MATH · GRADES 6-8 · 35-45 MIN
Probability Walk: Experimental vs. Theoretical Probability lesson plan
Students predict landing probabilities from room areas, run fair random trials with coordinates, and compare experiment with theory.
At a glance
Time: 35-45 min
Grades: Grades 6-8
Subject: Math: Experimental vs. Theoretical Probability
Grouping: Teams of 3.
Space: A plan with 4-6 rooms inside a rectangle.
Teacher prep: 15 min printing + 40 min taping
Included: Teacher guide, student sheet, checks for understanding, exit ticket with answers
Open the printable lesson: teacher guide and student sheet →
Essential question
If the kitchen is a fifth of the house, does a random landing hit it a fifth of the time?
What students learn and show
Learn: Theoretical probability for a uniform random point is area share; experimental results vary and tend to stabilize with more trials.
Show: Compute a theoretical probability from areas and compare it with experimental frequency, explaining the gap.
Objectives
Compute theoretical probability from area ratios
Run repeated random trials and record outcomes
Compare experimental results to theoretical predictions
Explain why more independent, uniformly sampled trials often stabilize relative frequencies, without guaranteeing monotonic convergence
Before this lesson students should: Fractions, decimals and percents; area.
Materials and prep
The 1:1 print
Tape measures
A soft tossable (beanbag or paper ball)
Trial tally sheets
Calculators
Independent random-number source or shuffled coordinate cards, with a marked sampling boundary
Prep (about 15 min to print, 40 min to tape)
Measure the plan's rooms and draw a rectangular sampling boundary around the included area.
Use the random-number table on the student sheet (or random-number dice).
Copy tally sheets.
Space: A plan with 4-6 rooms inside a rectangle.
Grouping: Teams of 3.
Ways to run it: Painter's tape on the floor; 1:1 printed floor plan; Paper only (no floor space needed)
Lesson procedure
Teacher model: 'Kitchen 120 of 600 sq ft = 1/5 = 20%. Random pair (37, 82): walk 37% across, 82% up. Which room? Tally it.'
Guided practice: The class runs five trials together before teams continue.
The theory: compute each room's area share of the whole. The kitchen's 120 of 600 sq ft = 1/5 = a 20% prediction.
Trial run: generate 20 or more independent uniform coordinate pairs in the bounding rectangle. Resample points outside the included plan area. Walk to each accepted location with eyes open, identify the room and tally the result.
Face the gap: kitchen predicted 20%, landed 35%? Discuss — small samples wobble, and was the toss really random from that spot?
Pool the class: combine the accepted trials and compare relative frequencies with the area ratios. Larger unbiased samples often stabilize estimates, but a particular added batch can move farther from the prediction.
Bias extension: compare the coordinate method with gentle, supervised beanbag tosses into a clear area. Explain why toss locations are not uniform, so area alone is not their probability model.
Checks for understanding
Ask: A 90 sq ft bedroom in a 450 sq ft plan: theoretical probability? Look for: 90/450 = 1/5 = 20%.
Ask: 20 trials gave the bedroom 7 landings. Experimental probability? Look for: 7/20 = 35%.
Ask: Theory says 20%, experiment says 35%. Who is wrong? Look for: Neither necessarily: small samples vary.
Exit ticket
A plan is 500 sq ft; the bathroom is 50 sq ft. Theoretical probability? Answer: 10%.
The class pooled 200 trials and got 11% for the bathroom. Does this support the model? Answer: Yes, it is close; larger samples tend to be closer.
Scoring: Item 1: 1 point. Item 2: 2 points.
Common misconception
Students may think: Tossing a beanbag gives every spot an equal chance.
Address it: Throws are biased toward the thrower's aim; that is why we use random coordinates.
Supports and extensions
Learning support: The tally-keeper role with a clicker; prediction table pre-filled with the theoretical values to compare against.
Multilingual learners: Percentages and tallies carry it; 'predict,' 'trial,' 'result' pre-taught with three tosses.
Mobility and access: Trials can be run on a tabletop copy with a ruler; probability reasoning stays the goal.
Extension: Redesign the toss to be MORE random (spin first? higher arc?) and test whether the gap shrinks.
Transfer task: Estimate the probability that a raindrop lands on a garden bed in a yard.
Spaced review: Next unit: design a spinner with a 25% region.
Standards connections
CCSS 7.SP.C.6 (standard)
CCSS 7.SP.C.7 (standard)
Connections show which skills the activity practices. They are not a claim of full coverage; see the library for scope notes.
Why this approach
A 10-week program delivered by classroom teachers to 337 students in grades 3-6 (15 classes) improved spatial reasoning relative to standard mathematics instruction. Limit: The study measured spatial reasoning only, not mathematics achievement, and did not test this lesson.
Ready to teach it?
The printable version has the full teacher guide, a student recording sheet, prep steps and an exit ticket. Open the printable lesson →
Need the floor plan itself? Get a 1:1 print quote.
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