The Border Crew
A tiling crew quotes borders for square pools — and four different ways of counting the same tiles produce four different-looking expressions that must be equivalent. Students expand, factor, and combine to prove it, then use the algebra to settle a pricing rivalry.
Common Core standards covered
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Teacher-facing overview
Big idea: Apply properties of operations to add, subtract, factor, and expand linear expressions with rational coefficients.
Student target: “I can expand, factor, and combine linear expressions with rational coefficients, and show two expressions are equivalent both by algebra and by the structure they count.”
Success criteria: I can write an expression from a counting strategy and point to what each term counts. · I can expand with the distributive property and combine like terms. · I can factor a common factor back out. · I can subtract one expression from another, distributing the minus sign through.
Essential question
Four crew members count the same border four ways and write four different expressions. Who's right — and what does 'the algebra agrees' actually prove?
Learning target: I can expand, factor, and combine linear expressions with rational coefficients, and show two expressions are equivalent both by algebra and by the structure they count.
Success criteria
- I can write an expression from a counting strategy and point to what each term counts.
- I can expand with the distributive property and combine like terms.
- I can factor a common factor back out.
- I can subtract one expression from another, distributing the minus sign through.
Why this matters
7.EE.A.1 is the grammar of algebra, and the border makes every rule visible: expanding is regrouping a count, factoring is spotting the repeated chunk, and combining like terms is merging two crew members' tallies. The equivalence habit — same value for EVERY input, provable by properties rather than spot-checks — is exactly what equation solving (next unit) and every later algebra course stand on. And the rival-quote subtraction, with its distributed minus sign, is the single most error-prone move in Grade 7 algebra, met here with money on the line.
Students should already be able to
- Evaluating expressions at a value.
- The distributive property with numbers — The Salvage Dive's check ritual.
- Fraction addition — Food Truck Friday.
Vocabulary to define explicitly
- Equivalent expressions
Expressions that produce the same value for EVERY input — not just one lucky test number.
Example: 4s + 4 and 4(s + 1): equal at s = 4 (20), at s = 10 (44), at every s.
Not an example: 4s + 4 and 5s equal at s = 4 only — one shared value is a coincidence, not equivalence.
🔒 3 more terms with examples and non-examples — Expand (distribute), Factor, Like terms — 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 counting strategies and sketches.
- Quote-sheet expansions during crew work — the subtraction line especially.
- The flyer-form debate (which form and why).
Summative — the graded evidence: The exit ticket (expand, factor, subtract, equivalence verdict), scored with the four-criterion rubric.
The task students work on
The Border Crew tiles one-tile borders around square pools of side s. Four crew members count the border tiles their own way: Ada counts four sides of s and 4 corners; Ben counts four overlapping side-plus-corner runs of (s + 1); Cara counts top and bottom rows of (s + 2) plus two side columns of s; Dev counts s + s + s + s + 4. The boss needs proof all four quotes agree for EVERY pool size. Then pricing: tiles cost $2.50 each plus a $40 service fee — build the price formula, factor it, and settle the rivalry with BorderPro, whose quote is 12s + 38: for which pools is the crew cheaper?
Four counts, one border — then the money.
| Crew member | Counting story | Expression | Check at s = 10 |
|---|---|---|---|
| Ada | four sides, then 4 corners | 4s + 4 | |
| Ben | four runs of side-plus-corner | 4(s + 1) | |
| Cara | top & bottom rows (s + 2), two side columns | 2(s + 2) + 2s | |
| Dev | one side at a time, corners last | s + s + s + s + 4 | |
| Price formula | $2.50 per tile + $40 fee | 2.50(4s + 4) + 40 = ? |
Anticipated student strategies
Pictures establish WHY equivalence must hold, algebra proves THAT it holds, spot-checks get demoted to supporting evidence — the epistemology the standard is quietly teaching.
- Picture-to-terms matching. The student colors each crew member's chunks on the grid and labels each term of the expression with its chunk.
Share first — it makes 'the expressions must agree' obvious before any algebra: they count the same tiles.
🔒 2 more anticipated strategies with select-and-sequence guidance are in the full lesson. See plans →
Lesson script
Collect strategies on the board with sketches: 4·4 + 4, 4·5, 2·6 + 2·4, 4+4+4+4+4. All give 20. Plant the flag: today we prove these agree for EVERY pool, not just this one.
Listen for: Guaranteed — they counted the same physical tiles. Different bookkeeping, same border.
Dev's s + s + s + s + 4 collapses by combining. Then the factoring direction: seeing 4s + 4, pull the 4 out: 4(s + 1) — factoring is expanding read right-to-left. Post the anchor: expand to prove, factor to compress, combine to tidy.
- Every term must point at tiles — algebra with a story can't silently go wrong.
- Equivalent means equal for EVERY s; the properties are the proof.
Listen for: The expressions encode the bookkeeping, but expanding strips it away — both reduce to the same tile total because the tiles were the same all along.
🔒 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
- “4(s + 1) = 4s + 1.” The 4 multiplies everything inside — count Ben's runs on the picture: four runs, each with a corner, means 4 corners. 4s + 4.
- “They're equivalent — I checked s = 10 and both gave 44.” — the teacher response is in the full lesson.
- “(12s + 38) − (10s + 50) = 2s + 88 — or 2s − 12 with the +38 flipped too.” — the teacher response is in the full lesson.
- “4s + 4 simplifies to 8s.” — 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 doProve Cara's count 2(s + 2) + 2s is equivalent to Ada's 4s + 4, and check both at s = 10.
🔒 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
- Expand and simplify: 3(2s + 5) + 4s.
🔒 3 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: Count one border four ways; write four expressions. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 2: Prove all four equivalent by expanding and combining. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 3: Build and factor the crew's price formula. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 4: Subtract two quotes to find the break-even pool. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
Differentiation and inclusion
Support: Every expression comes with its picture — support by always drawing the count first. Extend with the half-tile trim (rational coefficients) and the rival-quote subtraction.
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 expression proved equivalent to 4s + 4 two ways: by structure (the counting story) and by algebra.
Mastery: The student completes the task accurately and explains the strategy with limited prompting.