A class displays two calculator entries, (12 + 6) × 7 and the same symbols with parentheses removed. Compare their outcomes and explain which operation the grouping changes.
Application-and-critique task: Two engineers compare calculator entries, (18 + 5) × 5 and the same symbols with parentheses removed. Compare their outcomes and explain which operation the grouping changes. Independently prepare two annotated grouping trees, one for each interpretation; do not simply repeat the first application question’s computation. A student wrote this solution: “I erased grouping from (18 + 5) × 5 and wrote 18 + 5 × 5 = 43. I claimed parentheses do not change the answer.” Is that reasoning valid? Name the first incorrect step; demonstrate a corrected computation and independently check it.
A science report records its score using 16 + 2 × 8. Determine the score and mark the operation that must occur before addition.
Application-and-critique task: A laboratory quality-control card records its score using 24 + 7 × 3. Determine the score and mark the operation that must occur before addition. Independently prepare a numbered operation-precedence tree; do not simply repeat the first application question’s computation. A student wrote this solution: “24 + 7 × 3 = (24+7)×3 = 93. I completed addition before multiplication.” Is that reasoning valid? Name the first incorrect step; demonstrate a corrected computation and independently check it.
A classmate begins simplifying 18 − 2 + 7 from the right. Evaluate the unchanged expression correctly and explain the first left-to-right operation.
Application-and-critique task: A spreadsheet assistant starts simplifying 24 − 7 + 7 from the right. Evaluate the unchanged expression correctly and explain the first left-to-right operation. Independently prepare a left-to-right evaluation chain explicitly rejecting right-association; do not simply repeat the first application question’s computation. A student wrote this solution: “24 − 7 + 7 = 24 − (7 + 7) = 10. I changed the grouping of addition and subtraction.” Is that reasoning valid? Name the first incorrect step; demonstrate a corrected computation and independently check it.
A library prepares 4 identical classroom boxes. Every box contains 19 poetry books and 3 biographies. Write TWO equivalent expressions (one grouped, one distributed), then calculate the books in all boxes.
Application-and-critique task: A book donation team assembles 8 identical classroom boxes. Every box contains 28 poetry books and 6 biographies. Write TWO equivalent expressions (one grouped, one distributed), then calculate the books in all boxes. Independently prepare a separate distributive layout with all partial terms; do not simply repeat the first application question’s computation. A student wrote this solution: “I erased grouping from (28 + 6) × 8 and wrote 28 + 6 × 8 = 76. I claimed parentheses do not change the answer.” Is that reasoning valid? Name the first incorrect step; demonstrate a corrected computation and independently check it.
A fundraiser buys 6 identical welcome baskets. Each basket needs 25 notebooks costing $1 each and 6 pencils costing $1 each. Write a single expression for total spending and calculate it.
Application-and-critique task: An orientation committee orders 7 identical welcome baskets. Each basket needs 24 notebooks costing $1 each and 2 pencils costing $1 each. Write a single expression for total spending and calculate it. Independently prepare a grouped expression and an ungrouped counterexample; do not simply repeat the first application question’s computation. A student wrote this solution: “I erased grouping from (24+2)×7 and wrote 24 + 2 × 7 = 38. I claimed parentheses do not change the answer.” Is that reasoning valid? Name the first incorrect step; demonstrate a corrected computation and independently check it.
At each of 9 nature stations there are 22 maps and 3 cards. Volunteers remove 3 items from EACH station. Write a grouped expression and determine the total items left across all stations.
Application-and-critique task: During a map-making workshop, at each of 7 nature stations there are 14 maps and 4 cards. Volunteers remove 2 items from EACH station. Write a grouped expression and determine the total items left across all stations. Independently prepare a bar-per-station diagram marking removals before multiplication; do not simply repeat the first application question’s computation. A student wrote this solution: “I erased grouping from 7×(14+4−2) and wrote 7 × 14 + 4 − 2 = 100. I claimed parentheses do not change the answer.” Is that reasoning valid? Name the first incorrect step; demonstrate a corrected computation and independently check it.
Answer from first principles. Include a model where requested. These questions are distinct from taught worked examples.
Evaluate (36 ÷ (3 × 2)) + 7.
A student writes 7 × (3 + 5) = 7 × 3 + 5. State both sides and repair it.
Calculation items: 2 points for correct mathematics and units plus 1 for valid working. Notebook diagnosis tasks: 4 points, one each for identifying a specific faulty step, explaining the repair, solving accurately, and checking units/logic. Targeted transfer tasks: 4 points with detailed anchors in private teacher keys. Equivalent exact representations accepted; reviewers must examine explanations.