Phone Plan Showdown
Students compare phone plans using tables, graphs, and equations to determine the best value.
Common Core standards covered
- 8.F.B.4Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
- 8.F.A.2Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
- 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.
- 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.
© Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
Teacher-facing overview
Big idea: Represent and compare linear relationships in multiple forms.
Student target: “I can model each phone plan as a linear function, compare the plans using tables, graphs and equations, find break-even points, and recommend a plan for a specific customer with evidence.”
Success criteria: I can write y = mx + b for a plan from its monthly fee and per-GB rate, and say what m and b each mean in dollars. · I can complete a cost table and graph all three plans on shared axes. · I can find a break-even point from the graph and confirm it with the equations. · I can recommend a plan for a customer's usage and defend it with slope, intercept and intersection evidence.
Essential question
Three companies all claim to be the cheapest phone plan — and all three are telling the truth. How?
Learning target: I can model each phone plan as a linear function, compare the plans using tables, graphs and equations, find break-even points, and recommend a plan for a specific customer with evidence.
Success criteria
- I can write y = mx + b for a plan from its monthly fee and per-GB rate, and say what m and b each mean in dollars.
- I can complete a cost table and graph all three plans on shared axes.
- I can find a break-even point from the graph and confirm it with the equations.
- I can recommend a plan for a customer's usage and defend it with slope, intercept and intersection evidence.
Why this matters
This is linear modeling doing its day job: every phone plan, gym membership, ride fare and utility bill is y = mx + b wearing a logo, and 'which is cheapest' has no single answer — it depends on x. The break-even points are the student's first real system of equations: the intersection is the shared solution, which is next lesson's headline. And the consumer skill is genuine: companies advertise the b and hide the m, or advertise the m and hide the b, on purpose.
Students should already be able to
- Write and interpret y = mx + b — Why Slope Never Changes.
- Evaluate a linear function and build its table — Streaming Growth Studio.
- Solve a linear equation, including collecting terms — Balance the Books.
Vocabulary to define explicitly
- Slope (per-GB rate)
The cost added for each additional gigabyte — the m in y = mx + b.
Example: Plan A adds $8 per GB: its line climbs 8 for every 1 across.
Not an example: The monthly fee is not the slope — it is charged once, not per gigabyte.
- y-intercept (monthly fee)
The cost at 0 GB — the b, where the line meets the y-axis.
Example: Plan C costs $50 before a single gigabyte is used.
Not an example: The cheapest plan at 0 GB is not automatically the cheapest plan — that is the whole lesson.
- Break-even point
A usage level where two plans cost exactly the same — the intersection of their lines.
Example: Plans A and B both cost $60 at 5 GB: their lines cross at (5, 60).
Not an example: The point where one line crosses the y-axis is its intercept, not a break-even between plans.
- Intersection as a shared solution
A point on both lines satisfies both equations at once — that is why crossing means equal cost.
Example: (5, 60) works in y = 8x + 20 AND y = 5x + 35.
Not an example: A point on just one of the lines tells you about one plan only.
Where this shows up outside the classroom
Assessment plan
Formative — for feedback during the lesson
- The warm-up meanings of 6 and 25 — the m/b interpretation is the lesson's foundation stone.
- Cheapest-cell circles in the table during team work: the region structure appearing early.
- Customer clinic reasoning: costs computed under all plans BEFORE the recommendation.
Summative — the graded evidence: The exit ticket (Plan D), scored with the four-criterion rubric.
The task students work on
Three carriers compete for the same customers. Plan A: $20 monthly fee plus $8 per GB. Plan B: $35 plus $5 per GB. Plan C: $50 plus $3 per GB. Your consumer-advocate team will model all three as equations, tables and graphs, find every break-even point, and then match real customers to their best plan — with evidence a skeptical customer would accept.
Cost comparison table — complete every cell in dollars.
| Data used | Plan A: 8x + 20 | Plan B: 5x + 35 | Plan C: 3x + 50 |
|---|---|---|---|
| 0 GB | $20 | $35 | $50 |
| 2 GB | |||
| 5 GB | |||
| 8 GB | |||
| 10 GB | |||
| 15 GB |
Anticipated student strategies
Table, then picture, then algebra — concrete to exact. The B=C break-even at x = 7.5 is chosen so the graph alone cannot settle it: the equation-solvers finish what the graph starts, which is precisely the relationship between representations the standard wants.
- Table sweep. The student fills the cost table and watches the cheapest column change: A at 2 GB, tie at 5, B at 8, C from 10 on.
Share first — the lead changing hands inside one table is the lesson's plot twist, visible without any graph.
- Graph and read the crossings. The student plots all three lines and reads break-evens where they cross, estimating (5, 60), (6, 68) and (7.5, 72.50).
Connect to the table: the crossing at 5 GB is the table's tie row. Graphs estimate; note that (7.5, 72.50) needs the algebra to nail.
- Set the equations equal. The student writes 8x + 20 = 5x + 35 and solves: 3x = 15, x = 5 — the exact break-even, no graph required.
