Make the Calculation Plan Visible

At a glance

• A: 83, with 2 ×6 =12 and 8 ÷4 =2 completed before the remaining addition and subtraction.
• B: One correct version is {90 −[2 ×(6 +3) +8]} ÷4.
• C: 9 →26 →64 →16.
• D: Printed expression: 2 ×6 happens first. Completed expression: 6 +3 happens first.

Key idea

The numbers and operation signs stay in the same order, but grouping marks change which results must be completed before later operations act on them.

A. Evaluate the expression as printed

Complete multiplication and division first:

Then evaluate the remaining expression from left to right:

Equivalent work is accepted if it visibly shows both required multiplication-and-division results and gives 83.

B. Add the grouping marks

The plan requires these nested groups:

• inner: 6 +3;
• middle: 2 ×(6 +3) +8;
• outer: 90 −[2 ×(6 +3) +8], ending before ÷4.

One compliant expression is:

Every number and operation sign remains in its supplied position. Step 2 maps directly to 2 ×(6 +3) +8: multiply 2 by the result of 6 +3, then add 8. The three grouping pairs match, are properly nested, and use parentheses, brackets, and braces exactly once each.

C. Follow the completed groups

Work from the innermost group outward:

Therefore the trail is:

D. Compare the first operation involving 6

A complete labeled response is:

• Printed expression: 2 ×6 happens first.
• Completed expression: 6 +3 happens first.

A complete sentence expressing the same comparison is also accepted. A generic statement such as “grouping symbols come first” is not precise enough.

Part D is a brief consolidation check because the preceding directions already identify the two relevant calculations. Give it modest evidentiary weight. It does not substitute for the Part B construction and does not independently establish the full insight about later operations acting on entire grouped results.

Check

The completed expression satisfies every construction condition: it preserves the token order, marks the three required spans, keeps ÷4 outside the outer group, uses one matching pair of each delimiter type, and evaluates to 16.

Accepted constructions and notation

Any one-to-one assignment of parentheses, brackets, and braces to the three required spans is valid. There are 3 ×2 ×1 =6 such assignments; the key shows only one.

Proof that these are all six assignments

The calculation plan fixes exactly three nested spans: the inner, middle, and outer groups. A compliant construction assigns the three supplied delimiter types one-to-one to those spans. There are three choices for the inner span, two remaining choices for the middle span, and one remaining choice for the outer span, so there are

assignments in all. Each assignment is valid because changing only the delimiter shapes preserves the three boundaries and their nesting. Any other labeling must repeat a delimiter type or omit one, so it does not satisfy the stated inventory rule.

Score the three boundary locations separately from notation compliance:

• Clearly intended endpoints retain boundary credit even when a handwritten glyph is imperfect.
• A recognizable imperfect glyph counts as its intended delimiter type.
• If the opening and closing glyphs are different types, a clearly located span may retain boundary credit, but that pair does not earn matching-pair or delimiter-inventory compliance.
• Crossed pairs do not earn well-formed-nesting credit.
• Genuinely ambiguous endpoints require clarification or redrawing.
• Inventory compliance does not replace or outweigh evidence for the three mathematical boundaries.

A typed expression must preserve every supplied token and use one pair of each delimiter type. When delimiter entry is inaccessible, an equivalent structured response must name the three required spans and assign a different supplied delimiter type to each.