Surface students' understanding of fraction magnitude, regrouping, and construction rules through conceptual questions that activate spatial reasoning.
Subtract mixed fractions
- Subtract mixed numbers with unlike denominators. 5.NF.A.2
Before You Play
Which is larger: 7⅝ or 5⅞? How can you tell without computing?
Listen for: "7 is bigger than 5, so 7⅝ is bigger"—recognizing the whole number dominates even though ⅞ is a larger fraction than ⅝.
Watch for: Hand gestures showing relative size—holding hands farther apart for the larger number, closer together for the smaller. This spatial gesture reveals magnitude intuition before any formal reasoning.
Try this: Ask students to show you the size difference with their hands while keeping eyes closed. The physical spacing often matches their conceptual understanding better than their verbal explanations.
On your paper, trace with your finger how you'd subtract 8⅚ from 12. Where do you need to regroup?
Watch for: Students circling or tapping the 12, then tracing the conversion to 11⁶⁄₆. This physical marking shows they grasp "breaking apart" a whole to create fractional parts.
Listen for: Clear regrouping language: "I need to make 12 into 11 and ⁶⁄₆" or "I can't take ⅚ from nothing, so I have to borrow."
Notice: Students who physically cross out and rewrite numbers internalize regrouping better than those who just talk through it. The hand movement reinforces the transformation.
If you have digits 3, 5, 7, 8, what mixed fractions could you build? What's the rule about where digits can go?
Listen for: "The denominator has to be bigger than or equal to the numerator" or "You can't have an improper fraction."
Watch for: Students pointing to different arrangements while testing possibilities—7⅜, 5⅜, 3⅝ (valid) versus 8⁷⁄₅ (invalid). Finger movement between positions shows systematic exploration rather than guessing.
Try this: Have students physically arrange digit cards on their desk, sliding them into different positions to test validity. The tactile manipulation makes the constraint more concrete than verbal explanation alone.
Setup Tip:
Seat students in pairs or groups of 3-4 where everyone can see the number generator screen. Physical proximity matters—students need to lean in together to read digits, which naturally encourages collaboration. Keep scorecards and pencils within arm's reach so recording happens quickly without breaking flow.
During Gameplay
Watch for strategic decision-making and computational accuracy. Use targeted questions to build estimation habits, verification routines, and fluency with mixed fraction operations.
Building Fractions
Before you write anything down, estimate: will your fraction be closer to 5, closer to 10, or somewhere in between?
Listen for: Magnitude predictions: "I'm making about a 7" or "This should be around 4-something." Clear estimates show strategic size thinking, not random digit arrangement.
Watch for: Hand spacing showing relative size while estimating. The gesture reveals magnitude intuition before computation happens.
Notice: Students who point to imaginary positions on the table between 5 and 10 are spatially locating their target. This physical anchoring helps them construct fractions more strategically.
How are you deciding which digit should be the whole number and which two make the fraction?
Listen for: Strategic reasoning: "I'm putting the biggest digit as the whole number so I can subtract more" or "I'm trying to make a fraction that will get me close to a certain number after subtracting from 12."
Watch for: Fingers moving between digits, testing arrangements before committing. This active testing shows exploration. Students who write immediately may be choosing without strategy.
Try this: Ask students to hover their pencil over each digit position before writing, explaining their choice out loud. The pause between thinking and writing builds deliberate decision-making.
⚡ Physical Checking
Students who point to each digit while deciding placement engage in productive physical reasoning. Ask them to explain their arrangement out loud—verbalization of physical testing builds both strategic awareness and fraction construction fluency.
First Subtraction
Before you calculate, estimate your answer. About what number will you get?
Listen for: Reasonable estimates based on magnitude: "If I subtract about 7, I'll get about 5" or "I'm taking away 9-something, so I should land around 3." Estimation creates a checkpoint for verifying calculations.
Common error: Students who skip estimation and jump to computation often miss regrouping errors. The act of predicting creates a reference point for noticing when calculations go wrong.
Try this: Have students hold up fingers showing their estimate (5 fingers = about 5, etc.) before computing. The physical commitment makes the prediction concrete and harder to ignore when checking work.
Show me on your scorecard where you're regrouping. What are you changing the 12 into?
Watch for: Students crossing out and rewriting the 12 as 11 with a matching fraction. This visible transformation demonstrates regrouping. Without this step, computation hits a wall.
Listen for: Precise transformation language: "I'm turning the 12 into 11 and six-sixths" or "I need to borrow one whole and make it into fifths."
Notice: Students who physically draw arrows from the 12 to show where the borrowed whole goes are building a mental map of the regrouping process. Encourage this visual marking.
⚡ Collaboration Strategy
Have partners verify each other's regrouping before subtracting. The working student points to their regrouping step while the partner confirms the transformation. Builds verification habits in both students.
Second Subtraction
Look at your result from the first subtraction. What do you need to subtract now to get as close to zero as possible?
Listen for: Strategic targeting: "I'm at 4⅛, so I need to make about 4" or "I want to get really close, so I'll try to match what I have."
Strategic insight: Students who think ahead about their target play more sophisticatedly than those who just make whatever fraction their digits allow without considering the goal.
Watch for: Students using their finger to measure the gap between their current result and zero on a mental number line, then planning their second fraction to bridge that gap. This spatial planning is mathematical reasoning in action.
Your two fractions have different denominators. How will you subtract them?
Listen for: Common denominator strategy: "I need to make both be twelfths" or "I'm converting thirds into ninths and fourths into ninths."