Last and loudest: this is Balance the Books aimed at two plans, and it is the doorway to systems. Have this student derive B=C's x = 7.5 that the graph could only estimate.
Lesson script
Collect: $6 per gigabyte, $25 monthly fee. Ask which quantity a light user cares about, and which a heavy user cares about — the lesson's entire tension in one question.
Listen for: The 6 scales with usage (slope); the 25 is fixed at any usage (intercept).
Show the three claims: 'Only $20/month!' (A), 'Just $5 per GB!' (B), 'Rates as low as $3!' (C). Vote for the cheapest. Then the reveal: every ad quotes its best number and hides the other. The team's job is to find where each claim is actually true.
Listen for: Each plan IS cheapest for somebody — the ad targets its own best case. Low fee attracts light users; low rate attracts heavy users.
Build A's table rows for 0, 2, 5 GB (20, 36, 60), plot (0, 20), then rise-8-run-1 to draw the line. Note the graph trick for the day: the class grid marks cost in TENS of dollars, so $20 sits at height 2.
- Intercept first, slope second — the two-step graphing routine.
- Say units at every step: dollars, gigabytes, dollars per gigabyte.
Listen for: Higher start ($35 > $20) but shallower climb ($5 < $8): the lines must eventually cross.
Teams of 3 with Plan Cards and the graphing grid. Required output: full cost table, three labeled lines, and all three break-even points — from the graph where possible, from equations where the graph is ambiguous. B=C at 7.5 GB is deliberately between gridlines.
- Break-even protocol: estimate from the crossing, then set the two equations equal and solve exactly.
- A=B: 8x + 20 = 5x + 35 → x = 5, $60. A=C: x = 6, $68. B=C: x = 7.5, $72.50.
Listen for: Both plans cost $60 for 5 GB; the point (5, 60) satisfies both equations at once — one point, two truths.
Teams draw Customer Cards: Emma (3 GB, budget under $60), Alex (18 GB), the Family (42 GB — off the grid, equations only), the Traveler (2 GB most months, 12 GB twice a year), the Streamer (unsure, 'somewhere around 6 to 9 GB'). Each recommendation needs a computed cost, a comparison, and one sentence a customer would understand.
Listen for: The region answer: below 5 GB take A, 5–7.5 take B, above 7.5 take C — their range straddles a break-even, so the recommendation is conditional.
Teams post recommendations. Build the class number line of x: A's region, B's region, C's region, fenced by the three break-evens. Close the loop to the hook: all three ads were true — on their own region.
Listen for: Start lowest AND climb slowest — but then the others would never win anywhere, and in this market each line wins somewhere.
Students work alone: model Plan D ($15 plus $7 per GB), evaluate it, interpret its slope, and compare it with Plan B at 9 GB.
Misconceptions and how to respond
- “The lowest monthly fee is the cheapest plan.” Compute all three at 10 GB: the $20 plan costs $100 and the $50 plan costs $80. The fee is one-time; the rate compounds with usage.
- “The slope is the monthly fee, since it's the bigger deal.” Evaluate at 0 GB: only b survives — that is the fee. The slope shows up exactly when usage does.
- “Lines crossing means the plans have the same data usage.” Both plans always allow any usage. Crossing means same COST at that usage: the x is the usage, the y is the shared price.
- “Whichever line is on top is winning.” Top means most expensive — this graph's y is cost. The winner is the LOWEST line in each region.
- “Compare plans at one usage level and you're done.” A wins at 2 GB, C wins at 10. A single x proves nothing global — the regions and their fences are the full answer.
Scaffolded practice
The workbook follows gradual release rather than a page of blank prompts: 1 fully worked, 1 partially completed, then 4 independent.
- I doFind Plan A's cost at 6 GB, from the equation.
- We doComplete the 8 GB row of the cost table.
- You doFind the exact A=B break-even by setting the equations equal, and state what the point means.
- You doFind the B=C break-even. Why was the graph not enough for this one?
- You doThe Family uses 42 GB. Compute all three plans, recommend one, and give the ANNUAL savings over the runner-up.
- You doDesign Plan D to beat Plan A for heavy users but lose to it for light users. Give the equation and its break-even against A.
Worked solutions and scaffolds for every prompt are in the free sample kit — download it below.
Exit ticket
- Plan D costs $15 per month plus $7 per GB. Write its equation and find the cost of 9 GB.
- In one sentence: what does Plan D's slope mean, in dollars?
- At 9 GB, is Plan D cheaper or pricier than Plan B ($35 + $5/GB)? Show the evidence.
Worked answers, the completed answer tables, and the scoring rubric are in the free sample kit — download it below.
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: Model one plan as y=mx+b. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 2: Teams represent three plans in tables, graphs, and equations. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 3: Find break-even points. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 4: Write a recommendation for different customer profiles. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
Differentiation and inclusion
Support: Provide partially completed tables; extend with nonlinear add-ons.
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: Recommendation must cite slope, intercept, and intersection evidence.
Mastery: The student completes the task accurately and explains the strategy with limited prompting.