Learn on PengiReveal Math, AcceleratedUnit 5: Solve Problems Involving Operations with Integers and Rational Numbers

Lesson 5-6: Use Properties of Operations to Add, Subtract, Multiply, and Divide

In this Grade 7 lesson from Reveal Math, Accelerated, students apply the order of operations to evaluate expressions involving integers and rational numbers, including fractions, decimals, and negative exponents. Using the Fahrenheit-Celsius conversion formula as a real-world context, students practice substituting values into algebraic expressions and simplifying step by step. The lesson is part of Unit 5 and builds fluency with mixed operations across all rational number types.

Section 1

Order of operations

Property

1 Perform any operations inside parentheses, or above or below a fraction bar.

2 Compute all indicated powers.

3 Perform all multiplications and divisions in the order in which they occur from left to right.

Section 2

Raising a Number to a Power

Property

An even exponent on a negative base gives a positive result, while an odd exponent gives a negative result. Note the difference: (a)n(-a)^n is not the same as an-a^n.

Examples

(5)2=(5)(5)=25(-5)^2 = (-5)(-5) = 25
(5)3=(5)(5)(5)=125(-5)^3 = (-5)(-5)(-5) = -125
52=(55)=25-5^2 = -(5 \cdot 5) = -25

Explanation

Parentheses are VIP passes! In (3)4(-3)^4, the negative is included in the repeated multiplication, resulting in a positive. But in 34-3^4, the negative is an outsider, applied only after the power is calculated, making the final answer negative. Always check if the negative sign is inside the parentheses to see if it gets to join the party!

Section 3

Evaluating Expressions with Rational Numbers

Property

When evaluating expressions containing rational numbers (positive and negative fractions, decimals, and integers), apply the standard Order of Operations and Properties of Operations (such as the Commutative, Associative, and Distributive Properties) to simplify and solve.

Examples

Section 4

Evaluate Variable Expressions

Property

To evaluate a variable expression means to substitute a number for the variable in the expression and then simplify the resulting expression. When substituting negative numbers, it is often helpful to use parentheses to ensure the operations are performed correctly.

Examples

  • To evaluate 3x2+5x13x^2 + 5x - 1 when x=2x = -2, substitute to get 3(2)2+5(2)13(-2)^2 + 5(-2) - 1. This simplifies to 3(4)101=12101=13(4) - 10 - 1 = 12 - 10 - 1 = 1.
  • To evaluate (a+b)2(a+b)^2 when a=10a = -10 and b=7b = 7, substitute to get (10+7)2(-10+7)^2. This simplifies to (3)2=9(-3)^2 = 9.
  • To evaluate 50y50 - y when y=15y = -15, substitute to get 50(15)50 - (-15). This simplifies to 50+15=6550 + 15 = 65.

Explanation

Evaluating an expression is like plugging a specific value into a template. The variable is a placeholder, and you replace it with the given number. Using parentheses, especially for negative numbers, prevents mistakes with signs and exponents.

Book overview

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Unit 5: Solve Problems Involving Operations with Integers and Rational Numbers

  1. Lesson 1

    Lesson 5-1: Add Integers and Rational Numbers

  2. Lesson 2

    Lesson 5-2: Understand Additive Inverses

  3. Lesson 3

    Lesson 5-3: Subtract Integers and Rational Numbers

  4. Lesson 4

    Lesson 5-4: Multiply Integers and Rational Numbers

  5. Lesson 5

    Lesson 5-5: Divide Integers and Rational Numbers

  6. Lesson 6Current

    Lesson 5-6: Use Properties of Operations to Add, Subtract, Multiply, and Divide

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Order of operations

Property

1 Perform any operations inside parentheses, or above or below a fraction bar.

2 Compute all indicated powers.

3 Perform all multiplications and divisions in the order in which they occur from left to right.

Section 2

Raising a Number to a Power

Property

An even exponent on a negative base gives a positive result, while an odd exponent gives a negative result. Note the difference: (a)n(-a)^n is not the same as an-a^n.

Examples

(5)2=(5)(5)=25(-5)^2 = (-5)(-5) = 25
(5)3=(5)(5)(5)=125(-5)^3 = (-5)(-5)(-5) = -125
52=(55)=25-5^2 = -(5 \cdot 5) = -25

Explanation

Parentheses are VIP passes! In (3)4(-3)^4, the negative is included in the repeated multiplication, resulting in a positive. But in 34-3^4, the negative is an outsider, applied only after the power is calculated, making the final answer negative. Always check if the negative sign is inside the parentheses to see if it gets to join the party!

Section 3

Evaluating Expressions with Rational Numbers

Property

When evaluating expressions containing rational numbers (positive and negative fractions, decimals, and integers), apply the standard Order of Operations and Properties of Operations (such as the Commutative, Associative, and Distributive Properties) to simplify and solve.

Examples

Section 4

Evaluate Variable Expressions

Property

To evaluate a variable expression means to substitute a number for the variable in the expression and then simplify the resulting expression. When substituting negative numbers, it is often helpful to use parentheses to ensure the operations are performed correctly.

Examples

  • To evaluate 3x2+5x13x^2 + 5x - 1 when x=2x = -2, substitute to get 3(2)2+5(2)13(-2)^2 + 5(-2) - 1. This simplifies to 3(4)101=12101=13(4) - 10 - 1 = 12 - 10 - 1 = 1.
  • To evaluate (a+b)2(a+b)^2 when a=10a = -10 and b=7b = 7, substitute to get (10+7)2(-10+7)^2. This simplifies to (3)2=9(-3)^2 = 9.
  • To evaluate 50y50 - y when y=15y = -15, substitute to get 50(15)50 - (-15). This simplifies to 50+15=6550 + 15 = 65.

Explanation

Evaluating an expression is like plugging a specific value into a template. The variable is a placeholder, and you replace it with the given number. Using parentheses, especially for negative numbers, prevents mistakes with signs and exponents.

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Unit 5: Solve Problems Involving Operations with Integers and Rational Numbers

  1. Lesson 1

    Lesson 5-1: Add Integers and Rational Numbers

  2. Lesson 2

    Lesson 5-2: Understand Additive Inverses

  3. Lesson 3

    Lesson 5-3: Subtract Integers and Rational Numbers

  4. Lesson 4

    Lesson 5-4: Multiply Integers and Rational Numbers

  5. Lesson 5

    Lesson 5-5: Divide Integers and Rational Numbers

  6. Lesson 6Current

    Lesson 5-6: Use Properties of Operations to Add, Subtract, Multiply, and Divide