Check students' understanding of factors and their strategies for systematic factor-finding. These questions reveal prerequisite knowledge and where students need support.
Find factor pairs
- Find all factor pairs for a whole number in the range 1-100. (4.OA.B.4)
Before You Play
What are factors, and how are they different from multiples? Give me an example of each using the number 12.
Listen for: Factors divide evenly into a number (1, 2, 3, 4, 6, 12 for 12) while multiples are products (12, 24, 36). Students often confuse these. Watch for: Hand gestures showing "factors go into" (hands moving together) versus "multiples build from" (hands expanding outward)—these physical representations show conceptual understanding.
How would you find all the factor pairs of 24? Walk me through your strategy.
Listen for: Systematic approaches ("Start with 1, then try 2, then 3...") versus random guessing. Strong responses mention testing small numbers first and recognizing when to stop. Watch for: Finger counting or other physical tracking—students organizing their search bodily often develop systematic thinking.
Point to the suspects on this detective notebook. If I give you a clue that says "the thief's number is divisible by 7," which suspects would you eliminate?
Watch for: Whether students point systematically through the grid or jump around. Methodical pointing shows organized thinking. Listen for: Understanding that "divisible by 7" means 7 is a factor. Students might say "70 works because seven times ten equals seventy" or use skip counting while pointing to each number in sequence.
Setup Tip: Position the digital mystery map where all team members can see and reach the screen. Give each student their own detective notebook with space to mark eliminations clearly. If sharing notebooks, seat partners side by side (not across) so both can access it during gameplay. This physical arrangement prevents one partner from dominating while the other watches passively.
During Gameplay
The game requires strategic thinking and precise factor work. Prompt mathematical reasoning, catch errors early, and help students develop systematic approaches to factor-finding and elimination.
How did you decide which room to search? Does your team have a strategy?
Listen for: Intentional strategies ("We're going around the arena in order") versus random selection. Watch for: Students who physically trace paths on the screen with their finger before deciding—this spatial planning often precedes verbal strategy articulation.
⚡ Strategy Development:
Room selection doesn't affect math outcomes (each gives a random riddle), but systematic selection builds organized problem-solving habits. Deliberate choices in low-stakes moments transfer to higher-stakes mathematical decisions.
You're looking for all factor pairs of 56. How will you make sure you don't miss any?
Listen for: Organized approaches: "Start with 1, then 2, 3, 4..." or "Test divisibility by small numbers first." Strong students mention stopping at the square root or when factors repeat. Watch for: Physical tracking—tapping the desk, counting on fingers, making checkmarks. Even without prompting, students invent ways to track what they've tested.
How do you know when you've found all the factor pairs?
Listen for: Understanding that factor pairs mirror each other and meet in the middle: "When the numbers start repeating" or "When I get past the square root." Watch for: Hand gestures showing symmetry—both hands coming together at a middle point demonstrates this understanding physically.
⚡ Common Error:
Students miss factor pairs by testing randomly. Have struggling teams test divisibility by 1, 2, 3, 4, 5... in order, recording each successful pair. This structured approach prevents omissions.
The clue says the thief's number is divisible by 7. How will you test each suspect?
Listen for: Clear testing methods: "Check if 7 goes into this number evenly" or "Is this a multiple of 7?" Strong responses reference specific multiplication facts ("Seven times eight is fifty-six, so 56 stays"). Watch for: Students touching or pointing to each suspect as they test—this tactile tracking helps them avoid skipping numbers.
Why do we keep suspects whose numbers ARE divisible by the clue factor, rather than eliminating them?
Listen for: Understanding the logic: "The clue tells us what the thief's number IS, not what it ISN'T" or "We eliminate the ones that don't match." This reversal confuses many students initially—clarifying early prevents persistent errors.
⚡ Watch For:
Students who develop marking systems on their notebook—crossing out eliminated suspects, circling remaining ones, using check marks. These organizational systems reveal mathematical thinking. If errors occur, have students point to the suspect and verbally test divisibility again before correcting—this physical re-engagement often catches the mistake.
After three clues, which suspects remain? How do you know you've applied all clues correctly?
Listen for: Students who can reconstruct their logical chain: "This one got eliminated by the first clue, these two by the second clue..." This reveals systematic tracking. Watch for: How students scan their notebook—methodical versus scattered eye movements signal organized thinking. Some students run their finger across each row to verify eliminations.
⚡ Collaboration:
In pairs, encourage division of labor: one partner finds factor pairs while the other verifies, or alternate who tests divisibility for each clue. This coordination builds accuracy through peer checking.
