 # Replacing Numbers with Letters

Videos and solutions to help Grade 6 students learn how to replace numbers with letters in an expression. They will also discover the commutative properties of addition and multiplication, the additive identity property of zero, and the multiplicative identity property of one.

New York State Common Core Math Module 4, Grade 6, Lesson 8

Related Topics:
Lesson Plans and Worksheets for Grade 6
Lesson Plans and Worksheets for all Grades

Lesson 8 Student Outcomes

Students understand that a letter in an expression or an equation can represent a number. When that number is replaced with a letter, an expression or an equation is stated.

Students discover the commutative properties of addition and multiplication, the additive identity property of zero, and the multiplicative identity property of one. They determine that g ÷ 1 = g, g ÷ g = 1 and 1 ÷ g = 1/g

Opening Exercise

How many of these statements are true?
4 + 0 = 4
4 × 1 = 4
4 ÷ 1 = 4
4 × 0 = 0
1 ÷ 4 = 1/4

How many of those statements would be true if the number 4 was replaced with the number 7 in each of the number sentences?
Would the number sentences be true if we were to replace the number 4 with any other number?
What if we replaced the number 4 with the number 0? Would each of the number sentences be true?
Division by zero is undefined. You cannot make zero groups of objects, and group size cannot be zero.
It appears that we can replace the number 4 with any non-zero number and each of the number sentences will be true.
A letter in an expression can represent a number. When that number is replaced with a letter, an expression is stated.

Example 1: Additive Identity Property of Zero

g + 0 = g

Remember a letter in a mathematical expression represents a number. Can we replace with any number?
Choose a value for g and replace with that number in the number sentence. What do you observe?
Is the number sentence true for all values of g?
Write the mathematical language for this property below:
Additive Identity Property of Zero: Any number added to zero equals itself. The number’s identity does not change.

Example 2: Multiplicative Identity Property of One

g × 1 = g

Remember a letter in a mathematical expression represents a number. Can we replace with any number?
Choose a value for g and replace with that number in the number sentence. What do you observe?
Is the number sentence true for all values of g?
Write the mathematical language for this property below:
Multiplicative Identity Property of One: Any number multiplied by 1 equals itself. The number’s identity does not change.

Example 3: Commutative Property of Addition and Multiplication

3 + 4 = 4 + 3
3 × 4 = 4 × 3
3 + 3 + 3 + 3 = 4 × 3
3 ÷ 4 = 3/4

Replace the 3s in these equations with the letter a.
Choose a value for and replace a with that number in each of the number sentences. What do you observe?
Are the number sentences true for all values of ? Experiment with different values before making your claim.
Now write the number sentences again, this time replacing the number 4 with a variable, b.
Are the first two number sentences true for all values of and ? Experiment with different values before making your claim.
Write the mathematical language for this property below:
Commutative property of addition and commutative property of multiplication. These are sometimes called the “any-order properties”. Problem Set
1. State the commutative property of addition using the variables a and b.

2. State the commutative property of multiplication using the variables a and b.

3. State the additive property of zero using the variable b.

4. State the multiplicative identity property of one using the variable b.

5. Demonstrate the property listed in the first column by filling in the third column of the table.
Commutative Property of Addition: 25 + c =
Commutative Property of Multiplication: l × w =
Additive Property of Zero: h + 0 =
Multiplicative Identity Property of One: v × 1 =

6. Why is there no commutative property for subtraction or division? Show examples.

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