Identify the rule behind a linear function

Graf(x) reverses the usual direction of function problems. One player draws a card with a linear function—say, "Multiply by 3 and Add 10"—and keeps it hidden. Teammates call out numbers; the player applies the rule and announces only the output. Input: 4, output: 22. Input: 7, output: 31. From these pairs, the team reconstructs the function.

This setup creates what's called a "black box" problem: students can probe the function's behavior but can't see its internal structure. They're working inductively, building the rule from observations rather than evaluating a given rule—closer to how mathematicians actually discover patterns than how algebra is typically taught.

Students reconstruct functions from behavior, not just evaluate them.

The game works exclusively with linear functions in slope-intercept form: f(x) = mx + b. Each function multiplies the input by some value m, then adds (or subtracts) some constant b. Simple structure, but figuring out both parameters from limited trials requires systematic thinking about how linear functions behave.

The m coefficient represents slope—how much the output changes per unit input. Students develop intuition about rate of change through direct experience: a function with m = 8 produces outputs that grow much faster than one with m = 2. The b constant shifts everything vertically without changing the rate of growth.

Each Graph Card shows the function's visual representation alongside the rule, so after discovering a function, students see what it looks like graphed. This connects their algebraic reasoning to geometric intuition—another link that formal instruction sometimes skips over too quickly.

Teams balance speed with systematic data collection.

The four-minute timer creates an interesting tension. Test too few values and you're guessing; test too many and you run out of time. Students learn to judge when they have sufficient evidence—a useful metacognitive skill that extends well beyond this specific game.

The collaborative structure matters pedagogically. One student might notice "it's going up by threes," while another tests the hypothesis deductively: "If it's times three, then seven should give twenty-one, but we got twenty-three, so something's being added." Different cognitive approaches strengthen each other.

Because one player holds the card and evaluates the function while others discover it, everyone practices both skills: careful function evaluation under time pressure, and pattern recognition from limited data. Roles rotate each round.

The pedagogical move here is reversing the typical problem direction. Standard algebra gives you f(x) = 3x + 7 and asks for f(4). Graf(x) gives you the pairs (4, 19), (2, 13), (6, 25) and asks you to find the function. Working backward from effects to causes builds different and complementary skills.

This reversal connects to how functions appear in real contexts. Scientists observe data and infer relationships. Economists see patterns in markets and propose models. Graf(x) simulates this discovery process at an accessible scale.

The game naturally pushes students toward systematic experimentation. Random testing is inefficient, so they develop strategies: test zero first, then consecutive integers, look for constant differences. This evolution from trial-and-error to systematic approach mirrors the development of scientific method.

Mental arithmetic gets practiced incidentally but purposefully. To spot patterns, students calculate differences, notice multiplicative relationships, work backward from outputs. The math practice happens in service of the goal rather than as isolated drill.

Strategic thinking replaces random guessing through experience.

Students also develop communication skills around mathematical reasoning. Saying "I think it's times five" evolves into "I think it's times five because consecutive inputs give outputs that differ by five." The game creates natural motivation to articulate and justify conjectures clearly.

The format implicitly introduces inverse operations. If you know the output is 29 and you've determined the function adds 4, you work backward: 29 - 4 = 25, so if my input was 5, the slope would be 5. This back-and-forth between operations builds algebraic flexibility.

Collaborative problem-solving exposes students to different approaches. One person focuses on differences between consecutive outputs (constant rate of change). Another tests specific strategic values. A third might try algebraic reasoning. Seeing multiple valid paths to the same answer builds cognitive flexibility.

Graf(x) fits well after introducing function notation and slope-intercept form but before students have developed real fluency. It provides repeated exposure to how m and b affect function behavior, making these parameters concrete through experience rather than memorization.

Students need basic prerequisite understanding: functions map inputs to outputs, each input gets exactly one output, and they should be comfortable with multiplication and addition of integers. Graphing experience isn't necessary—the game can help build intuition before formal graphing instruction.

After the first round or two, pause for strategy discussion. "What numbers were most helpful to test?" "How did you figure out the rule?" Students who discovered that zero isolates the constant can share that insight, accelerating everyone's strategic thinking.

Four minutes creates productive urgency for most students.

Timing can be adjusted based on student needs. Classes still building arithmetic fluency might need five or six minutes initially. Advanced students can be challenged with three-minute rounds or more complex functions (larger multipliers, negative constants).

While circulating, watch for strategic patterns. Who tests systematically versus randomly? Who keeps organized tables versus scattered notes? Who dominates versus observes quietly? These observations inform both immediate help ("Try testing zero") and longer-term instructional planning.

Some functions prove harder to crack. Larger multipliers (like ×9) or negative constants (like -7) require more careful analysis. Early on, you might sequence cards from simple to complex, then mix difficulties once students build confidence.

Extensions can deepen engagement. Have students create their own function cards following the game's pattern—this reversal from discovering to creating requires understanding what makes a function challenging but solvable. Or challenge teams to find the function using the minimum number of test values.

For struggling students: provide organizers with strategic prompts ("Try 0 first"), use functions with smaller numbers, extend time limits, or pair with stronger students who explain their process. The goal is productive engagement, building confidence before removing supports.

Fast learners can tackle non-linear functions (quadratic, absolute value), functions with division, or three-operation rules. Or have them write explanations of their strategy—articulating mathematical thinking deepens understanding.

Post-game analysis connects gameplay to formal function concepts.

After playing, discuss the mathematics explicitly. Display several discovered functions: "Which grows fastest? How can you tell?" "What if we changed 'Add 5' to 'Add 10'?" "Which part controls steepness versus vertical position?" Connect the game experience to the formal properties you're teaching.

Multiple rounds reveal growth. Students move from random testing to strategic choices, from needing ten values to deducing functions from four or five, from vague language to precise mathematical vocabulary. These observable improvements provide useful formative assessment of developing algebraic reasoning. The activity drives the mathematics.