Problem 2: Recall the expression we studied in class: 48/2(9 + 3), and consider the following calculations performed by a student to evaluate the expression: = 2 2(9 + 3) + 29 + 3) (9 + 3) (a) This student equated the original horizontal alignment with vertical alignment: 48/2(9 + 3) = 209 + 3 Explain how this might make sense. (b) In the calculations shown on the line labeled “the student strictly adhering to the order of operations,” respond yes or no and then justify your response by appealing to the student’s work.
The Correct Answer and Explanation is:
Let’s break down the given expression and the student’s work step by step.
Expression:
48/2(9 + 3)
The standard approach is to evaluate this using the order of operations (PEMDAS/BODMAS), which states we should:
- Parentheses: Solve any expressions inside parentheses first. (9+3)=12(9 + 3) = 12(9+3)=12
- Exponents: There are none in this problem.
- Multiplication and Division (from left to right): This is where some confusion can arise. 48/2×1248 / 2 \times 1248/2×12 According to the order of operations, multiplication and division are done from left to right, not strictly multiplication before division.
So, we evaluate it as: 48/2=24and then24×12=28848 / 2 = 24 \quad \text{and then} \quad 24 \times 12 = 28848/2=24and then24×12=288
Thus, the correct answer is 288.
(a) Equating Horizontal Alignment with Vertical Alignment:
The student seems to interpret the expression incorrectly by writing:48/2(9+3)=209+348 / 2(9 + 3) = 209 + 348/2(9+3)=209+3
This transformation doesn’t make sense because 48 / 2(9 + 3) is a single mathematical expression that should be evaluated based on the order of operations, not by arbitrarily changing its structure. The student appears to have misunderstood the placement of parentheses, potentially associating 2(9 + 3) as a separate term that should be evaluated as 2 × 12 = 24, but this is not correct. The division and multiplication should be performed from left to right, leading to a final result of 288, not 209 + 3.
(b) Strictly Adhering to the Order of Operations:
No. The student’s work on the line labeled “the student strictly adhering to the order of operations” is not correct. The order of operations explicitly requires that division and multiplication be performed from left to right. By treating 48/2(9 + 3) as 48 / (2 × 12), the student has violated the correct procedure of evaluating from left to right. Thus, the student’s interpretation leads to an incorrect result.
In conclusion, the correct way to evaluate 48/2(9 + 3) is to perform division and multiplication from left to right, leading to the final answer of 288.
