Rational Functions: Where Graphs Go Wild
Students work in pairs to build a fast paper or digital prototype for a real-world rational-function situation. They choose a situation with a rate, a limit, or a restriction, then write the rational function they think fits best. On a class evidence board, each pair adds the situation, its equation, a sketch of the graph, and a short note about the asymptote, domain limits, and what the model shows. Students test their model with feedback questions from a STEM career speaker from a museum, makerspace, or tech nonprofit in Hudson Valley NY. They revise their thinking, compare at least one second model, and prepare a partner-ready pitch board with their best model and a short justification for why it fits the situation best. Adapted from the New York State Portrait of a Graduate emphasis on critical thinking, communication, and creative innovation.
Steps
| # | What teachers do | Time |
|---|---|---|
| 1 | Launch the Model Makers Kickoff by posing the essential question and inviting students to act as creative innovators, critical thinkers, and effective communicators while modeling a Hudson Valley situation involving a rate, limit, or restriction. | 4 min |
| 2 | Pair students, assign rotating modeler and checker roles, and guide each pair to create a fast paper or digital prototype containing a labeled situation, variable definitions, units, and a proposed rational function. | 8 min |
| 3 | Set up a class evidence board—a shared wall or digital display where pairs add their situation, equation, graph sketch, asymptotes, realistic domain limits, and a brief explanation of what the model shows—and prompt pairs to examine one another’s reasoning. | 9 min |
| 4 | Facilitate a live or teacher-delivered review using feedback questions prepared by a STEM career speaker from a Hudson Valley museum, makerspace, or tech nonprofit about assumptions, constraints, graph behavior, and usefulness. | 7 min |
| 5 | Direct each pair to revise its original model, create or evaluate at least one alternative rational-function model, and compare how well both models represent rates of change, asymptotes, domain restrictions, and the real-world context. | 9 min |
| 6 | Guide pairs to finish a partner-ready pitch board featuring the strongest model, graph, constraints, comparison evidence, and concise justification, then rehearse and deliver a brief pitch for teacher, peer, or STEM speaker review. | 8 min |
| Total | 45 mins |
Preparation
- Arrange pairs and prepare paper, markers, rulers, graph paper, sticky notes, calculators or graphing tools, and optional shared slides for fast paper or digital prototypes and final pitch boards.
- Set up a class evidence board—a wall, whiteboard, or shared digital display—with labeled spaces for each pair’s situation, rational-function equation, graph sketch, asymptotes, domain constraints, interpretation, and comparison model.
- Confirm a Hudson Valley STEM speaker from a museum, makerspace, or tech nonprofit and provide the essential question in advance; request two or three feedback questions that help students test assumptions, compare models, and explain real-world constraints.
- Prepare a visible 45-minute schedule and success criteria requiring a meaningful real-world situation, two compared rational-function models, accurate graphs and constraints, revisions based on speaker feedback, and a concise justification; identify critical thinking, creative innovation, and effective communication as competencies drawn from the New York State Portrait of a Graduate.
Teacher Notes
Display the essential question and prepare a class evidence board—a shared wall or digital slide where each pair adds its situation, equation, graph, asymptotes, domain constraints, and model interpretation. Plan approximately 5 minutes for the kickoff, 15 minutes for prototyping, 8 minutes for speaker feedback, 10 minutes for comparing and revising models, and 7 minutes for pitches or an exit check. Ask the Hudson Valley STEM speaker to use prompts such as “What does each variable represent?”, “Where does this model stop making sense?”, and “Why does this model fit better than your alternative?”; if the speaker cannot attend live, obtain written or recorded questions beforehand. Confer with pairs to ensure both students contribute and assess whether the final pitch board is mathematically accurate, compares two models, uses feedback, and communicates a clear justification, reflecting the Portrait of a Graduate competencies of creative innovation, critical thinking, and effective communication (portrait-of-a-graduate.pdf, p. 1).
Student Instructions
With a partner, choose a real-world Hudson Valley situation involving a rate, limit, or restriction, and ask: “How can rational functions model and compare its rates of change, asymptotes, and constraints?” Build a quick paper or digital prototype, then add your situation, equation, graph sketch, domain restrictions, asymptotes, and interpretation to the class evidence board—a shared display where each pair posts its model and reasoning. Test your model using feedback questions from a STEM career speaker from a museum, makerspace, or tech nonprofit; compare it with at least one alternative rational model and revise based on evidence. Create a partner-ready pitch board presenting your best model and a brief justification of why it fits best, then divide speaking roles and communicate your reasoning clearly while listening respectfully to feedback.
Success Criteria
- I can define a real-world situation’s variables, constraints, and reasonable domain.
- I can write a rational function that represents the situation and sketch a graph consistent with the model.
- I can explain what the model’s asymptote, domain limits, and rate of change mean in context.
- I can compare at least two possible models using mathematical evidence and justify which one fits the situation best.
- I can use a STEM career speaker’s feedback questions to revise my model or strengthen my explanation.
- I can clearly present my equation, graph, assumptions, and justification with my partner on a pitch board.
Differentiation
- Offer three teacher-prepared situation choices with realistic values—such as average cost per visitor, time to complete a shared task, or signal strength over distance—and prompt pairs to identify the input, output, rate or constraint, and reasonable domain before writing a function.
- Provide a visible model checklist: situation and units; rational-function equation; labeled graph; domain restriction; vertical or horizontal asymptote and its meaning; comparison with a second model; and evidence-based justification.
- Assign flexible partner roles such as model builder and evidence checker, then switch roles before revision so both students calculate, interpret, listen to feedback, and contribute to the pitch.
- Preteach key terms with symbols, sketches, and plain-language meanings: domain as allowable inputs, asymptote as a line the model approaches, restriction as an excluded or unrealistic value, and rate as one quantity compared with another.
- Provide sentence frames: “Our model, ___, represents ___ because ___.” “The domain is limited to ___ because ___.” “As ___ increases, the model approaches ___, which means ___.” “Model ___ fits better than ___ because ___.”
- Allow students to plan ideas in their home language and use a labeled graph, equation, gestures, and brief speaking notes during the STEM speaker review; pair multilingual learners with supportive partners without assigning them only presentation or recording duties.
- Ask students to estimate parameters from two or more realistic data points, then explain how changing one parameter affects an asymptote, intercept, or rate of change.
- Have students compare their preferred model with a second rational model by testing both at boundary values and several realistic inputs, then identify where each model becomes misleading.
- Invite students to add a sensitivity note to the pitch board explaining how a small change in an assumption or constraint could change a decision made by a Hudson Valley museum, makerspace, or tech nonprofit.
- Chunk the task into timed checkpoints—choose a situation, define variables and domain, build and graph two models, interpret features, and prepare the pitch—and provide a written agenda with completion boxes.
- Allow approved tools and formats, including a graphing calculator, digital graphing tool, enlarged grid paper, speech-to-text, typed labels, or an audio-recorded justification; assess the same mathematical reasoning and model comparison.
- Reduce copying and visual load by giving teacher-prepared situation cards and a partially labeled pitch-board layout while retaining all essential decisions; provide extended time or a quieter presentation setting according to each student’s IEP or 504 plan.