div 2) = 8 \div 2 = 4. Not equal!

Confusing identity numbers: Using 1 for addition or 0 for multiplication

Why it's wrong: Adding 1 changes a number, and multiplying by 0 gives 0, not the original number.

Correct: For addition, the identity is 0 (a+0=aa + 0 = a). For multiplication, the identity is 1 (a×1=aa \times 1 = a).

Why It Matters

Understanding number properties helps you:
  • Calculate faster: Rearrange numbers to make mental math easier
  • Check your work: Verify answers using different groupings
  • Solve equations: Apply properties to simplify complex problems
  • Understand algebra: These same properties apply to variables and expressions
For example, instead of calculating 7+38+37 + 38 + 3, you can use the commutative property to reorder it as 7+3+38=10+38=487 + 3 + 38 = 10 + 38 = 48 - much easier!

Real World Applications

Shopping Calculations

Use the commutative and associative properties to add prices quickly.

Example:

Items cost 7 dollars, 5 dollars, and 13 dollars. Instead of adding 7+5+137 + 5 + 13 in order, swap and group: (7+13)+5=20+5=25(7 + 13) + 5 = 20 + 5 = 25 dollars.

1Try It Yourself

You're buying items that cost 8 dollars, 12 dollars, 6 dollars, and 4 dollars.

Rearrange to find the total using friendly numbers.

Step 1: Write the mathematical expression

Group numbers that add to 10 or 20:

Calculating Areas

The distributive property helps calculate areas of combined shapes.

Example:

A garden is 5 meters wide. One section is 8 meters long, another is 2 meters long. Total area: 5×(8+2)=5×10=505 \times (8 + 2) = 5 \times 10 = 50 square meters.

2Try It Yourself

A floor has two sections: 4 meters by 6 meters and 4 meters by 9 meters.

Use the distributive property to find the total area.

Step 1: Write the mathematical expression

Write as: 4×(6+9)4 \times (6 + 9)

Key Takeaways

  • 1Commutative Property: Order doesn't matter for + and ×\times (a+b=b+aa + b = b + a)
  • 2Associative Property: Grouping doesn't matter for + and ×\times ((a+b)+c=a+(b+c)(a+b)+c = a+(b+c))
  • 3Identity Property: a+0=aa + 0 = a and a×1=aa \times 1 = a
  • 4Distributive Property: a×(b+c)=a×b+a×ca \times (b + c) = a \times b + a \times c
  • 5The commutative and associative properties do NOT work for subtraction or division

Frequently Asked Questions

Why don't the commutative and associative properties work for subtraction and division?

Subtraction and division are not commutative: 5−3≠3−55 - 3 \neq 3 - 5, because the order decides what is taken away from what, and 12÷3≠3÷1212 \div 3 \neq 3 \div 12. They are not associative either: (10−5)−2=3(10 - 5) - 2 = 3, but 10−(5−2)=710 - (5 - 2) = 7. Multiplication does distribute over subtraction, though: a×(b−c)=a×b−a×ca \times (b - c) = a \times b - a \times c.

What is the zero property of multiplication?

Any number multiplied by zero equals zero: a×0=0a \times 0 = 0. This is different from the identity property, which keeps the number unchanged.

How do these properties help in algebra?

The same properties apply to variables! For example, x+y=y+xx + y = y + x (commutative) and 2(x+3)=2x+62(x + 3) = 2x + 6 (distributive). Understanding these makes algebra much easier.

Glossary

Commutative Property
A property stating that changing the order of numbers in addition or multiplication does not change the result
Associative Property
A property stating that changing the grouping of numbers in addition or multiplication does not change the result
Identity Property
Adding 0 or multiplying by 1 leaves a number unchanged
Distributive Property
Multiplying a number by a sum is the same as multiplying by each addend and then adding
Addend
A number being added to another number
Factor
A number being multiplied by another number

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