Review of Basic Probability and Counting Principles

Active learning builds confidence with counting principles by letting students physically manipulate examples. Sorting tasks and tree diagrams turn abstract rules into visual, kinesthetic experiences that correct misconceptions faster than lectures.

• 12th Grade
• Mathematics
• 4 activities
• 25 min–45 min

Learning Objectives

1. Classify counting problems as permutations or combinations based on whether order is significant.
2. Calculate the number of possible outcomes for sequential events using the multiplication principle.
3. Construct a probability model for a simple random event, including defining the sample space and assigning probabilities.
4. Analyze the relationship between permutations and combinations in solving counting problems.
5. Apply fundamental probability rules to determine the likelihood of simple and compound events.

Think-Pair-Share

⏱ 25 min·Pairs

Pairs Sort: Permutations vs Combinations

Provide pairs with 20 scenario cards, such as 'line up for photos' or 'choose toppings.' Students sort cards into permutation or combination piles and justify each choice with formulas. Pairs then swap piles with another pair to check and discuss discrepancies.

Prepare & details

Differentiate between permutations and combinations in various counting scenarios.

Facilitation Tip: During Pairs Sort, circulate and ask each pair to justify one card placement before moving on.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Think-Pair-Share

⏱ 35 min·Small Groups

Small Groups: Multiplication Principle Trees

Groups receive event prompts like clothing choices or lock combinations. They draw tree diagrams to count total outcomes, calculate using the multiplication principle, and verify by listing smaller cases. Share one tree with the class for feedback.

Prepare & details

Analyze how the multiplication principle applies to sequential events.

Facilitation Tip: For Small Groups: Multiplication Principle Trees, provide blank paper and colored pencils so groups can visually track branches of choices.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Think-Pair-Share

⏱ 45 min·Whole Class

Whole Class: Probability Model Simulations

Use a spinner or dice for a simple event like drawing marbles. Class predicts the model, runs 50 trials collectively recording data on a shared board, then revises the model based on empirical probabilities and discusses variances.

Prepare & details

Construct a probability model for a simple random event.

Facilitation Tip: Before Whole Class: Probability Model Simulations, demonstrate how to tabulate results in a shared class table to model precision.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Think-Pair-Share

⏱ 40 min·Individual

Individual: Counting Challenge Stations

Set up stations with problems varying difficulty, from basic to contextual like lottery odds. Students solve three per station, using manipulatives like beads for combinations, then rotate and self-check with answer keys.

Prepare & details

Differentiate between permutations and combinations in various counting scenarios.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Teaching This Topic

Teachers should avoid rushing to formulas before students experience the need for them. Start with concrete objects or scenarios, then formalize the language of outcomes and events. Research shows that tree diagrams and tables reduce errors by making the sample space explicit before calculations begin.

What to Expect

Students will confidently label scenarios as permutations or combinations, apply the multiplication principle to count outcomes, and construct accurate probability models using tables or trees. Evidence of learning includes correctly reasoned choices and clear representations.

Watch Out for These Misconceptions

Common Misconception

During Pairs Sort: Permutations vs Combinations, watch for pairs who treat both scenarios as the same count.

What to Teach Instead

Ask them to physically arrange three labeled cards in different orders, then group the same cards without regard to order. Have them write the counts and compare, leading to a discussion of why one scenario requires dividing by 3!.

Common Misconception

During Small Groups: Multiplication Principle Trees, watch for groups who incorrectly assume outcomes are identical or dependent.

What to Teach Instead

Have each group present their tree to the class and explain why each branch represents an independent choice. If they hesitate, ask them to list all possible outcomes for a simpler version, like two coin flips, to rebuild their understanding.

Common Misconception

During Whole Class: Probability Model Simulations, watch for students who assign probabilities based on intuition rather than observed frequencies.

What to Teach Instead

After rolling dice 50 times, ask students to compare their predicted probabilities to the actual frequencies in their table. Challenge them to explain discrepancies and adjust their model accordingly.

Assessment Ideas

Exit Ticket

After Pairs Sort: Permutations vs Combinations, provide an exit ticket with two scenarios: 1) Selecting 3 students from a class of 20 to form a committee. 2) Awarding gold, silver, and bronze medals to 3 runners out of 8. Ask students to identify whether each scenario involves a permutation or combination and briefly explain why.

Quick Check

After Small Groups: Multiplication Principle Trees, present a scenario: 'A restaurant offers 5 appetizers, 10 main courses, and 4 desserts. How many different meal combinations (one appetizer, one main, one dessert) can a customer order?' Ask students to write down the principle used and show their calculation.

Discussion Prompt

During Whole Class: Probability Model Simulations, pose the question: 'Imagine you are designing a lottery game. What are the key decisions you need to make regarding counting principles and probability? Consider how many numbers are chosen, the range of numbers, and whether the order matters.' Facilitate a class discussion on their choices.

Extensions & Scaffolding

• Challenge: Ask students to design a new password policy that uses both permutations and combinations, then calculate the total number of possible passwords under their rules.
• Scaffolding: Provide partially completed trees or tables with some branches already filled to guide students who struggle with starting from scratch.
• Deeper: Introduce dependent events and let students extend their probability models to situations where choices affect later probabilities.

Key Vocabulary

Permutation	An arrangement of objects in a specific order. The order of selection matters, so different orderings are counted as distinct outcomes.
Combination	A selection of objects where the order of selection does not matter. Only the group of objects selected is considered, not their arrangement.
Multiplication Principle	Also known as the fundamental counting principle, this states that if there are 'm' ways to do one thing and 'n' ways to do another, then there are m * n ways to do both.
Sample Space	The set of all possible outcomes of a random experiment or event. It is often denoted by S.
Probability	A measure of the likelihood that an event will occur, expressed as a number between 0 and 1, inclusive.