The Catch-Up Question
Two savers, one question: when does the faster one catch up? Students solve simple systems by graphing the crossing point and by setting equations equal — including the never-catches (parallel) and secretly-the-same-plan cases. The gentler on-ramp to Phone Plan Showdown's three-way decision.
Common Core standards covered
- 8.EE.C.8Analyze and solve pairs of simultaneous linear equations.
- 8.EE.C.8.aUnderstand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
- 8.EE.C.8.bSolve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
- 8.EE.C.8.cSolve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.
© Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
Teacher-facing overview
Big idea: Analyze and solve pairs of simultaneous linear equations: understand the solution as the intersection point, solve simple cases by inspection and by setting expressions equal, and solve real-world problems leading to two linear equations.
Student target: “I can solve a pair of linear equations by graphing the intersection and by setting the two expressions equal, check the solution in BOTH equations, and recognize the no-solution and infinitely-many cases by inspection.”
Success criteria: I can explain the crossing point as the one (week, balance) that makes both plans true. · I can solve by setting the two mx + b expressions equal. · I can verify a solution in both equations. · I can spot parallel plans (no solution) and identical plans (infinitely many) before solving.
Essential question
Two savings plans grow at different speeds from different starts. What does the point where their graphs cross actually SAY — and what if they never cross?
Learning target: I can solve a pair of linear equations by graphing the intersection and by setting the two expressions equal, check the solution in BOTH equations, and recognize the no-solution and infinitely-many cases by inspection.
Success criteria
- I can explain the crossing point as the one (week, balance) that makes both plans true.
- I can solve by setting the two mx + b expressions equal.
- I can verify a solution in both equations.
- I can spot parallel plans (no solution) and identical plans (infinitely many) before solving.
Why this matters
The crossing point is the big idea of 8.EE.C.8, and 'when does Ben catch Ava' gives it a story a developing learner can hold: before week 4 Ava leads, at week 4 they tie at $80, after week 4 Ben leads forever. Solving by setting expressions equal is exactly the standard's 'simple cases algebraically,' and the parallel and same-plan pairs cover 8.EE.C.8.a-b's inspection cases. Phone Plan Showdown then plays this skill at full speed with three plans and a recommendation to defend.
Students should already be able to
- Building y = mx + b from a savings story — The Savings Tracker.
- Graphing a line from start and rate.
- Solving one linear equation — Pan-Balance Puzzles or Balance the Books.
Vocabulary to define explicitly
- System of equations
Two (or more) equations about the same two variables, asked to be true at once.
Example: Ava's y = 10x + 40 and Ben's y = 15x + 20, both about weeks and dollars.
Not an example: Two equations about unrelated quantities (Ava's savings, Ben's height) aren't a system — no shared story.
🔒 3 more terms with examples and non-examples — Solution of a system, No solution (parallel plans), Infinitely many (same plan) — 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 catch-up predictions and the 20 ÷ 5 argument.
- Match-up board rows during floor work — the two inspection calls especially.
- Both-equation checks on the fresh headliner.
Summative — the graded evidence: The exit ticket (solve-and-check, before/after reasoning, inspection call), scored with the four-criterion rubric.
The task students work on
The Savings Club posts weekly match-ups. Today's headliner: Ava ($40 start, $10 a week) vs Ben ($20 start, $15 a week) — Ben swears he'll catch her. Undercard: Cam ($60 start, $10 a week) claims Ava will NEVER catch him, and new member Dee's plan card reads '2y = 20x + 80,' which looks suspiciously familiar. Your team settles every match-up: when (if ever) do the balances tie, and who leads after that?
Match-up board — settle each one.
| Match-up | Equations | Tie at (week, $)? | Who leads after? |
|---|---|---|---|
| Ava vs Ben | y = 10x + 40 vs y = 15x + 20 | ||
| Cam vs Ava | y = 10x + 60 vs y = 10x + 40 | ||
| Dee vs Ava | 2y = 20x + 80 vs y = 10x + 40 |
Anticipated student strategies
Graph first to give the solution a location, algebra second to give it a method, table last as the check that also motivates the algebra.
- Graph and read the crossing. The student graphs both plans from start and rate and reads off the intersection (4, 8) in tens — week 4, $80.
Share first — the picture defines what 'solution' means before algebra compresses it.
🔒 2 more anticipated strategies with select-and-sequence guidance are in the full lesson. See plans →
Lesson script
Take gut answers with reasons. Ben gains $5 on Ava every week and starts $20 behind — some students will already say 'four weeks.' Hold that: the graph and the algebra now have a prediction to hit.
Listen for: The gap between the lines shrinks $5 per step until it's zero — at the crossing. Catch-up rate = difference of slopes.
Then the check ritual: a solution must satisfy BOTH equations — 80 = 40 + 40 ✓ and 80 = 60 + 20 ✓. Before week 4 Ava leads; after, Ben leads forever.
- Set-equal works because both expressions ARE y.
- Check in both equations, every time — one check only proves half.
Listen for: Every point on Ava's line passes her check. The solution is special because it passes BOTH — that's the definition.
🔒 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
- “Finding x = 4 finishes the problem.” The solution is a PAIR — week 4 AND the $80 both plans hit. Report (4, 80), then check it in both equations.
- “A point that checks in one equation is a solution.” — the teacher response is in the full lesson.
- “Getting 60 = 40 means I made an algebra mistake.” — the teacher response is in the full lesson.
- “The saver with the bigger start wins forever.” — 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 doAva (y = 10x + 40) vs Ben (y = 15x + 20): when do they tie, and who leads after? Solve by setting equal; check in both.
🔒 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
- Two savers: y = 5x + 30 and y = 10x + 10. Find the week and balance where they tie, and check in both equations.
🔒 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: Graph both savers and find the crossing point. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 2: Get the same answer by setting the equations equal. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 3: Meet the never-catches pair and the same-plan pair. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 4: Solve a catch-up story from scratch. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
Differentiation and inclusion
Support: One match-up at a time, whole-number crossing points, and every answer checked in both equations. Extend to a three-plan decision with a recommendation — Phone Plan Showdown's arena.
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 system solved both ways — graph and algebra — with the crossing point interpreted.
Mastery: The student completes the task accurately and explains the strategy with limited prompting.