Teaching Guide - Graf(x) | 10story Learning

Identify the rule behind a linear function

Understand that a function is a rule that assigns to each input exactly one output. (8.F.A.1)
Interpret the equation y = mx + b as defining a linear function. (8.F.A.3)

Before You Play

Graf(x) asks students to discover patterns from limited data rather than apply formulas. Surface their understanding of how relationships follow predictable rules and catch misconceptions about functions early.

What does it mean for a relationship to follow a "rule"? Give an example from everyday life.

Listen for students who describe consistent patterns—"If you buy more items, you pay more" or "More work hours equals more money." They should see that rules produce predictable outcomes, not just abstract formulas. Watch for hand gestures showing "if this, then that" connections—these physical movements reveal causal thinking.

If I give you input-output pairs like (1, 5), (2, 8), (3, 11), how would you find the rule connecting them?

Listen for students who spot the constant difference between outputs (5→8→11 shows +3). Some test multiplication: "1 times what gives 5?" Notice students who organize pairs vertically on paper or use their finger to track down the list—this spatial arrangement helps them see patterns. Students who gesture the "jump" between numbers are thinking about rate of change.

Why would testing the input "0" be especially useful when finding a function rule?

Listen for students who recognize that zero isolates the constant term—"Whatever you get when x is 0, that's what you're adding." This insight speeds up pattern discovery. If they don't see it yet, they'll discover it through gameplay.

Setup Tip: Position the timer where everyone can see it. Students should sit where they can reach the Graph Cards pile easily and have space to write on organizers without elbowing each other. Cramped setups kill collaboration—students need room to point at each other's work.

During Gameplay

Graf(x) develops systematic testing strategies and pattern recognition from organized data. Watch how students choose inputs, record pairs, and collaborate—their actions reveal developing algebraic reasoning.

Card Selection & Timer Start

How confident does the card-holder seem with their rule? What helps them prepare?

Watch for card-holders who take a moment to internalize the two-step operation—multiply first, then add/subtract. Some cover the graph side to focus on the rule text. Those who rush make computational errors under pressure. Notice students who use finger-counting or trace the operations in the air—they're rehearsing the sequence physically.

Testing Phase: Calling Numbers & Recording

What strategies are students using to choose test numbers? Random or systematic?

Listen for students who start with zero, then consecutive integers (0, 1, 2, 3...). Those calling random numbers (7, 15, 23) struggle to spot patterns. After two rounds, pause to discuss: "Which number choices helped you find the rule fastest?" Students learn strategy faster through reflection than trial and error alone.

How are partners dividing the work? Who records, who analyzes, who chooses inputs?

Watch for effective teams where one person records while another spots patterns in the growing table. Weaker teams compete for the same role or work in parallel without talking. Listen for explicit coordination: "You write, I'll watch for the pattern." Notice teams who lean in to point at the organizer together—physical proximity supports collaborative thinking.

⚡ Facilitation Move: If students struggle with patterns, prompt them to scan the Y column: "Run your finger down. What's happening between each number?" This focuses attention on constant differences—the key to identifying slope. Students who physically trace with their finger often spot patterns faster than those who just look.

Pattern Recognition

When someone proposes a rule, how do they justify it? Do they reference specific pairs?

Listen for students who support claims with evidence: "It's times 5 because when we jump from 1 to 2, the output jumps by 5" or "We tested zero and got 10, so it's adding 10." Evidence-based reasoning beats guessing. Notice students who point to specific rows on their organizer while explaining—they're grounding abstract rules in concrete data.

Can students explain the difference between the multiplier (slope) and constant term (y-intercept)?

Listen for students who distinguish the two parts: "The times-3 makes it go up, the plus-10 is where it starts" or "The constant stays the same, but the multiplier depends on x." Watch for students who gesture vertically (showing rise) when talking about the multiplier and horizontally (showing starting point) for the constant—these spatial gestures reveal structural understanding of linear functions.

⚡ Watch For: Teams stuck after finding one parameter (they know it's "times 4" but can't find what's being added). Prompt strategic testing: "You know the multiplier—what input would reveal the other part?" Testing zero directly reveals the constant.

Verification: Revealing the Graph Card

When students see the graph, what connections do they make to their table?

Listen for students who connect representations: "Those points on the graph match what we tested" or "The line goes through all our numbers." Watch for students who trace the graphed line with their finger while referencing their organizer—this visual-tabular connection shows they understand multiple representations describe the same relationship. Students who rotate their organizer to align with the graph's orientation are making spatial sense of the connection.

⚡ Materials: After revealing the graph, have students place it next to their organizer. Ask them to point to where specific table values appear on the graph—this spatial comparison makes abstract connections concrete.