Watch for: Students drawing visual models or writing out equivalent fractions on scratch paper. Physical representation supports thinking about denominator relationships.
Try this: Have students tap out the rhythm of equivalent fractions—two taps for ½, four taps for 2/4, six taps for 3/6. The auditory pattern reinforces that these fractions are equal despite different numbers.
⚡ Common Error Alert
Students who try subtracting denominators directly ("4 minus 3 is 1") reveal a fundamental misunderstanding. Stop them immediately and ask: "Can you subtract thirds from fourths without converting them first?" This question surfaces the need for common units.
Scoring
Where does your answer fall in the score key? How many points did you earn?
Watch for: Students running their finger down the score key, checking each range until they find their answer. This physical tracking prevents scoring errors and builds careful reading habits.
Listen for: Precise categorization: "I got 1⅙, so that's in the 1 to 1½ range, which is 40 points."
Notice: Students who use a pencil tip or fingertip to bracket their result on the scoring table are engaging in productive precision. The physical marking helps them verify they're reading the correct row.
After You Play
Help students articulate strategies and reflect on their mathematical thinking. Focus on what worked, what didn't, and what patterns emerged. These consolidation questions build metacognitive awareness that transfers beyond the game.
What strategy did you develop for estimating whether your answer would be close to zero? Did your approach improve over the rounds?
Listen for: Strategy evolution: "At first I just guessed, but then I learned to think about what's halfway between my number and zero" or "I started checking my first answer before making the second fraction so I could aim better."
Metacognitive value: Students who articulate how their thinking changed demonstrate learning awareness that transfers to other mathematical contexts.
Watch for: Students using hand gestures to show the narrowing gap between their results and zero across rounds. The closing hand motion reveals their spatial understanding of improvement.
Look across all six rounds. Which round did you score best? What did you do differently?
Watch for: Students examining their scorecards, tracing back through work to identify successful moves. Physical review of artifacts helps connect decisions to outcomes.
Listen for: Specific tactical insights: "In Round 3, I made a medium-sized first fraction, which left me with more options" or "When I got ½ after the first subtraction, it was easy to subtract exactly ½ in the second."
Try this: Have students circle their best round on the scorecard and draw arrows to the specific moves that worked. The visual marking makes strategy concrete and memorable.
What was the hardest part of this game for you? What helped you figure it out?
Listen for: Specific challenges: "Finding common denominators" or "Remembering to regroup" or "Estimating what number to make."
Listen for: Problem-solving approaches: "My partner helped me see..." or "I figured out that..." or "After a few tries, I noticed..."
Compare your scorecard with three other students. What patterns do you notice in rounds where people scored well?
Watch for: Students physically arranging multiple scorecards to compare strategies. Collaborative material examination helps them see patterns across different approaches.
Listen for: Pattern recognition: "People who made fractions around 6 or 7 tended to score better" or "When you get really close on your first subtraction, you have less room for error on the second one."
Notice: Students who line up scorecards side-by-side and run their finger across corresponding rounds are conducting comparative analysis. This physical organization supports strategic thinking.
Extensions & Variations
Target Shift
Change the goal from zero to 2 or 5. Students subtract from 12 trying to land as close as possible to the new target. Scoring shifts accordingly. Recalibrates estimation and strategic thinking for different objectives.
Restricted Denominators
For struggling students, modify the digit generator to only produce 2, 3, 4, and 6. Limits possible denominators to numbers with easy common multiples. Lets students focus on estimation and strategy without getting stuck on complex denominator arithmetic.
Three Subtractions
Advanced variation: Students perform three subtractions per round instead of two. Start from 15, subtract a mixed fraction, then subtract two more. Increases strategic complexity—students must plan further ahead and manage a longer operation sequence.
Fraction Construction Challenge
Before each round, give students a target fraction to build (like "make a fraction close to 4½"). They must strategically arrange their four digits to hit the target. Reverses the usual process and emphasizes construction strategy over subtraction.
Partner Collaboration Mode
Pairs work together rather than competing. One partner does the first subtraction while the other does the second. They share a scorecard and must coordinate their thinking to achieve the best combined result.
Tournament Play
Run a class tournament with cumulative scoring across multiple games. Students track their averages and improvement over time. Meta-level competition rewards consistency and strategic development, not just single-round success.
Practical Notes
Timing
The first round typically takes 8-10 minutes as students learn the mechanics. Let this happen—rushing early rounds creates confusion and computational errors. By Round 3, most students settle into a 4-5 minute rhythm. Plan for 35-40 minutes total including setup and brief discussion.
Grouping
Pairs or groups of 3-4 work best. Students need to sit close enough to see each other's scorecards and the number generator simultaneously. Physical access matters—larger groups create wait time and reduce engagement.
Materials
Each student needs a scorecard and pencil. The gray "Spin #2" area on scorecards serves as a visual cue to copy their first answer—make sure students understand this before starting. Have calculators available for students who want to verify arithmetic without derailing strategic thinking.
Common Struggles
Watch for regrouping errors (skipping the step entirely), denominator confusion (trying to subtract unlike denominators directly), and estimation gaps (no prediction before computing). Each error type needs different intervention—regrouping needs conceptual work, denominators need procedural review, estimation needs strategic prompting.
Assessment
Scorecards provide written documentation of computational work—look for patterns across rounds. Students who consistently make regrouping errors need place value support. Students who score well show both computational accuracy and strategic thinking. Use post-game discussions to assess verbal articulation of fraction concepts.