Functions Tell Stories

Overview

Students investigate how functions model real-world relationships, moving from a verbal situation to an equation, table, graph, and interpretation. The lesson connects modern function notation to ancient Babylonian problem-solving and Greek geometric reasoning.

Big Ideas: A function assigns exactly one output to each input; equations, tables, and graphs represent the same relationship; key features communicate meaning in context.

Essential Questions: How can we represent a relationship in multiple ways? What can a graph reveal that an equation or table may hide?

Prior knowledge: Students should understand ordered pairs, coordinate-plane quadrants, rate of change, slope, arithmetic patterns, and solving simple linear equations.

Learning intentions

Students will be able to:

• Write a linear function from a real-world description or two input-output pairs.
• Represent a function with an equation, table, and graph.
• Interpret intercepts, rate of change, and increasing/decreasing behavior in context.
• Justify whether a representation is a function.

Success criteria

• I can define the input and output quantities and their units.
• I can write a function such as (C(x)=mx+b) and explain what (m) and (b) mean.
• I can use a table or graph to identify and interpret key features.
• I can support my answer with evidence from the situation.

Curriculum links

• Interpreting functions: describe key features of graphs and tables in terms of the quantities.
• Building functions: write an explicit function from a context and determine steps for calculation.
• Arithmetic and geometric sequences: connect a constant additive change to an explicit and recursive rule.
• Linear and exponential models: construct a linear function from a description, graph, or input-output pairs.

Lesson structure (45 minutes)

1. 0–5 min · Motivation and history. Open with the opening mystery and history image: “A Babylonian tablet gives a rule for calculating a field’s area. Is that a function?” Briefly explain that Babylonian mathematicians used organized numerical procedures, while Greek mathematicians such as Euclid and Pythagoras emphasized relationships and geometric patterns. Students predict what modern functions add to these ideas and record one real-world relationship.

2. 5–12 min · I do—model a function. Using the function representation slides, model a bike rental costing $8 to unlock plus $3 per hour: (C(h)=3h+8). Identify (h) as input, (C(h)) as output, units, initial value, rate of change, table values, and graph features. Students annotate the functions practice worksheet and answer: “What does the y-intercept mean?”

3. 12–20 min · We do—translate representations. Display the scenario “A gym charges $25 to join and $12 per month.” As a class, complete (G(m)=12m+25), a table for 0–3 months, and a graph. Ask: “Which feature tells us the starting cost?” and “Is the function increasing or decreasing?” Students explain answers to a partner, then complete the matching section of the functions practice worksheet.

4. 20–31 min · You do—real-world practice. Students work independently for five minutes, then compare with a partner. They solve: “A school club has $150 and raises $18 each week. Write a function, find the amount after 6 weeks, and interpret the slope and intercept.” They then analyze “A phone battery starts at 92% and loses 7% per hour”: write the function, identify when it reaches 50%, and state a reasonable domain. Teacher circulates and checks input/output labels, units, and equations.

5. 31–37 min · Common mistake clinic. Show the error (C(h)=8h+3) for the bike rental. Students identify the mistake and correct it. Guide with: “What happens at zero hours?” and “Which quantity changes repeatedly?” Address the frequent error of reversing input and output or treating the intercept as the rate. Students correct one answer in a different color.

6. 37–42 min · Formative and summative check. Formatively, cold-call or use mini-whiteboards for “What does a negative slope mean in context?” and “Does the table (1,4; 2,7; 3,10) represent a linear function?” For the summative task, students independently solve: “A water tank contains 40 liters and fills at 6 liters per minute. Write (W(t)), find (W(10)), and explain the meaning of 40 and 6.” Collect responses for evidence of modeling and interpretation.

7. 42–45 min · Exit ticket. Finish with the three-question exit ticket. Students complete all three questions silently and submit before leaving.

Resources

• the functions and history slide deck
• the functions practice worksheet
• Graph paper or calculators
• Mini-whiteboards or response cards
• Projector and markers
• Exit-ticket answer key

Assessment

• Formative: listen for correct identification of inputs, outputs, units, slope, and intercept during partner explanations; check whiteboard responses.
• Summative: score the independent water-tank task using accuracy of equation, calculation, interpretation, and units.
• Exit ticket: use the key below; reteach students who confuse slope and intercept or omit contextual meaning.

Exit ticket questions and answer key

1. A taxi charges $4 plus $2 per mile. Write the function and find the cost of 5 miles. Answer: (C(m)=2m+4); (C(5)=14) dollars.
2. A savings account starts with $60 and increases by $15 each week. Write the function and explain the intercept. Answer: (S(w)=15w+60); the intercept means $60 is saved initially.
3. A candle is 24 cm tall and burns 1.5 cm per hour. Write the function, find its height after 8 hours, and state when it reaches 12 cm. Answer: (H(t)=24-1.5t); (H(8)=12) cm; it reaches 12 cm after 8 hours.

Differentiation

• Support/IEP: provide a partially completed table, color-code slope and intercept, allow graph paper and calculator use, chunk the worksheet, and offer sentence frames: “The slope means ___ per ___” and “The intercept means ___ when ___ is zero.”
• ELL: preteach rate, initial value, input, output, increase, and decrease with visuals and units; permit labeled diagrams, partner rehearsal, and bilingual dictionaries.
• Extension: students create a context for (y=-4x+30), identify a sensible domain, and explain why values outside that domain may not make sense.
• Cognitive demand: primarily application and analysis, progressing to evaluation when students critique and correct a flawed model.

Appendix: AI Use

• AI tool used: Generative AI assistant.
• Prompt used: “Create a 45-minute Grade 9 U.S. mathematics lesson in which students model real-world relationships with linear functions, including history, I do/We do/You do practice, differentiation, assessment, and answer keys.”
• Student engagement with AI: After completing the exit ticket, students may ask an approved AI tool to generate a similar function scenario. They must identify the input, output, slope, intercept, and domain, then verify every claim using their own table or graph. Students may not submit AI-generated work as their own.
• Assessment: Student learning is assessed through the independently completed modeling task, exit ticket, written explanations, and the accuracy of verification. AI use is evaluated for reasoning, not originality of wording.
• Reflection: AI helped organize the lesson sequence and generate varied contexts efficiently. Teacher review remained essential to check mathematical accuracy, age appropriateness, historical claims, accessibility, and alignment with New York expectations.