Surface students' conceptual understanding of fraction multiplication before equation building. Focus on the key misconception that multiplication always makes things bigger. Mix traditional questions with spatial reasoning checks.
Multiply a fraction by a whole number
- Multiply a fraction by a whole number. (5.NF.B.4)
Before You Play
Which is bigger: 3/4 or 5/6? How do you know without finding a common denominator?
Listen for: References to proximity to one whole ("five-sixths only misses one piece, three-fourths misses one too, but sixths are smaller") or benchmark fractions ("both are more than 1/2, but 5/6 is closer to 1").
Watch for: Hand gestures showing fraction size—hands spreading wider or narrower reveals spatial thinking about magnitude. Some students cup their hands like holding water, adjusting the "container size" to show different fractions.
If I multiply 6 × 4, my answer is 24—bigger than both numbers. What happens when I multiply 6 × 3/4?
Listen for: Recognition that multiplying by a fraction less than one reduces the original value ("you're taking part of the 6, not all of it"). This is the critical foundation.
Watch for: Students who hold up fingers showing "6" then fold down one or two to show "taking away part." This physical reduction mirrors the mathematical concept. Initial predictions of "bigger" followed by self-correction show valuable thinking—don't rush past it.
Point to something in the room that's about 2/3 the width of your desk. How did you estimate?
Watch for: Students who stand up and physically align themselves with objects, turning their bodies to compare widths. Hand framing (forming rectangles with fingers) helps them visualize proportional relationships.
Listen for: Estimation strategies like "I looked for something smaller than my desk" or "I thought about dividing the desk into thirds." Squinting or tilting heads while comparing distances shows active spatial processing.
Setup Tip: Place the game board between partners so both can reach the hiker token and sun tracker. Position recording sheets side-by-side rather than stacked—when students see both sheets simultaneously, they discuss more and compare approaches naturally.
During Gameplay
Mathematical thinking happens during equation construction and strategic estimation. Students decide which numbers to use where, estimate whether their equation will hit the target, and adjust based on results.
You have five numbers. Which two would make the biggest possible fraction? The smallest?
Listen for: Recognition that large numerator with small denominator creates a big fraction, while small numerator with large denominator creates a small one.
Watch for: Fingers sliding across the recording sheet, pairing numbers mentally. Students who physically cover or bracket number combinations are testing possibilities visually without calculating yet.
⚡ Watch For: Students who immediately calculate before considering options. Prompt: "Before you write, which numbers look promising?" The pre-calculation pause is where strategic thinking develops.
Phase 2: Building Equations
Your target is 8. You're building 7 × 3/4 + something. Will 7 × 3/4 be more or less than 8?
Listen for: Estimation reasoning like "three-fourths of 7 is a little more than half of 7, so maybe around 5." Students estimating before calculating are developing number sense, not just following procedures.
Watch for: Students who gesture "cutting" 7 into pieces with hand motions, or who draw invisible number lines in the air. These spatial gestures reveal estimation thinking before formal calculation.
You have two equations that might work. How do you decide which to try first?
Listen for: Strategic thinking like "this one uses a bigger fraction" or "this one has larger addition, giving more control." Articulated decision-making builds metacognitive awareness.
Watch for: Partners pointing back and forth between potential equations, physically indicating with fingers which numbers they're discussing. One student may hold their hand over one equation while the partner points to another—this turn-taking gesture shows collaborative negotiation.
⚡ Collaboration: Encourage natural division of labor—one student manages the recording sheet while the other handles tokens. This physical split keeps both engaged while building interdependence.
Phase 3: Comparing to Target
Your result is 6.5 and the target is 8. How far off? What does that tell you about your next equation?
Listen for: Calculation of difference (1.5) followed by strategy: "We need to make our next equation bigger, maybe use a larger fraction." This iterative adjustment shows learning from results.
Watch for: Students who physically measure the gap with their fingers on the game board or recording sheet, holding that distance as a concrete representation of how much they missed by.
