The Set Builders
Opening night is Friday and the stage crew's notebook is full of mixed numbers: center a 9¾-ft banner on a 27½-ft stage, price a painter's 10% raise in your head, and stretch a $150 budget. Multi-step problems in any form — with an estimate standing guard over every answer.
Common Core standards covered
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Teacher-facing overview
Big idea: Solve multi-step real-life problems with positive and negative rational numbers in any form, converting between forms as appropriate and assessing reasonableness with mental computation and estimation.
Student target: “I can solve multi-step problems in any numeric form, choose the form that makes each step easiest, and guard every exact answer with a mental estimate.”
Success criteria: I can estimate a multi-step answer before computing it. · I can convert between fractions, decimals, and percents when it helps. · I can execute the compute path: what to subtract, what to halve, what to multiply. · I can state answers in the form the situation wants — a tape measure wants 8 7/8, not 8.875.
Essential question
The notebook mixes fractions, decimals, and percents on one page. When do you convert, when do you estimate, and how do you know an answer deserves trust?
Learning target: I can solve multi-step problems in any numeric form, choose the form that makes each step easiest, and guard every exact answer with a mental estimate.
Success criteria
- I can estimate a multi-step answer before computing it.
- I can convert between fractions, decimals, and percents when it helps.
- I can execute the compute path: what to subtract, what to halve, what to multiply.
- I can state answers in the form the situation wants — a tape measure wants 8 7/8, not 8.875.
Why this matters
7.EE.B.3 is the standard that says fluency means FLEXIBILITY: its own examples are the $25-an-hour raise done as a mental tenth and the towel bar centered by estimate-then-exact. The set crew lives in exactly that world — measurements in halves and quarters, money in decimals, discounts in percents — and every card rewards choosing the kind form over the given form. This is also the estimation habit of Food Truck Friday promoted from ritual to strategy: here the estimate sometimes IS the answer a stage crew needs.
Students should already be able to
- All four operations on rationals — Units 1–2.
- Percent as multiplier — The Price Tag Decoder.
- Fraction/decimal conversion — The Salvage Dive's gauge station.
Vocabulary to define explicitly
- Compute path
The ordered plan of operations a multi-step problem needs, chosen before arithmetic starts.
Example: Banner: whole stage MINUS banner, then HALVE the leftover — subtract, then divide by 2.
Not an example: Diving into digits with no plan — the most common source of plausible-looking wrong answers.
🔒 3 more terms with examples and non-examples — Form switching, Reasonableness check, Answer form — are defined in the full lesson. See plans →
Where this shows up outside the classroom
Assessment plan
Formative — for feedback during the lesson
- Warm-up ballparks and the stated compute path.
- Notebook trios during crew calls — the form column especially.
- The worst-error postmortem in the production meeting.
Summative — the graded evidence: The exit ticket (centering, mental percent, fraction-of-amount), scored with the four-criterion rubric.
The task students work on
The spring play opens Friday and the tech crew's notebook holds the week's jobs. CENTER IT: the 9¾-ft title banner must hang centered on the 27½-ft stage — how far from each edge? PAYROLL: the scenic painter earns $25 an hour and just got a 10% raise — new rate, computed in your head. LUMBER: five braces, each 2¾ ft, cut from one 16-ft plank — does it fit, and what's left? BUDGET: $150 total; two-thirds went to lumber, then $37.85 to paint — what remains for opening-night flowers? ROPE: the rigging rope is ¼ off its $24.60 price. Every job: estimate first, exact second, answer in the form the job wants.
The crew notebook — estimate before exact, form chosen last.
| Job | Numbers | Estimate | Exact | Form for the job |
|---|---|---|---|---|
| Center the banner | (27 1/2 − 9 3/4) ÷ 2 | |||
| Painter's raise | $25/h + 10% | |||
| Lumber cut | 16 − 5 × 2 3/4 | |||
| Budget check | 150 − 2/3 of 150 − 37.85 | |||
| Rope discount | 1/4 off $24.60 |
Anticipated student strategies
The plan disciplines the work, the form-switch makes it kind, and the estimate keeps the whole pipeline honest — three habits stacked in the order they should fire.
