Mathematics · Grade 4 · The Power of Place Value and Large Numbers · Term 1

Introduction to Factors and Multiples

Students explore factors and multiples through hands-on activities, identifying factor pairs and determining if a number is a multiple of another.

• Ontario Curriculum Expectations
• CCSS.MATH.CONTENT.4.OA.B.4

About This Topic

Factors are whole numbers that multiply together to equal a given number, and multiples are the results of multiplying a number by whole numbers. Grade 4 students identify factor pairs for numbers up to 100, such as finding (1, 36), (2, 18), (3, 12), (4, 9), (6, 6) for 36. They generate lists of multiples, like 3, 6, 9, 12 for 3, and classify numbers as prime, with exactly two distinct factors (1 and itself), or composite, with more than two.

This topic aligns with Ontario's Grade 4 mathematics expectations for operational sense and builds on place value by showing how numbers decompose. Students answer key questions like explaining the factor-multiple relationship through division checks and constructing complete factor pair lists. These skills prepare for fractions, where understanding divisors is essential, and encourage systematic listing using divisibility rules.

Concrete manipulatives reveal these relationships clearly. Students arrange tiles into arrays to discover pairs or use number lines to plot multiples visually. Active learning benefits this topic because exploration with materials helps students notice patterns independently, strengthens retention through kinesthetic engagement, and turns abstract ideas into tangible experiences that boost confidence.

Key Questions

1. Explain the relationship between factors and multiples.
2. Construct a list of all factor pairs for a given number.
3. Differentiate between prime and composite numbers based on their factors.

Learning Objectives

• Identify all factor pairs for any whole number up to 100.
• Calculate the first ten multiples for any given whole number.
• Classify whole numbers up to 100 as prime or composite based on their factors.
• Explain the inverse relationship between factors and multiples using division and multiplication examples.
• Construct arrays using manipulatives to represent factor pairs of a number.

Before You Start

Introduction to Multiplication and Division

Why: Students need a solid understanding of basic multiplication facts and the concept of division as sharing or grouping to grasp factors and multiples.

Number Sense and Numeration

Why: Familiarity with whole numbers and their properties is essential for identifying factors and generating multiples.

Key Vocabulary

factor	A factor is a number that divides evenly into another number. For example, 3 and 4 are factors of 12 because 3 x 4 = 12.
multiple	A multiple is the result of multiplying a number by any whole number. For example, 12, 15, and 18 are multiples of 3.
factor pair	A factor pair is a set of two numbers that multiply together to equal a given number. The factor pair for 12 could be (3, 4) or (2, 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.

Watch Out for These Misconceptions

Common Misconception

1 is a prime number.

What to Teach Instead

Prime numbers have exactly two distinct positive factors: 1 and the number itself. One has only a single factor. Sorting cards into categories with partner justification helps students test several numbers and recognize this unique case through repeated practice.

Common Misconception

Factors exclude 1 and the number itself.

What to Teach Instead

Every number's complete factor list starts with 1 and ends with itself. Array activities with tiles demonstrate the 1xn rectangle, making students see these endpoints visually and include them consistently.

Common Misconception

Factors and multiples mean the same thing.

What to Teach Instead

Factors divide evenly into a number; multiples result from multiplying out. Skip-counting games paired with division checks clarify the inverse relationship, as students physically mark multiples then divide back to find factors.

Active Learning Ideas

Tile Arrays: Factor Pairs

Give each small group interlocking cubes and number cards from 16 to 48. Students build all possible rectangular arrays for a number, record side lengths as factor pairs, then test completeness by multiplying pairs. Groups share one unique array with the class.

30 min·Small Groups

Skip-Count Number Lines: Multiples

Pairs draw number lines from 1 to 100. One partner calls a number like 7; the other marks multiples by skipping equal intervals. They compare with another pair's line for 7 and discuss patterns like every seventh number.

25 min·Pairs

Prime-Composite Sort: Card Challenges

Prepare cards numbered 1-50. Small groups sort into prime, composite, and neither piles, listing factors for each. Rotate cards among groups for verification, then hold a class vote on tricky numbers like 1 or 25.

35 min·Small Groups

Factor Bingo Boards: Whole Class Game

Students create 4x4 bingo cards with numbers 20-60. Call factor pairs like '3 and 8'; students mark multiples of 3 or 8, or numbers with those factors. First to connect four wins and explains their marks.

20 min·Whole Class

Real-World Connections

• Bakers often divide ingredients into equal portions, requiring them to understand factors. For instance, if a recipe needs to be shared among 6 people, a baker might look for factors of the ingredient quantities to ensure fair distribution.
• Event planners use multiples when scheduling activities for a large group. If a bus holds 40 people, they will need to calculate multiples of 40 to determine how many buses are needed for 120 guests (3 buses).

Assessment Ideas

Quick Check

Present students with a number, such as 24. Ask them to write down all the factor pairs for 24 and list the first five multiples of 4. Review responses to gauge understanding of both concepts.

Exit Ticket

Give each student a card with a number (e.g., 17, 20, 30). Ask them to write: 1) two factor pairs for the number, 2) three multiples of the number, and 3) whether the number is prime or composite, explaining why. Collect and review.

Discussion Prompt

Pose the question: 'If 5 is a factor of a number, what do you know about that number?' Guide students to explain that the number must be a multiple of 5 and therefore end in a 0 or 5. Discuss how this connects to divisibility rules.

Frequently Asked Questions

How do you teach factors and multiples in grade 4 math?

Start with concrete examples using numbers like 24. Guide students to list pairs by testing divisors from 1 up to the square root, then connect to multiples via skip-counting. Use real contexts like sharing 24 cookies equally. Reinforce with daily practice problems that build toward prime/composite classification, ensuring fluency by term's end.

What is the difference between prime and composite numbers?

Prime numbers greater than 1 have exactly two distinct factors: 1 and themselves, like 13. Composite numbers have more than two factors, like 15 (1, 3, 5, 15). Teach by having students list factors systematically; primes yield short lists, composites longer ones. This distinction supports later topics in factorization.

How can active learning help students understand factors and multiples?

Active approaches like building tile arrays or marking number line multiples let students discover factor pairs and patterns through touch and movement. This kinesthetic input reduces abstraction, as they see why 36 forms six rectangles but 37 forms only one. Group discussions during activities refine thinking, build peer teaching skills, and make concepts stick better than worksheets alone.

What are common misconceptions about factors for grade 4?

Students often omit 1 and the number itself from factor lists or confuse 1 as prime. They may think only even numbers have multiple factors. Address with hands-on sorts and arrays that force complete lists, plus class talks to air ideas. Regular low-stakes checks catch errors early and track growth.