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6-A-2-1

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Saved by Levi
on December 5, 2016 at 10:03:00 am

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 real-world and mathematical problems.

In a Nutshell

In the recent past students have been taught to use GEMA (preferred over PEMDAS of BEDMAS) to solve order of operations problems. This is effective in the short term but as a stand alone teaching technique it leaves out basic arithmetic properties that will be utilized heavily in Algebra. By teaching commutative, associative and distributive properties first, the techniques in the acronym GEMA have more meaning. When students understand that multiplication and division are the same operation, “solve multiplication and division left to right” makes sense. Similarly, when students understand that subtraction is simply adding a negative number, “solve addition and subtraction left to right” makes sense. Students are beginning to understand how to represent real world problems and numerical situations with expressions such as four times a number is 4x and three less than a number is x - 3. Once these techniques are mastered, evaluating expressions for a missing positive rational variable is a simple substitution process. Models and pictorial examples are useful to help students understand the properties and how they relate to the expressions.

Student Actions	Teacher Actions
Given an expression students can use accurate and efficient procedures to simplify to a single number. Students will develop an understanding of how and when to apply the arithmetic properties. Demonstrate a deep and flexible understanding that algebraic expressions behave in the same way as numerical expressions. Using efficient and accurate procedures students will evaluate expressions given a positive rational number for the variable.	Build fluency with procedures by offering a wide variety of problems that use all levels of order of operations. Engage students in activities that assist with identifying opportunities to use the arithmetic properties to simplify an expression. Engage students in making connections among mathematical representations and numerical real-world examples. Build fluency and procedures for substitution of a variable by modeling order of operation methods as a culmination of arithmetic properties.
Key Understandings	Misconceptions
Understand the meaning and effect of arithmetic properties on numerical and algebraic expressions. Understand that a variable is a representation of a number and follows the same arithmetic properties as a number. Understand the the associative, commutative, and distributive properties will determine the order in which operations are simplified. Order of operations must be followed inside all grouping symbols first, all exponents must be evaluated, then all multiplication and division from left to right and finally all addition and subtraction from left to right. Recognizing that addition and subtraction are inverse operations, the order in which they appear in the equation does not matter. Both are evaluated from left to right as they appear. Recognizing that multiplication and division are inverse operations, the order in which they appear in the equation does not matter. Both are evaluated from left to right as they appear.	Students may not understand the function of grouping symbols. Students may not understand the distributive property of multiplication. Students may believe that a variable does not represent a number. Students may work arithmetic left to right regardless of the order of operations. Students may think parentheses are the only type of grouping symbols. (See Nix the Tricks resource below.) Students may think all multiplication is done before all division. (See Nix the Tricks resource below.) Students may think all addition is done before all subtraction. (See Nix the Tricks resource below.) Students may think operations inside grouping symbols are done left to right rather than following order of operations.