Operations Ladder Worksheet
OPERATIONS LADDER
Part 1: Fundamental Grouping & Exponents • Page 1 of 2
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1. Groups \(( )\) \([ ]\) › 2. Powers \(x^n\) › 3. \(\times\) & \(\div\) (Left → Right) › 4. \(+\) & \(-\) (Left → Right)
Tier 1 • Mid Level Four Operations & Single Grouping
30 PTS
Problem 1:
\( 42 - 6 \times (14 - 9) + 18 \div 3 \)
Simplify parentheses first • then multiply/divide
Show Step-by-Step Simplification: Rewrite the expression line-by-line
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Execution Tip
Once the group \( (14 - 9) \) is simplified, evaluate multiplication and division from left to right before adding or subtracting.
Watch out: Do not compute \( 30 + 6 \) first! Addition does NOT precede subtraction.
Final Value:
Tier 2 • High Level Grouping + Exponential Powers
35 PTS
Problem 2:
\( 70 - 2 \times (3 + 1)^2 + 45 \div 3^2 \)
Parentheses sum • compute powers before multiplying
Show Step-by-Step Simplification: Rewrite the expression line-by-line
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Order Trap Alert
The exponent applies strictly to the base \((3+1)=4\). Do NOT multiply \( 2 \times 4 \) before squaring!
Key Step: \( 4^2 = 16 \) and \( 3^2 = 9 \). Simplify both powers before multiplying.
Final Value:
Completed Page 1? Turn over to solve the Nested Double Group challenge.
Continue to Page 2
OPERATIONS LADDER
Part 2: Double Grouping & Mathematical Analysis • Page 2 of 2
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Tier 3 • Challenger Nested Double Grouping \([ \; ( \; ) \; ]\)
35 PTS
Problem 3:
\( 100 - \left[\, 4 \times (15 - 2^3) + 7 \,\right] \times 2 \)
Work from the inside out • inner group first
Show Step-by-Step Simplification: Work from inside parentheses outward
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Double Group Roadmap
1 Power inside group: \( 2^3 = 8 \)
2 Parentheses: \( 15 - 8 = 7 \)
3 Brackets: \( [4 \times 7 + 7] \)
4 Multiply bracket by 2 • subtract
Final Value:
Error Detective • Find the Mistake Left-to-Right Misconception Analysis
BONUS
Case Study: A student evaluated \( 36 \div 3 \times 2 + (8 - 2^2) \) and arrived at 10.
Incorrect Result
1. Identify & Explain the Error: What rule did the student violate?
2. Correct Evaluation: Solve correctly step-by-step:
Correct Answer:
Mastery Self-Check: Innermost evaluated first • powers squared • left-to-right operations respected
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