Multiplicative Thinking and Algebraic Patterns · Weeks 1-9

Factors, Multiples, and Primes

Students will find all factor pairs for a whole number in the range 1-100 and determine whether a given whole number is prime or composite.

About This Topic

Factors, multiples, and prime numbers form a foundational number theory unit in fourth grade that deepens students' understanding of how whole numbers are built and related. CCSS 4.OA.B.4 asks students to find all factor pairs for numbers 1-100 and classify numbers as prime or composite. These concepts are prerequisites for fraction work (finding common factors and multiples) and for understanding multiplication and division relationships more deeply.

Finding all factor pairs requires systematic thinking , students learn that starting with 1 and working upward to the square root of a number ensures no pairs are missed. Prime numbers, which have exactly two factors (1 and themselves), are often called the building blocks of all other numbers because every composite number can be expressed as a product of primes. This idea, while not formalized until later grades, is worth introducing conceptually in fourth grade.

Active learning benefits this topic because factor and multiple concepts are rich with patterns worth investigating. When students use arrays to find factor pairs, sort numbers by category, or explore patterns in multiples, they discover relationships that make the concepts memorable and connected. Inquiry-based tasks and collaborative sorting activities are particularly productive entry points.

Learning Objectives

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

Before You Start

Multiplication Facts

Why: Students need fluency with multiplication facts to efficiently find factor pairs and recognize multiples.

Division Concepts

Why: Understanding division is essential for determining if one number divides evenly into another, a key aspect of finding factors.

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.

Active Learning Ideas

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.

⏱ 30 min·Pairs

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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.

⏱ 20 min·Small Groups

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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?

⏱ 25 min·Pairs

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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?

⏱ 20 min·Small Groups

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Real-World Connections

Gardeners often plan planting arrangements in rows and columns, which relates to finding factor pairs. For instance, planting 36 seeds might involve arrangements like 4 rows of 9 or 6 rows of 6.

Musicians use multiples when counting beats in music. A common time signature like 4/4 means there are 4 beats per measure, and understanding multiples helps in recognizing rhythmic patterns and subdivisions.

Watch Out for These Misconceptions

Common Misconception

Students confuse factors and multiples, using the terms interchangeably or reversing their meanings.

What to Teach Instead

Anchoring each term to a concrete action helps: factors are the numbers you multiply together (factors go in), multiples are the products you get out (multiples come out). Sorting activities and factor pair recording give students repeated, meaningful practice with each concept in its proper context. Regular vocabulary checks during activities address confusion early.

Common Misconception

Students think 1 is a prime number because it has only one factor (itself).

What to Teach Instead

Prime numbers are defined as having exactly two distinct factors: 1 and the number itself. The number 1 has only one factor (1), so it meets neither the prime nor composite definition. It is a special case. Addressing this directly during sorting activities prevents the misconception from solidifying.

Common Misconception

Students stop looking for factor pairs too early, missing pairs where both factors are larger (e.g., missing 4 x 6 for 24 after finding 1 x 24, 2 x 12, and 3 x 8).

What to Teach Instead

Teach the systematic strategy of starting with 1 and incrementing one factor until the two factors would cross (i.e., when the smaller factor exceeds the square root). Array building makes this systematic search visual and reveals when all rectangles have been found. Asking 'how do you know you have them all?' promotes this habit.

Assessment Ideas

Quick Check

Provide students with a list of numbers (e.g., 12, 17, 24, 29). Ask them to write down all factor pairs for the composite numbers and to circle the prime numbers, writing 'prime' next to them.

Exit Ticket

On one side of an index card, write a number (e.g., 18). Ask students to list all its factors. On the other side, write a different number (e.g., 13) and ask them to explain if it is prime or composite and why.

Discussion Prompt

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 the original number.

Frequently Asked Questions

How do I teach factors and multiples to 4th graders?

Start with concrete representations: arrays for factors and skip-counting sequences for multiples. Explicitly contrast the two with examples and non-examples. Sorting tasks and factor pair hunts give students active practice distinguishing the concepts. Avoid introducing both in the same lesson initially; let students solidify one before adding the other.

Why is 1 not a prime number?

Prime numbers have exactly two distinct factors: 1 and the number itself. The number 1 only has one factor, so it does not fit the definition of prime. It is also not composite, which requires more than two factors. The number 1 is simply a special case, and it is worth addressing this directly in class since many students assume it is prime.

What active learning strategies help students understand factors, multiples, and primes?

Array building is particularly powerful for factors because students can physically construct every rectangle for a given area and record the dimensions as factor pairs. Hundreds chart coloring activities make multiples visual and reveal patterns. Sorting prime and composite numbers into categories , especially debating edge cases , builds genuine understanding of the definitions.

How do factors and multiples connect to fraction work in 4th grade?

Finding common denominators relies on finding multiples of two numbers. Simplifying fractions relies on finding common factors. Building fluency with factor pairs and multiples in the number theory unit makes these fraction procedures much more accessible later in the year. Pointing out these connections as they arise helps students see the coherence of the curriculum.