Mathematics · Grade 9

Active learning ideas

Experimental Probability and Relative Frequency

Active learning works for experimental probability because students need to experience chance directly. When they flip coins or roll dice themselves, they see firsthand how probabilities emerge from repetition, making abstract concepts concrete. This hands-on approach builds intuition and reveals why chance behaves the way it does.

• Ontario Curriculum Expectations
• CCSS.MATH.CONTENT.7.SP.C.6
• CCSS.MATH.CONTENT.7.SP.C.7.B

• 30–45 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Experiential Learning
• 45 min · Whole Class

Whole Class: Coin Flip Marathon

Have the whole class flip coins simultaneously for 10, 50, and 100 trials per student, recording heads in a shared spreadsheet. Calculate relative frequencies after each round and plot on a class graph. Discuss how results change with more trials.

Compare experimental probability to theoretical probability, explaining potential discrepancies.

Facilitation Tip

During the Coin Flip Marathon, circulate to ensure groups record results accurately and use consistent counting methods.

What to look for

Present students with a scenario: 'A die was rolled 50 times, and the number 3 appeared 12 times.' Ask them to calculate the experimental probability of rolling a 3 and the theoretical probability of rolling a 3. Then, ask them to explain any difference.

• Apply
• Analyze
• Evaluate
• Self-Awareness
• Self-Management
• Social Awareness

Activity 02

• Experiential Learning
• 35 min · Small Groups

Small Groups: Dice Probability Relay

Groups roll a die 200 times total, passing it relay-style, and tally outcomes on a group chart. Compute experimental probabilities for each face and compare to theoretical 1/6. Predict outcomes for fewer trials and test.

Analyze how the number of trials affects the convergence of experimental probability to theoretical probability.

Facilitation Tip

For the Dice Probability Relay, place a timer at each station and have students rotate quickly to keep energy high.

What to look for

Students are given a bag with 5 red marbles and 5 blue marbles. Ask them to: 1. State the theoretical probability of drawing a red marble. 2. Describe a simple experiment they could conduct to find the experimental probability. 3. Predict how the experimental probability might change if they drew 100 times instead of 10.

• Apply
• Analyze
• Evaluate
• Self-Awareness
• Self-Management
• Social Awareness

Activity 03

• Experiential Learning
• 40 min · Pairs

Pairs: Custom Spinner Design

Pairs create spinners divided into unequal sections, spin 100 times, and record data. Calculate probabilities, then swap spinners with another pair to verify. Graph results to show convergence.

Construct a simple experiment to determine the experimental probability of an event.

Facilitation Tip

In the Custom Spinner Design activity, provide protractors and colored pencils so students can precisely label their spinners.

What to look for

Facilitate a class discussion using the prompt: 'Imagine you flip a coin 10 times and get heads 7 times. Is this result surprising? Why or why not? How would your confidence in the coin being fair change if you flipped it 1000 times and got heads 700 times?'

• Apply
• Analyze
• Evaluate
• Self-Awareness
• Self-Management
• Social Awareness

Activity 04

• Experiential Learning
• 30 min · Individual

Individual: Marble Jar Simulation

Each student draws marbles from a jar with known ratios 50 times with replacement, tracking frequencies. Calculate probabilities and reflect on discrepancies in a journal entry.

Compare experimental probability to theoretical probability, explaining potential discrepancies.

Facilitation Tip

During the Marble Jar Simulation, give each pair a sealed container so they cannot peek and change their sampling method.

What to look for

Present students with a scenario: 'A die was rolled 50 times, and the number 3 appeared 12 times.' Ask them to calculate the experimental probability of rolling a 3 and the theoretical probability of rolling a 3. Then, ask them to explain any difference.

• Apply
• Analyze
• Evaluate
• Self-Awareness
• Self-Management
• Social Awareness

A few notes on teaching this unit

Teach this topic by letting students run experiments before defining terms. Start with a few trials to show unpredictability, then scale up to hundreds to reveal patterns. Avoid lectures on probability formulas early on. Instead, let students discover the law of large numbers through their own data. Research shows this approach builds durable understanding better than abstract explanations alone.

Successful learning looks like students connecting their observations to probability language. They should fluently calculate both experimental and theoretical probabilities, explain variability in results, and recognize when sample sizes matter. Most importantly, they should use evidence from their experiments to discuss reliability and chance.

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Watch Out for These Misconceptions

• During the Coin Flip Marathon, watch for students assuming that after 10 flips with 6 heads, the next flip is more likely to be tails.

  Use the pooled class data to show how the proportion of heads stabilizes around 0.5, and ask students to explain why the law of large numbers makes individual flips unpredictable.

• During the Dice Probability Relay, listen for students believing that a small number of rolls (e.g., 20) gives a reliable estimate of the probability of rolling a 4.

  Ask students to graph their results on a line plot and discuss how the graph flattens out as trials increase, making variability more visible.

• During the Custom Spinner Design, notice students thinking that a section's size affects the outcome of the next spin.

  Have students spin their spinners 50 times and compare the experimental probability to the theoretical value, then ask them to reflect on why the spinner has no memory.

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Methods used in this brief

• Experiential Learning