Learn on PengienVision, Mathematics, Grade 5Chapter 2: Use Models and Strategies to Add and Subtract Decimals

Lesson 1: Mental Math

In this Grade 5 lesson from enVision Mathematics Chapter 2, students learn how to use mental math strategies — including the Commutative Property, Associative Property, and compensation — to add and subtract decimals. Students practice identifying compatible numbers and adjusting values to simplify calculations without paper or a calculator. The lesson builds fluency with decimal operations through real-world problems involving money and multi-addend expressions.

Section 1

Grouping Compatible Numbers Using Properties of Addition

Property

The Commutative Property lets you add numbers in any order: a+b=b+aa + b = b + a.
The Associative Property lets you change the grouping of numbers being added: (a+b)+c=a+(b+c)(a + b) + c = a + (b + c).
Together, these properties allow you to rearrange and group compatible numbers to simplify mental addition.

Examples

Section 2

Using Compensation for Addition

Property

To mentally calculate the sum a+ba + b, you can change one addend by an amount cc to make it an easier number to work with, and then adjust the sum by subtracting cc.

a+b=a+(b+c)−ca + b = a + (b + c) - c

Examples

Section 3

Using Compensation for Subtraction

Property

To mentally calculate a difference like a−ba - b, you can change the subtrahend (bb) to a nearby, easier number (cc). Then, subtract cc from aa and adjust the result by adding the difference between cc and bb.

a−b=(a−c)+(c−b)a - b = (a - c) + (c - b)

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Grouping Compatible Numbers Using Properties of Addition

Property

The Commutative Property lets you add numbers in any order: a+b=b+aa + b = b + a.
The Associative Property lets you change the grouping of numbers being added: (a+b)+c=a+(b+c)(a + b) + c = a + (b + c).
Together, these properties allow you to rearrange and group compatible numbers to simplify mental addition.

Examples

Section 2

Using Compensation for Addition

Property

To mentally calculate the sum a+ba + b, you can change one addend by an amount cc to make it an easier number to work with, and then adjust the sum by subtracting cc.

a+b=a+(b+c)−ca + b = a + (b + c) - c

Examples

Section 3

Using Compensation for Subtraction

Property

To mentally calculate a difference like a−ba - b, you can change the subtrahend (bb) to a nearby, easier number (cc). Then, subtract cc from aa and adjust the result by adding the difference between cc and bb.

a−b=(a−c)+(c−b)a - b = (a - c) + (c - b)

Examples