6.A.2.1 Generate equivalent expressions and evaluate expressions involving positive rational numbers by applying the commutative, associative, and distributive properties and order of operations to solve realworld and mathematical problems.
In a Nutshell
The commutative property and associative property can be applied to expressions involving addition and multiplication. The distributive property can be used to generate equivalent expressions involving multiplication over addition and subtraction. By teaching students how to apply the commutative, associative and distributive properties, students will be able to generate equivalent expressions that will make it easier for students to solve mental math problems like 2 3 1/4 = 2(3 + 1/4) = (2 3) + (2 1/4) = 6 + 1/2 = 6 1/2. Students come to sixth grade with prior knowledge of solving problems involving the commutative, associative, and distributive property with whole numbers, but at this grade level, students will learn how to use these properties with positive rational numbers. Students will also generate equivalent expressions using the order of operations which is: 1. grouping symbols and evaluating expression above and below a fraction bar, 2. exponents, 3. multiplication, including division, left to right, and 4. addition, including subtraction, left to right. Student have experience solving order of operations from fifth grade, but expressions involving exponents is new to them in sixth grade. Once these techniques are mastered, evaluating an algebraic expression given a positive rational number for a variable is a simple substitution process.
Student Actions

Teacher Actions


Develop a deep and flexible conceptual understanding of how and when to apply the commutative, associative, and distributive property by exploring the application of these skills in realworld and mathematical situations.

Develop accurate and appropriate procedural fluency by exploring the use of order of operations including exponents to generate equivalent expressions in realworld and mathematical situations.

Develop accurate and appropriate procedural fluency by engaging in real world and mathematical tasks that require evaluating algebraic expressions given a positive rational number for a variable.
 Develop the ability to communicate mathematically through discussion and writing about strategies used to generate and evaluate expressions by applying the commutative, associative, and distributive property and order of operations.


Build procedural fluency by providing real world tasks that allow students to decide when to apply the commutative, associative, and distributive property.

Build procedural fluency by facilitating student exploration in generating equivalent expressions from realworld or mathematical situations using order of operations including exponents.

Build procedural fluency by facilitating student exploration with evaluating algebraic expressions in realworld and mathematical situations.
 Pose purposeful questions to assess studentsâ€™ reasoning and understanding about generating and evaluating expressions by applying the commutative, associative, and distributive property and order of operations to realworld and mathematical situations. For example, how would the distributive property be helpful in evaluating 2 3 1/4?

Key Understandings

Misconceptions


Students will understand how to apply and identify the commutative, associative, and distributive property to generate equivalent numerical expressions. For example, 2 3 1/4 = 2(3 + 1/4) = (2 3) + (2 1/4) = 6 + 1/2 = 6 1/2.

Students will understand that an exponent tells how many times to multiply the base to itself.

Students will understand how to apply the order of operations including exponents to generate equivalent expressions involving positive rational numbers.

Students will understand that a variable is a representation of a number.

Students will understand how to substitute positive rational numbers for variables in algebraic expressions in order to evaluate the expressions.


Students may work arithmetic left to right regardless of the order of operations.

Students may not understand that a variable represents an unknown number.

Students may forget to multiply the number outside the parentheses to every number inside the parentheses when using the distributive property. For example, students may say 4(2 + 7) is equivalent to 8 + 7.

Students may think that 3 x 5 is equivalent to 3 x 3 + 2, not recognizing the need for parentheses.

Students may misinterpret exponents. For example, students may think 4^{2} is equivalent to 4 x 2.
 Students may evaluate an expression like 5x to be 53 when given x = 3 instead of the correct expression, 5 3.

Resources
OKMath Framework Introduction
6th Grade Introduction
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