Investigating Multiples and Factors

Active learning helps students see the difference between factors and multiples as actions rather than abstract definitions. When students build arrays or move in timed relays, they feel the difference between dividing into equal parts and counting forward in jumps. This physical and social engagement turns number theory into something they can manipulate and explain.

• Year 6
• Mathematics
• 4 activities
• 20 min–45 min

Learning Objectives

1. Differentiate between multiples and factors of whole numbers up to 100.
2. Calculate the least common multiple (LCM) for pairs of numbers using listing or prime factorization.
3. Determine the greatest common factor (GCF) for pairs of numbers using listing or prime factorization.
4. Solve problems involving grouping and sharing by applying the concepts of LCM and GCF.
5. Explain why a number has an infinite set of multiples but a finite set of factors.

Collaborative Problem-Solving

⏱ 35 min·Pairs

Array Building: Factor Pairs

Provide square tiles or grid paper. Students build rectangles for numbers like 12 or 18, listing dimensions as factor pairs. Pairs compare arrays to identify common factors and compute GCF. Share findings on class chart.

Prepare & details

Differentiate between a multiple and a factor of a number.

Facilitation Tip: During Array Building, have students rotate partners so each pair explains their rectangle to a new group before switching roles.

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

Collaborative Problem-Solving

⏱ 25 min·Small Groups

Multiples Relay: LCM Challenges

Divide class into teams. Call out pairs of numbers; first team member runs to board, writes multiples until common ones appear, tags next for LCM. Discuss efficient strategies post-race.

Prepare & details

Analyze how finding the least common multiple can help solve real-world problems.

Facilitation Tip: In Multiples Relay, set a visible timer and call out the next multiple so students practice listening while moving.

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

Collaborative Problem-Solving

⏱ 45 min·Small Groups

Real-World Sharing Stations

Set up stations with problems: divide 48 cookies by 6 and 8 kids (GCF), bus schedules (LCM). Groups solve with drawings or counters, rotate, and explain solutions.

Prepare & details

Justify why every number has an infinite number of multiples but a finite number of factors.

Facilitation Tip: At Real-World Sharing Stations, station signs should include visuals like plates or baskets so students connect the math to the objects.

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

Collaborative Problem-Solving

⏱ 20 min·Whole Class

Factor Bingo Boards

Students create bingo cards with factors of 36. Call multiples; they mark factors. First to line wins, then justify choices in pairs.

Prepare & details

Differentiate between a multiple and a factor of a number.

Facilitation Tip: With Factor Bingo Boards, require students to write the factor pair in the corner of each square before marking it to reinforce the definition.

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

Teaching This Topic

Teachers approach this topic by starting with concrete models before moving to abstract lists. Arrays help students visualize factor pairs as the sides of a rectangle, while multiples become the increasing rows or columns. Avoid rushing to algorithms; let students discover patterns through repeated building and counting. Research shows that students who explain their arrays to peers retain the difference between factors and multiples better than those who only complete worksheets.

What to Expect

Successful learning looks like students using physical tools to model factors as rectangle sides and multiples as growing chains. They discuss why some numbers have only two factors, and they apply LCM and GCF to real sharing problems without mixing up the terms. Clear explanations from students show they see the connection between arrays, lists, and real-world use.

Watch Out for These Misconceptions

Common Misconception

During Array Building, watch for students labeling the total number of tiles as a factor instead of the side lengths.

What to Teach Instead

Ask them to point to the two numbers that multiply to make the total and label those sides as the factors. Have them explain to a partner why the total is not a factor.

Common Misconception

During Multiples Relay, watch for students assuming all multiples are even because they only use even starter numbers.

What to Teach Instead

Have them run the relay with odd starters like 3 or 5 and record the multiples. After the round, ask the group to describe the pattern and why it differs from even multiples.

Common Misconception

During Multiples Relay, watch for students thinking the LCM is always the larger number.

What to Teach Instead

Stop the relay at 12 for 4 and 6, then ask students to compare the timeline to both numbers. Ask them to find the next common point and explain why it is larger than both.

Assessment Ideas

Quick Check

After Array Building, present pairs of numbers like 18 and 24. Ask students to build arrays for each, list all factor pairs, and then identify the GCF from their arrays. Collect their boards to check for accurate labeling of sides as factors.

Exit Ticket

During Real-World Sharing Stations, give each student a card with a number like 12 or 18. Ask them to write the possible numbers of friends Sarah can share with by listing the factors. Then have them solve the LCM problem for chores on the back of the same card before handing it in.

Discussion Prompt

During Factor Bingo Boards, pause the game and ask students to compare prime and composite numbers. Ask why a number like 7 has only two factors but an infinite number of multiples. Have students share examples from their boards to justify their answers.

Extensions & Scaffolding

• Challenge early finishers to find the smallest number with exactly 6 factors using their array models.
• Scaffolding for struggling students: Provide pre-made arrays with some tiles missing so they can count and complete the factor pairs.
• Deeper exploration: Ask students to create a number with a GCF of 5 when paired with 20, then prove it with arrays.

Key Vocabulary

Factor	A factor is a number that divides exactly into another number without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
Multiple	A multiple is the result of multiplying a number by an integer. For example, the multiples of 5 are 5, 10, 15, 20, and so on.
Greatest Common Factor (GCF)	The GCF is the largest factor that two or more numbers share. It is also known as the highest common factor (HCF).
Least Common Multiple (LCM)	The LCM is the smallest positive number that is a multiple of two or more numbers. It is also known as the lowest common multiple.