The Goal Line
The fundraiser has a target, and every question runs backward from it: how many boxes, how many cars, how many sales AT LEAST? Students build and solve px + q = r equations and px + q > r inequalities — then graph the solution set and defend what it means.
Common Core standards covered
- 7.EE.B.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
- 7.EE.B.4.aSolve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
- 7.EE.B.4.bSolve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.
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Teacher-facing overview
Big idea: Construct simple equations and inequalities from word problems, solve px + q = r, p(x + q) = r, and px + q > r fluently, compare algebraic to arithmetic solutions, and graph and interpret solution sets.
Student target: “I can translate a goal story into px + q = r or px + q ≥ r, solve it fluently, compare the algebraic steps to the arithmetic ones, and graph an inequality's solution set with the right endpoint.”
Success criteria: I can name p, q, and r in a goal story and build the equation. · I can solve px + q = r and p(x + q) = r, checking by substitution. · I can explain how the algebraic steps mirror the arithmetic solution in reverse. · I can solve an inequality, graph its solution set, and interpret the endpoint (open or closed) in context.
Essential question
Arithmetic runs forward from what you know; algebra runs backward from what you want. When the goal is fixed and the unknown is the effort — how do you solve, and what does 'at least' do to the answer?
Learning target: I can translate a goal story into px + q = r or px + q ≥ r, solve it fluently, compare the algebraic steps to the arithmetic ones, and graph an inequality's solution set with the right endpoint.
Success criteria
- I can name p, q, and r in a goal story and build the equation.
- I can solve px + q = r and p(x + q) = r, checking by substitution.
- I can explain how the algebraic steps mirror the arithmetic solution in reverse.
- I can solve an inequality, graph its solution set, and interpret the endpoint (open or closed) in context.
Why this matters
7.EE.B.4 is the year's algebra summit: the px + q = r form is the skeleton of every linear model students will ever build, and the standard explicitly asks for the arithmetic-algebra comparison — the undo list IS the arithmetic solution read backward, which demystifies solving forever. The inequality half adds the idea that answers can be SETS with graphs, endpoints, and context (17 sales, because sales come in wholes). Unit 5's proportions, Grade 8's equations, and every break-even problem downstream reuse this exact machinery.
Students should already be able to
- Building and simplifying expressions — The Border Crew.
- Signed-number operations — Units 1–2.
- Compute paths and estimation — The Set Builders.
Vocabulary to define explicitly
- The goal equation px + q = r
Rate times effort, plus what's already in hand, equals the target.
Example: $4 per box, $28 in the jar, $100 goal: 4b + 28 = 100.
Not an example: 28b + 4 = 100 swaps the roles — the jar isn't earning per box.
🔒 3 more terms with examples and non-examples — The undo list, Solution set, Open vs closed endpoint — 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 mental solve and its stated path.
- Goal-board solves during floor work — endpoint tests especially.
- The boundary-vs-context distinction in the treasurer's report.
Summative — the graded evidence: The exit ticket (equation, inequality with graph, endpoint test), scored with the four-criterion rubric.
The task students work on
The club's fundraiser has one goal line: $300 by Saturday. The BAKE SALE sells boxes at $4 with $28 already in the jar — the practice target is $100. The TEAM DRIVE has 6 teams each bringing the same t items plus 3 bonus items; the drive's target is 54 items. The CAR WASH charges $8 a car with $76 in hand, and the club needs AT LEAST $300. And treasurer trivia from the salesperson want-ad: $50 a week plus $3 per sale, pay of at least $100 — how many sales? Every question runs backward from its goal: build the equation or inequality, solve, check, and graph the ones whose answers are sets.
Goal board — equation or inequality, solved and checked.
| Goal | Model | Solve | Check / graph |
|---|---|---|---|
| Bake sale: $100 | 4b + 28 = 100 | ||
| Team drive: 54 items | 6(t + 3) = 54 | ||
| Car wash: at least $300 | 8c + 76 ≥ 300 | ||
| Want-ad: at least $100 | 3s + 50 ≥ 100 |
Anticipated student strategies
Arithmetic-to-algebra makes solving familiar, the undo list makes it general, and pricing out guess-and-improve motivates why the general method earns its keep — exactly the fluency-with-understanding the standard wants.
- Arithmetic first, then translated. The student solves the bake sale as (100 − 28) ÷ 4 = 18, then watches the algebra do the same subtraction and division to both sides.
The standard's own move — sequence it first and make the mirror explicit: algebra is arithmetic with a paper trail.
🔒 2 more anticipated strategies with select-and-sequence guidance are in the full lesson. See plans →
Lesson script
Most heads compute (100 − 28) ÷ 4 = 18. Write that path on the board and name its two moves: subtract the jar, divide by the price. Those two moves are about to become algebra.
Listen for: The jar money isn't box money — only the remaining $72 comes in fours. The story ordered the operations.
Check ritual: 4(18) + 28 = 100 ✓. Then the grouped form: 6 teams each bring (t + 3) items, target 54: 6(t + 3) = 54. Two legal routes — divide by 6 first (t + 3 = 9, t = 6) or distribute first (6t + 18 = 54). Same answer; the ÷6 route is two steps, the distribute route three. Route choice is part of fluency.
- Undo in reverse order: +28 came last, so −28 goes first.
- p(x + q) = r: try ÷p first — when r/p is whole, it's the express lane.
Listen for: When r/p is ugly (say 6(t + 3) = 57) — then 6t + 18 = 57 keeps integers longer. Fluency means owning both routes and choosing.
🔒 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
- “Solve 4b + 28 = 100 by dividing everything by 4 first: b + 28 = 25.” Dividing both sides by 4 divides the 28 too: b + 7 = 25 (fine, b = 18!). The error is dividing only SOME terms. The undo list avoids the trap: +28 came last, so −28 leaves first.
- “An inequality's answer is the endpoint: 'c = 28.'” — the teacher response is in the full lesson.
- “s ≥ 16 2/3 means 16 sales is enough — round to the nearest.” — the teacher response is in the full lesson.
- “100 − 5d < 40 needs the flip rule and that's scary.” — 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 doBake sale: $4 boxes, $28 jar, $100 target. Solve arithmetically, then algebraically, and match the steps.
🔒 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
- T-shirts cost $5 each plus a $35 setup fee, and the budget is exactly $90. Model and solve for the number of shirts.
🔒 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: Solve the bake-sale goal arithmetically, then algebraically — and match the steps. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 2: Solve the grouped form p(x + q) = r. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 3: Turn 'at least' into an inequality and solve it. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 4: Graph the solution set and read it in context. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
Differentiation and inclusion
Support: The undo list (what was done to x, undone in reverse) scaffolds every solve. Extend with the fraction-answer salesperson problem and the drain inequality.
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 equation and one inequality built from words, solved, checked, and (for the inequality) graphed.
Mastery: The student completes the task accurately and explains the strategy with limited prompting.