The original expression is \(5 + 3 \times(8-2) \div 3\). Step 1 correctly evaluates the parentheses: \(8-2=6\), resulting in \(5 + 3 \times 6 \div 3\). Step 2 shows \(8 \times 6 \div 3\), which means the student incorrectly added \(5+3\) before performing multiplication and division. According to the order of operations, multiplication and division must be done before addition. Therefore, the first mistake was made in Step 2.

Following the order of operations, treat the absolute value bars as grouping symbols. First, evaluate the expressions inside them: \(|-12| = 12\) and \(|5 - 9| = |-4| = 4\). The expression becomes \(20 - 12 \div 3 + 4\). Next, perform the division: \(12 \div 3 = 4\). The expression is now \(20 - 4 + 4\). Finally, perform addition and subtraction from left to right: \(20 - 4 = 16\), and \(16 + 4 = 20\).

This question tests the ISEE Middle Level skill of evaluating expressions using the order of operations. The order of operations ensures consistent results in mathematical expressions, following the hierarchy of parentheses, exponents, multiplication and division, and addition and subtraction (PEMDAS). In this expression $(8 ÷ 2) × (3^2 - 5) - rac{1}{2}$, we first evaluate within each set of parentheses: $8 ÷ 2 = 4$ and $3^2 - 5 = 9 - 5 = 4$, then multiply $4 × 4 = 16$, and finally subtract $16 - rac{1}{2} = rac{32}{2} - rac{1}{2} = rac{31}{2}$. Choice C is correct because it reflects the proper application of the order of operations, computing the expression correctly to arrive at $rac{31}{2}$. The other choices represent common errors such as performing operations out of order or making arithmetic mistakes. Teaching strategies: Use practice problems that require identifying and correcting common errors, and have students verbalize each step as they work through the problem.

Follow the order of operations, starting with the innermost parentheses: \(9-3 = 6\). The expression becomes \(5 + 2[-4 + 6^2 \div 12]\). Inside the brackets, evaluate the exponent: \(6^2 = 36\). The expression is now \(5 + 2[-4 + 36 \div 12]\). Still inside the brackets, perform the division: \(36 \div 12 = 3\). The expression becomes \(5 + 2[-4 + 3]\). Evaluate inside the brackets: \(-4 + 3 = -1\). The expression is now \(5 + 2(-1)\). Perform the multiplication: \(2(-1) = -2\). Finally, perform the addition: \(5 + (-2) = 3\).

When you see a problem asking about parentheses placement, you're working with the order of operations. Parentheses change which calculations happen first, which can dramatically affect the final result. You need to evaluate each option systematically to find which gives you 20.

Let's work through the correct answer first. In choice D, $$6 \times(8 - 5) + 2$$, you solve the parentheses first: $$8 - 5 = 3$$. Then multiply: $$6 \times 3 = 18$$. Finally add: $$18 + 2 = 20$$. This matches our target value.

Now let's see why the other options don't work. Choice A gives us $$6 \times(8 - 5 + 2) = 6 \times 5 = 30$$, which is too large. Choice B, $$(6 \times 8) - 5 + 2$$, equals $$48 - 5 + 2 = 45$$, also too large. Choice C yields $$6 \times 8 - (5 + 2) = 48 - 7 = 41$$, again too large.

Notice that choices A, B, and C all produce values much larger than 20. This happens because they either multiply 6 by a larger number (choice A) or start with the full product of 48 (choices B and C). Only choice D reduces the multiplication factor by subtracting within the parentheses first.

When tackling order of operations problems, always work through each option completely rather than trying to shortcut. Small changes in parentheses placement create big differences in results, so careful calculation is essential.

According to the order of operations, multiplication and division have the same level of priority and should be performed from left to right. First, perform the division: \(48 \div 8 = 6\). The expression becomes \(6 \times 3 - 2\). Next, perform the multiplication: \(6 \times 3 = 18\). The expression becomes \(18 - 2\). Finally, perform the subtraction: \(18 - 2 = 16\).

Following the order of operations (PEMDAS): First, evaluate the expression in the parentheses: \(5-2 = 3\). The expression is now \(15 - 3^2 + 3^3 \div 9\). Next, evaluate the exponents from left to right: \(3^2 = 9\) and \(3^3 = 27\). The expression becomes \(15 - 9 + 27 \div 9\). Next, perform division: \(27 \div 9 = 3\). The expression is now \(15 - 9 + 3\). Finally, perform subtraction and addition from left to right: \(15 - 9 = 6\), and \(6 + 3 = 9\).

We are given \(8 < 20 - 2k < 12\). Let's test the answer choices for \(k\).

A) If \(k=3\), \(20 - 2(3) = 20 - 6 = 14\). 14 is not less than 12.

B) If \(k=5\), \(20 - 2(5) = 20 - 10 = 10\). \(8 < 10 < 12\) is true.

C) If \(k=6\), \(20 - 2(6) = 20 - 12 = 8\). 8 is not greater than 8.

D) If \(k=8\), \(20 - 2(8) = 20 - 16 = 4\). 4 is not greater than 8.

First, evaluate the expression \(4 \times(5+3)^2\). According to the order of operations, start with the parentheses: \(5+3 = 8\). The expression becomes \(4 \times 8^2\). Next, evaluate the exponent: \(8^2 = 64\). The expression becomes \(4 \times 64\). Finally, multiply: \(4 \times 64 = 256\). The question asks what number must be subtracted from this result to get 200. Let the number be \(n\). So, \(256 - n = 200\). Solving for \(n\), we get \(n = 256 - 200 = 56\).

When you encounter expressions with multiple operations, the order of operations (PEMDAS/BODMAS) determines which calculations to perform first. You must work through parentheses, then multiplication and division (left to right), then addition and subtraction (left to right).

Let's evaluate each expression systematically:

For choice A: $$10 + 20 \div 5 - 2$$

First divide: $$20 \div 5 = 4$$

Then work left to right: $$10 + 4 - 2 = 12$$

For choice B: $$(10 + 20) \div 5 - 2$$

Parentheses first: $$10 + 20 = 30$$

Then divide: $$30 \div 5 = 6$$

Finally subtract: $$6 - 2 = 4$$

For choice C: $$(10 + 20) \div(5 - 2)$$

Both sets of parentheses: $$30 \div 3 = 10$$

For choice D: $$10 + 20 \div(5 - 2)$$

Parentheses first: $$5 - 2 = 3$$

Then divide: $$20 \div 3 = 6\frac{2}{3}$$

Finally add: $$10 + 6\frac{2}{3} = 16\frac{2}{3}$$

Choice D gives the greatest value at $$16\frac{2}{3}$$. Choice A incorrectly ignores that division comes before addition and subtraction. Choice B reduces the dividend by putting addition in parentheses with division. Choice C reduces the divisor, but dividing the same number by a smaller divisor (3 instead of 5) doesn't increase the result enough to exceed choice D.

Remember: parentheses can dramatically change an expression's value by altering the order of operations. Always identify what's in parentheses first, then follow PEMDAS carefully.