⚡ Facilitation Move: When teams hit the target exactly, ask them to explain their equation to another team. Having to articulate successful strategy reinforces learning and spreads effective approaches.
Phase 4: Tracking Progress
Trace the path from START to where your hiker is now. About what fraction of the journey have you completed?
Watch for: Finger tracing the path while estimating—physical action helps visualize fractional position. Students who tap each space are counting; those who sweep their finger smoothly are estimating proportionally.
Listen for: Fraction language like "about halfway" or "maybe two-thirds done." Some students will physically divide the board into sections with their hands, marking imaginary boundaries.
⚡ Materials: The recording sheet provides a permanent record of strategic thinking. If students erase heavily, they may lack confidence. Encourage visible scratch work—seeing multiple attempts reveals the iterative nature of problem-solving.
After You Play
Post-game consolidation should focus on articulating strategies and generalizing patterns. Students have built dozens of equations—help them see what worked, what didn't, and why.
What strategy did you use to decide which number should be the whole number versus which should form the fraction?
Listen for: Evidence of strategic evolution ("At first we picked randomly, then we realized bigger whole numbers give us more control"). This metacognitive reflection shows learning.
Watch for: Students who flip through their recording sheet pages while explaining, physically pointing to rounds as examples. This connection between strategy and evidence strengthens their argument.
Looking at all your equations, which fraction sizes appeared most often—close to 0, close to 1/2, or close to 1?
Listen for: Pattern recognition like "we used lots of fractions close to 1 because those keep most of the whole number." Noticing strategic patterns reveals sophisticated thinking.
Watch for: Students scanning down their recording sheet, sometimes using a finger or pencil as a visual guide. Some may physically sort or group their fractions by size, revealing they're thinking about magnitude.
Find one equation where you got very close to the target. Point to it and explain why that combination worked.
Watch for: Selective use of materials to reconstruct thinking. Students who trace their finger from the numbers used to the result are making the connection visible.
Listen for: Explanations connecting structure to results: "We used 6 × 5/6 because five-sixths keeps most of the 6, giving us 5, then added 3 to get exactly 8." This precision shows understanding.
Extensions & Variations
Exact Target Only
Move forward only when hitting the target exactly. This increases difficulty and requires more precise strategic thinking about number combinations.
Restricted Operations
Play without addition—just whole number × fraction must equal target. Simplifies structure but makes hitting targets harder, forcing more careful thinking about fraction magnitude.
Target Path Navigation
Create a floor number line (0-15) with tape. Students physically stand at their result, then step toward the target. Distance walked represents the difference—making absolute value concrete through movement.
Multiple Equation Challenge
For each round, find two different equations that both get within 1 of target. Comparing which gets closer develops evaluation skills and shows multiple paths can lead to similar results.
Reverse Engineering
Teacher provides target and result (e.g., target 8, result 7.5). Students work backward to determine what equation produced that result. Builds algebraic thinking about operation relationships.
Cooperative Speed Round
Class works as one team with a timer. Each pair plays simultaneously, and the class advances only when 75% of teams hit their target. Builds collective strategy sharing and peer support.
Practical Notes
TIMING
Full game takes 15-20 minutes including setup. Allow 3-4 minutes for conceptual activation questions before playing—this prevents confusion during gameplay. Each round takes 2-3 minutes. Plan 5 minutes for post-game discussion.
GROUPING
Pairs work best because both students actively contribute to equation construction. Avoid groups of 3—collaborative negotiation works better with two voices. Solo play is mathematically viable but removes valuable discussion about strategy.
MATERIALS
Position recording sheets so both partners can write and see clearly—slight overlap at corners works better than side-by-side. Keep the game board centered so both can reach tokens. The recording sheet becomes a thinking tool, not just documentation—students who keep scratch work visible show more sophisticated problem-solving.
ASSESSMENT
Look for strategic development across rounds: Do students adjust their approach after missing targets? Do they discuss options before committing? Recording sheets show calculation accuracy, but listen to conversations to assess whether they understand fractions as scaling operations or just follow procedures. Watch for estimation before calculating—this reveals number sense.