- Plan-then-compute. The student writes the compute path in words first — 'stage minus banner, then halve' — then executes it.
Share first. The path is the thinking; the arithmetic is the typing.
🔒 2 more anticipated strategies with select-and-sequence guidance are in the full lesson. See plans →
Lesson script
Collect ballparks (≈ 8½–9 ft: leftover ≈ 17, halved ≈ 8½). Then the compute path in words: whole minus banner, halve the leftover. Exact: 27.5 − 9.75 = 17.75; ÷2 = 8.875 = 8 7/8 ft. The estimate certified it — and for a crew member on a ladder, 'about 9 feet' was already enough to start.
Listen for: Halving first also works — (13.75 − 4.875 is messier) — the centered gap is (stage − banner)/2 either way; the chosen path just keeps the numbers friendlier.
Board the form-switch table: percent ↔ decimal ↔ fraction, with 'when each is kindest' — tenths and quarters mental as fractions, register totals as decimals, comparisons as percents. Then the lumber card modeled: 5 × 2¾ = 13¾ (five 2s, five ¾s: 10 + 3¾), fits in 16 with 2¼ left.
- 10% = one tenth = slide the decimal — the fastest raise in mathematics.
- Multiply mixed numbers by splitting: whole parts, then fraction parts.
Listen for: The tenth wins mentally; ×1.10 wins on a register; all three are the same expression wearing different forms — Price Tag Decoder's lesson, applied.
🔒 The remaining 4 steps — fully scripted with teacher-says lines, discourse questions, look-fors, and interventions — are in the full lesson. See plans →
Misconceptions and how to respond
- “Center the banner by dividing the stage in half: 13¾ from each edge.” 13¾ + 9¾ + 13¾ is 37¼ — more stage than we own. The gaps share what's LEFT after the banner: (27½ − 9¾)/2. Draw the bar; the picture won't let you skip the subtraction.
- “A 10% raise means multiply by 0.10.” — the teacher response is in the full lesson.
- “Keep every number in the form it arrived in.” — the teacher response is in the full lesson.
- “Estimates are for people who can't compute exactly.” — the teacher response is in the full lesson.
🔒 Every misconception ships with a tested classroom response. See plans →
Scaffolded practice
The workbook follows gradual release rather than a page of blank prompts: 1 fully worked, 1 partially completed, then 5 independent.
- I doCenter the 9¾-ft banner on the 27½-ft stage: estimate, then compute the gap on each side, and report it for a tape measure.
🔒 6 more prompts across all three release tiers are in the full lesson. See plans →
Worked solutions and scaffolds for every prompt are in the downloadable kit, included with membership. See plans and pricing.
Exit ticket
- A 9¾-inch towel bar is centered on a 27½-inch door. About how far from each edge, and exactly?
🔒 2 more exit-ticket questions are in the full lesson. See plans →
Worked answers, the completed answer tables, and the scoring rubric are in the downloadable kit, included with membership. See plans and pricing.
Printable handouts in the kit
All handouts are in Printable_Handouts.pdf inside the kit download.
Materials and classroom resources
Student experience
Students enter through a low-floor prompt, build or sketch a model, test a strategy, revise their thinking, collaborate, and present a final solution. The task remains accessible while allowing for sophisticated reasoning.
Detailed facilitation notes
- Step 1: Estimate the banner gaps, then compute them exactly. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 2: Price the painter's raise mentally. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 3: Cut the lumber list and audit the budget. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 4: Convert forms wherever it makes the arithmetic kinder. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
Differentiation and inclusion
Support: Estimates are the built-in floor; the form-switch table (fraction ↔ decimal ↔ percent) is the built-in ladder. Extend with the budget audit and the design-your-own crew card.
Language support: “I started by…”, “My model shows…”, and “I know my answer is reasonable because…”.
Extension: Require a second solution, alter a condition, or have students write a challenge for another group.
Accessibility: Manipulatives, enlarged visuals, oral response choices, partner reading, and reduced-copy student pages.
Assessment plan
Formative checks: Strategy choice, accuracy, representation, collaboration, and explanation.
End evidence: One completed crew card with estimate, exact answer in a sensible form, and a reasonableness verdict.
Mastery: The student completes the task accurately and explains the strategy with limited prompting.