MA.5.AR.2 Algebraic Reasoning
Demonstrate an understanding of equality, the order of operations and equivalent numerical expressions
Evaluate multi-step numerical expressions using order of operations. Example: Patti says the expression 12 ÷ 2 × 3 is equivalent to 18 because she works each operation from left to right. Gladys says the expression 12 ÷ 2 × 3 is equivalent to 2 because first multiplies 2 × 3 then divides 6 into 12. David says that Patti is correctly using order of operations and suggests that if parentheses were added, it would give more clarity.
Clarification 1: Multi-step expressions are limited to any combination of arithmetic operations, including parentheses, with whole numbers, decimals and fractions.
Clarification 2: Within this benchmark, the expectation is not to include exponents or nested grouping symbols.
Clarification 3: Decimals are limited to hundredths. Expressions cannot include division of a fraction by a fraction.
Purpose and Instructional Strategies
The purpose of this benchmark is for students to use the order of operations to evaluate numerical expressions. In Grade 4, students had experience with numerical expressions involving all four operations (MA.4.AR.2.1/2.2), but the focus was not on order of operations. In Grade 6, students will be evaluating algebraic expressions using substitution and these expressions can include negative numbers (MA.6.AR.1.3).
Begin instruction by exposing student to expressions that have two operations without any grouping symbols, before introducing expressions with multiple operations. Use the same digits, with the operations in a different order, and have students evaluate the expressions, then discuss why the value of the expression is different. For example, have students evaluate 6 × 3 + 7 and 6 + 3 × 7.
In Grade 5, students should learn to first work to simplify within any parentheses, if present in the expression. Within the parentheses, the order of operations is followed. Next, while reading left to right, perform any multiplication and division in the order in which it appears. Finally, while reading from left to right, perform addition and subtraction in the order in which it appears.
During instruction, students should be expected to explain how they used the order of operations to evaluate expressions and share with others. To address misconceptions around the order of operations, instruction should include reasoning and error analysis tasks for students to complete (MTR.3.1, MTR.4.1, MTR.5.1).
Common Misconceptions or Errors
When students learn mnemonics like PEMDAS to perform the order of operations, they can confuse that multiplication must always be performed before division, and likewise addition before subtraction. Students should have experiences solving expressions with multiple instances of procedural operations and their inverse, such as addition and subtraction, so they learn how to solve them left to right.
Instructional Task 1
The two equations below are very similar. Are both equations true? Why or why not? Equation One: 4 × 6 + 3 × 2 + 4 = 34 Equation Two: 4 × (6 + 3 × 2 + 4) = 64
Instructional Task 2
Part A. Insert one set of parentheses around two numbers in the expression below. Then evaluate the expression. 40 ÷ 5 × 2 + 6 Part B. Now insert one set of parentheses around a different pair of numbers. Then evaluate this expression. 40 ÷ 5 × 2 + 6
Instructional Item 1
What is the value of the numerical expression below: (2.45 + 3.05) ÷ (7.15 − 2.15)
Instructional Item 2
A numerical expression is evaluated as shown. 1 2 × (3 × 5 + 1) − 2 In which step does the first mistake appear
a. Step 1: 1/2 × (15 + 1) − 2
b. Step 2: 1/2 × 14
c. Step 3: 14/2
d. Step 4: 7