Algebra 1 Lesson

Linear vs. Exponential Modeling: The Long Game

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8th or 9th GradeBuilding Thinking ClassroomsCCSS.MATH.CONTENT.HSF.LE.A.1, CCSS.MATH.CONTENT.HSF.LE.A.2, CCSS.MATH.CONTENT.8.F.A.360 min

Learning Objective

Students will identify linear vs. exponential relationships from tables and contexts by computing first differences and consecutive ratios, write equations for both model types, and explain why exponential growth with any base $b > 1$ eventually exceeds any linear function.

Lesson Overview

Students use Building Thinking Classrooms structures to discover the structural difference between additive (linear) and multiplicative (exponential) growth, then grapple with the counterintuitive truth that exponential growth always wins in the long run.

Materials

Vertical non-permanent surfaces (whiteboards or chart paper)
One dry-erase marker per group
Erasers or paper towels
Random grouping cards or app
Desmos-enabled device per group (for Part 3)

Scaffolded Task Progression

1Task 1

Low-Floor Launch (Anchored Problem): Two investors offer you a deal on a $1,000 investment. Plan A adds $200 in cold hard cash to your balance every single year. Plan B increases your current total balance by 15% every single year. Build a table tracking both plans from Year 0 up to Year 5. Which plan makes you more money at Year 1? Which plan makes you more money at Year 5?

2Task 2

Slice 1 — The Long Game: Extend your table or construct a mathematical shortcut rule for both plans. Find out who is winning at Year 10 and Year 20. Sketch both paths on a single coordinate grid.

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