Before you submit your guess, how confident are you?
Listen for: Discussion of certainty levels and reasoning about remaining suspects. If only one remains, they should be very confident. If multiple remain, listen for probabilistic thinking ("50/50 chance") or strategic guessing. Watch for: Teams that physically review their notebook one more time, running a finger down the list before guessing—this double-checking habit is valuable.
After You Play
Help students articulate what they learned and extract generalizable strategies. Focus on helping them verbalize their thinking and recognize patterns across games.
What strategy did you develop for finding all factor pairs? Did your approach change from the first game to later games?
Listen for: Strategic evolution: "At first I guessed, but then I realized I should test 1, 2, 3 in order" or "I started recognizing which factors would work faster." This metacognitive reflection is the key learning outcome. Watch for: Students demonstrating their physical tracking methods while explaining—showing how they counted on fingers or tapped to keep track.
Which clues eliminated the most suspects? Why do some clues eliminate more than others?
Listen for: Recognition of strategic patterns: "Clues about 7 or 11 eliminated more suspects because fewer numbers are divisible by those" versus "Clues about 2 didn't help much because lots of numbers are even." This reveals understanding that less common factors provide more information.
Point to a mistake you made on your detective notebook. What went wrong and how did you catch it?
Watch for: Students who can locate specific errors and explain faulty reasoning. Listen for: Self-correction strategies: "I thought 54 was divisible by 7, but when I checked again I realized 7×7 is only 49." The ability to identify and correct errors indicates mature mathematical thinking. Notice: Whether students point to the error while explaining—this physical reference helps them reconstruct their thinking process.
What's the difference between finding factor pairs for a riddle versus testing if a suspect's number is divisible by a clue?
Listen for: Connections between operations: "When I find factor pairs, I'm finding all the factors. When I test divisibility, I'm checking if a specific number is one of those factors" or "They're opposite directions—factor pairs build UP to the number, divisibility tests check if the number breaks DOWN evenly." Watch for: Students using directional hand gestures to show "building up" versus "breaking down"—these spatial representations clarify the mathematical distinction.
Extensions & Variations
Speed Round
Set a timer for 15 minutes. This adds pressure that encourages faster factor recognition and more efficient divisibility testing, building fluency through time constraints.
Minimum Rooms Challenge
Can students solve the mystery searching only 3 rooms instead of 5? This requires strategic thinking about which clues provide maximum information and encourages thinking ahead.
Create Your Own Mystery
Students design their own nine-suspect lineup using different 2-digit numbers, then create factor riddles that systematically eliminate suspects. This requires understanding factor relationships deeply enough to engineer specific outcomes.
Physical Suspect Grid
Create a 3×3 grid on the floor with tape. Place number cards (jersey numbers) in each space. Students physically walk to each suspect as they test divisibility. This spatial version makes elimination kinesthetic and helps students who benefit from movement. Watch how students develop walking patterns—some move systematically across rows, others spiral from the center.
Explain Your Reasoning
After each clue, one team member must explain aloud why specific suspects were eliminated and which remain. This verbal articulation strengthens mathematical communication and ensures both partners understand the logic.
Larger Number Set
Advanced students play with suspects using 3-digit numbers (101-999), requiring more sophisticated factor-finding and divisibility testing. Include factors beyond 12 to increase challenge.
Practical Notes
TIMING
Plan 25-30 minutes for the first game including setup and debrief, with subsequent games taking 15-20 minutes as students gain fluency. Factor-finding takes the most time initially but speeds up with practice. Allow a few minutes after each game for reflection questions—this consolidation time is where learning solidifies.
GROUPING
Pairs work best because both students stay actively engaged—one finding factor pairs while the other verifies, or alternating roles. In groups of 3-4, some students disengage. If using larger groups, assign explicit roles: Factor Finder, Checker, Elimination Marker, rotating with each new room.
MATERIALS & SPACE
The detective notebook is the central workspace for tracking logical progress. Position it where both partners can reach it. Provide erasable markers or light pencils for early games so students can correct errors without messy crossing-out. Position the digital mystery map centrally rather than in front of one student—this encourages shared decision-making and prevents one partner from controlling all interactions.
ASSESSMENT
Look for systematic factor-finding rather than random guessing. Listen for students articulating their strategy verbally. Check detective notebooks after gameplay—organized, sequential marking reveals systematic thinking; scattered marks suggest guessing. Students who struggle with divisibility testing may need additional practice with multiplication facts or skip-counting before returning to the game.