After You Play

Use consolidation to help students articulate discovered strategies and connect gameplay to formal mathematics. Make their intuitive approaches explicit and transferable.

What strategies worked best for discovering rules quickly? How did your approach change from first round to later rounds?

Listen for students who recognize strategic evolution—"At first we guessed numbers, but then we figured out that zero and then 1, 2, 3 helped us see patterns faster." This reflection makes learning visible. Celebrate students who explain why systematic testing beats random testing.

If I give you the rule "Multiply by 6 and Add 2," what would the graph look like? Steeper or flatter than "Multiply by 3 and Add 2"?

Listen for students who connect multiplier to steepness—"Times 6 is steeper because outputs go up faster." This understanding of slope as rate of change is foundational for graphical reasoning. Watch for students who gesture diagonal lines at different angles—strong spatial intuition about linear functions. Some may even tilt their hand to show different slopes.

Look at your organizer from one round. If I covered all but three consecutive pairs, could you still find the rule? How?

Listen for students who've internalized the algorithm: "Find the differences to get the multiplier, then use any pair to figure out the constant." Watch for students who point to specific rows while explaining—they understand the process is grounded in organized data. This is when gameplay strategies transfer to formal problem-solving.

How is discovering a function rule from pairs different from evaluating a function when you know the rule?

Listen for students who recognize that discovering requires inductive reasoning (building general rules from specific cases), while evaluation is direct application. "When discovering, you look for patterns. When evaluating, you follow steps." This helps students see that functions work both directions.

Extensions & Variations

Negative Constant Challenge

Introduce functions with negative constants like "Multiply by 4 and Subtract 3" (4x - 3). Students see that outputs still increase by a constant amount, but the starting point is lower. This deepens understanding of how constants shift functions vertically.

Mystery Coordinate Discovery

Give teams three pre-plotted points on graph paper (no table, no equation). They determine the function by measuring rise over run directly on the graph, then create a table to verify. This spatial-to-algebraic pathway shows graphs encode the same information as equations.

Minimum Data Challenge

Challenge students: "What's the smallest number of input-output pairs needed to find any linear rule?" Let them test theories, then discuss why two points determine a line. This connects Graf(x) to fundamental geometric principles.

Steepness Ranking

After multiple rounds, have students arrange graph cards from flattest to steepest slope. Then check: does the order match multipliers from smallest to largest? This visual-algebraic comparison clarifies how coefficient magnitude determines slope steepness. Students can gesture the slopes with their arms to feel the difference.

Backwards Engineering

Give teams a completed graph (rule hidden). They create an input-output table by reading coordinates from the graph, then derive the algebraic rule from their table. This full cycle—graph to table to equation—demonstrates all three representations are interconnected views of the same relationship.

Non-Integer Slopes

For advanced students, introduce functions like "Multiply by 1.5 and Add 4" or "Multiply by 0.5 and Subtract 1." Fractional and decimal multipliers challenge students to recognize patterns when differences aren't whole numbers, extending understanding beyond integer coefficients.

Practical Notes

Timing

Four minutes per round balances urgency with adequate testing time. Most teams identify 2-3 functions in early rounds, accelerating to 4-5 as strategies develop. If students struggle initially (random numbers, missing patterns), extend to 5-6 minutes and discuss strategy between rounds. Once efficient patterns emerge, return to 4 minutes. Setup requires 3-4 minutes—students need time to understand the card-holder role and organize materials.

Grouping

Teams of 3-4 optimize participation and collaboration. Pairs work but lack diversity; groups of 5+ create coordination challenges. The card-holder rotation is essential—students experience both sides: precise evaluation under time pressure (applying the rule accurately) and pattern discovery from organized data (inductive reasoning from limited information).

Materials & Space

Students need adequate table space to write on organizers without crowding. Cramped conditions make recording difficult and reduce interaction. Graph Cards should be large enough for teams to examine together—after discovering a function, students benefit from gathering around the revealed card to compare their table with the graphed representation. Some students naturally rotate their organizers 90° while working—this physical reorientation sometimes helps them see patterns they missed in the original layout.

Assessment

Collect organizers after gameplay to assess systematic versus random testing strategies. Look for strategic input choices (starting with zero, using consecutive integers) versus scattered numbers. Notice organizational quality—do students record pairs neatly in order, or skip around? Common errors: calculation mistakes by card-holders under time pressure, teams identifying slope but missing the constant term, groups testing too few values before guessing. During gameplay, listen for evidence-based reasoning—students who justify proposed rules with specific pairs show stronger algebraic thinking than those who guess without verification.