Factors, Multiples, and Prime Numbers

Activities that use visual and hands-on materials help students see the relationships between factors, multiples, and primes. Manipulatives and games make abstract concepts concrete, while problem-solving connects math to real-world situations. These approaches build lasting understanding through repetition and engagement.

• 6th-class
• Mastering Mathematical Reasoning
• 4 activities
• 30 min–45 min

Learning Objectives

1. Calculate the prime factorization of composite numbers up to 100.
2. Compare and contrast the sets of factors and multiples for two different numbers.
3. Explain the significance of prime numbers as the fundamental building blocks of integers.
4. Apply the concept of common multiples to solve problems involving synchronized events.
5. Determine if a number between 1 and 100 is prime using a systematic divisibility test.

Collaborative Problem-Solving

⏱ 30 min·Pairs

Manipulative: Factor Tile Arrays

Provide square tiles or grid paper for students to form rectangles matching a target number's area. Record side lengths as factor pairs, then compare arrays across different numbers. Extend by predicting factors before building.

Prepare & details

Explain why prime numbers are considered the fundamental building blocks of all other numbers.

Facilitation Tip: During Factor Tile Arrays, encourage students to arrange tiles in rectangles to physically confirm factor pairs, rotating pieces to test different configurations.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Simulation Game

⏱ 40 min·Pairs

Simulation Game: Sieve of Eratosthenes Hunt

Print number grids up to 100. Pairs cross out multiples of each prime starting from 2, circling remaining primes. Discuss the pattern and test new numbers systematically.

Prepare & details

Differentiate how identifying common multiples can help synchronize repeating events.

Facilitation Tip: In the Sieve of Eratosthenes Hunt, have students cross out multiples in the same color to reinforce the pattern of elimination and prime identification.

Setup: Flexible space for group stations

Materials: Role cards with goals/resources, Game currency or tokens, Round tracker

• Apply
• Analyze
• Evaluate
• Create
• Social Awareness
• Decision-Making

Collaborative Problem-Solving

⏱ 35 min·Small Groups

Problem Solving: Multiple Event Sync

Present scenarios like buses every 4 and 6 minutes. Small groups list multiples, find the least common multiple, and graph timelines. Share solutions and verify with counters.

Prepare & details

Apply a systematic method to determine whether a given number is prime.

Facilitation Tip: For Multiple Event Sync, provide students with a blank timeline to mark bus departures and visually align the two routes.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Stations Rotation

⏱ 45 min·Small Groups

Stations Rotation: Prime Factor Trees

Stations include dice rolls for numbers, tree-building templates, divisibility charts, and peer checks. Groups rotate, constructing and explaining factorizations before presenting one to the class.

Prepare & details

Explain why prime numbers are considered the fundamental building blocks of all other numbers.

Facilitation Tip: During Prime Factor Trees, ask students to write the prime factors in order at the bottom of their trees to reinforce the standard form of factorization.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Teaching This Topic

Teach this topic by starting with concrete examples before moving to abstract rules. Use guided questions to prompt students to notice patterns, such as how composite numbers break down into primes. Avoid rushing to definitions; instead, let students discover properties through exploration. Research shows that students grasp number theory better when they construct their own understanding through structured activities before formalizing concepts.

What to Expect

By the end of these activities, students should confidently identify factor pairs, generate multiples, and distinguish primes from composites. They should also explain how prime factorization works and apply common multiples to solve practical problems. Look for clear reasoning and accurate calculations in their work.

Watch Out for These Misconceptions

Common Misconception

During Factor Tile Arrays, watch for students who list 1 as a prime number and include it in their factor pairs for primes like 7 or 11.

What to Teach Instead

Have students create two columns on their paper: one for factor pairs and one for verifying if numbers are prime. Ask them to count the factors for 1 and primes, then discuss why 1 does not fit the prime definition.

Common Misconception

During Prime Factor Trees, watch for students who think composite numbers like 12 are not made of primes because they see 3 and 4 as separate factors.

What to Teach Instead

Prompt students to break down every composite number until only primes remain. Ask them to trace the branches back to the original number to confirm the product.

Common Misconception

During Multiple Event Sync, watch for students who assume the next common multiple of 15 and 20 must be even.

What to Teach Instead

Guide students to list multiples of both numbers and circle the smallest shared value, regardless of its parity. Ask them to compare the sequences to see the pattern.

Assessment Ideas

Exit Ticket

After Factor Tile Arrays and Prime Factor Trees, provide students with the number 48. Ask them to list all factor pairs, write the first five multiples, and draw a factor tree to show its prime factorization.

Quick Check

During Sieve of Eratosthenes Hunt, give students a mixed list of numbers including 1, 2, 9, 17, and 25. Ask them to circle the primes and explain why the others are not prime.

Discussion Prompt

After Multiple Event Sync, pose the question: 'Two classes share a hallway. One uses it every 6 minutes, the other every 8 minutes. How long until both classes use it at the same time again? Have students explain their method using multiples.'

Extensions & Scaffolding

• Challenge early finishers to find a number with exactly 8 factors and explain how they know their answer is correct.
• Scaffolding for struggling students: Provide a partially completed factor tree or allow the use of calculators to check multiplications.
• Deeper exploration: Ask students to research how prime numbers are used in real-world applications like cryptography or computer security.

Key Vocabulary

Factor	A number that divides exactly into another number without a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
Multiple	A number that can be divided by another number without a remainder; it is the product of a given number and any integer. For example, multiples of 5 are 5, 10, 15, 20, and so on.
Prime Number	A whole number greater than 1 that has only two factors: 1 and itself. Examples include 2, 3, 5, 7, and 11.
Composite Number	A whole number greater than 1 that has more than two factors. For example, 4 has factors 1, 2, and 4.
Prime Factorization	Expressing a composite number as a product of its prime factors. For example, the prime factorization of 12 is 2 x 2 x 3.