Factors, Multiples, and Primes

Active learning builds students’ number sense concretely. When learners physically arrange arrays, sort cards, or race to find factors, they turn abstract definitions into tangible experiences. These kinesthetic and visual tasks help fourth graders internalize the relationships between factors, multiples, and primes before moving to symbolic notation.

• 4th Grade
• Mathematics
• 4 activities
• 20 min–30 min

Learning Objectives

1. Identify all factor pairs for any whole number up to 100.
2. Classify whole numbers up to 100 as prime or composite, providing justification.
3. Calculate the first ten multiples for any given whole number.
4. Compare and contrast the concepts of factors and multiples for a specific number.
5. Explain why prime numbers are fundamental building blocks for composite numbers.

Stations Rotation

⏱ 30 min·Pairs

Format: Array Factor Hunt

Give each pair a set of square tiles or grid paper and assign them a number (e.g., 24). They build every possible rectangle with that many squares and record each dimension pair as a factor pair. Pairs share their factor lists and the class compares numbers with many factor pairs to numbers with only one, introducing composite vs. prime.

Prepare & details

Analyze what determines if a number can be broken down into equal smaller groups.

Facilitation Tip: During Array Factor Hunt, circulate with a checklist to note which students are starting arrays at 1 and moving up, rather than guessing.

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

Stations Rotation

⏱ 20 min·Small Groups

Format: Factor/Prime/Composite Sort

Small groups receive number cards 1-30 and sort them into categories: prime, composite, and (as a discussion challenge) the special case of 1. Groups compare their sorts and discuss any disagreements. Post a class anchor chart defining each category based on students' language from the discussion.

Prepare & details

Differentiate between factors and multiples of a given number.

Facilitation Tip: While students sort Factor/Prime/Composite cards, listen for precise language like 'exactly two factors' when they describe primes.

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

Stations Rotation

⏱ 25 min·Pairs

Format: Multiple Patterns on a Hundreds Chart

Each student colors multiples of an assigned number on a 1-100 chart. Pairs compare their charts and identify shared colored squares, which are common multiples. Discussion questions: What do the patterns look like? Why do some numbers appear on more charts than others? What do you notice about multiples of prime numbers?

Prepare & details

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

Facilitation Tip: On the Hundreds Chart for Multiple Patterns, ask students to describe the visual pattern they see before they color, to connect skip-counting with multiples.

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

Stations Rotation

⏱ 20 min·Small Groups

Format: Factor Pair Relay Race

Teams of four race to find all factor pairs for a given number by passing a recording sheet: each student adds one factor pair and passes it on. The first team to correctly list all factor pairs wins, but any team that misses a pair must keep working. Post-game discussion: how did you know you had found them all?

Prepare & details

Analyze what determines if a number can be broken down into equal smaller groups.

Facilitation Tip: In the Factor Pair Relay Race, stand at the finish line to watch students cross off found factors on their number cards to avoid duplicates.

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 through structured, systematic exploration first. Avoid rushing to rules like 'a number is prime if…' before students experience what primes look like through arrays and sorting. Use consistent vocabulary: factors go in, multiples come out. Research shows that repeated exposure to visual models and partner talk strengthens retention and corrects misconceptions before they take root.

What to Expect

By the end of these activities, students should confidently identify all factor pairs for numbers 1–100, classify numbers as prime or composite with reasons, and explain how factors and multiples connect. Clear vocabulary use and systematic strategies will be evident in their work and discussion.

Watch Out for These Misconceptions

Common Misconception

During Array Factor Hunt, watch for students who create only one array per number or stop searching once they find a pair.

What to Teach Instead

Prompt students to ask, 'Have I found all the rectangles?' and guide them to start at 1 and increment systematically, recording each array. Ask them to point to the missing arrays on their grid paper.

Common Misconception

During Factor/Prime/Composite Sort, watch for students who place 1 in the prime category or misclassify 2 as composite.

What to Teach Instead

Pause the sort and ask, 'How many factors does 1 have? How many does 2 have?' Use the card labels to remind them: primes have exactly two distinct factors, and composites have more than two.

Common Misconception

During Factor Pair Relay Race, watch for students who stop listing pairs once they reach a factor larger than 10, missing larger pairs.

What to Teach Instead

Have students use a whiteboard to track pairs and remind them to stop when the factors would cross (e.g., for 24, stop after 4 x 6 because 5 x 4.8 is not whole). Ask, 'How do you know you have them all?'

Assessment Ideas

Quick Check

After Array Factor Hunt, provide a list of numbers (e.g., 12, 17, 24, 29). Ask students to write all factor pairs for the composite numbers and circle the primes, writing 'prime' next to them.

Exit Ticket

During Factor Pair Relay Race, give each student an index card. On one side, write a number like 18 and ask them to list all its factors. On the other side, write 13 and ask them to explain if it is prime or composite and why.

Discussion Prompt

After Multiple Patterns on a Hundreds Chart, pose the question: 'If a number is a multiple of 6, what else do you know about its factors?' Guide students to discuss how multiples relate to the factors of 6 and other numbers.

Extensions & Scaffolding

• Challenge early finishers to create a new hundreds chart showing only prime numbers, then explain the pattern they notice.
• For students who struggle, provide circle cutouts to physically cover multiples on the chart, reducing visual clutter.
• Deeper exploration: Have students research why 2 is the only even prime number and present their findings to the class.

Key Vocabulary

Factor	A factor is a whole number that divides evenly into another whole number. For example, 3 and 5 are factors of 15.
Multiple	A multiple is the product of a whole number and any other whole number. For example, 12 and 18 are multiples of 6.
Prime Number	A prime number is a whole number greater than 1 that has only two factors: 1 and itself. Examples include 2, 3, 5, and 7.
Composite Number	A composite number is a whole number greater than 1 that has more than two factors. Examples include 4, 6, 8, 9, and 10.