Mathematics · Year 6 · The Power of Number Systems · Term 1

Investigating Multiples and Factors

Investigating common multiples and factors to solve problems involving grouping and sharing.

• ACARA Content Descriptions
• AC9M6N01

About This Topic

Investigating multiples and factors equips Year 6 students with tools to break down numbers and solve practical problems. Multiples of a number come from repeated addition or multiplication by integers, like 4, 8, 12 for 4. Factors are divisor pairs that multiply to the number, such as 1 and 24, 3 and 8 for 24. Students find common multiples and factors, calculate least common multiples (LCM) and greatest common factors (GCF), and apply them to grouping items or sharing resources equally.

This content supports AC9M6N01 in the Australian Curriculum's number strand, linking to prime factorization and divisibility rules from prior years. Students differentiate multiples from factors, explore why multiples extend infinitely while factors are finite, and justify LCM use in scenarios like synchronizing class rotations or dividing play equipment.

Active learning benefits this topic greatly because visual and kinesthetic methods reveal patterns that worksheets miss. When students arrange counters into arrays or compete in multiples relays, they internalize relationships through doing, correct errors in real time, and connect math to everyday decisions with lasting understanding.

Key Questions

1. Differentiate between a multiple and a factor of a number.
2. Analyze how finding the least common multiple can help solve real-world problems.
3. Justify why every number has an infinite number of multiples but a finite number of factors.

Learning Objectives

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

Before You Start

Introduction to Whole Numbers and Operations

Why: Students need a solid understanding of whole numbers and the basic operations of multiplication and division to grasp the concepts of factors and multiples.

Division with Remainders

Why: Understanding division with remainders is crucial for identifying factors, as a factor results in a remainder of zero.

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.

Watch Out for These Misconceptions

Common Misconception

Factors and multiples are the same.

What to Teach Instead

Students often reverse definitions, thinking factors are bigger numbers. Hands-on array building shows factors as rectangle sides while multiples form longer chains. Peer teaching reinforces distinctions as students explain their models.

Common Misconception

All multiples are even numbers.

What to Teach Instead

This stems from examples with even starters. Multiples games with odd numbers like 3 or 5 reveal patterns. Group discussions help students spot the error and generalize rules.

Common Misconception

LCM is always the larger number.

What to Teach Instead

Trial-and-error with timelines shows LCM can exceed both, like LCM(4,6)=12. Relay activities build intuition through repeated practice and comparison.

Active Learning Ideas

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.

35 min·Pairs

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.

25 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.

45 min·Small Groups

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.

20 min·Whole Class

Real-World Connections

• Event planners use the LCM to schedule recurring events, like coordinating a town's annual festival that happens every 3 years and a fireworks display that occurs every 4 years, to find when both will happen in the same year.
• Bakers use the GCF to divide ingredients equally when making multiple batches of cookies. If a recipe needs 24 eggs and they have 36 cups of flour, they find the GCF to determine the largest number of identical cookie batches they can make.
• Musicians use LCM to determine when different rhythmic patterns will align. If one drummer plays a beat every 2 counts and another plays a beat every 3 counts, the LCM (6) tells them when both beats will occur simultaneously.

Assessment Ideas

Quick Check

Present students with two numbers, such as 18 and 24. Ask them to list all factors of each number, identify the GCF, list the first five multiples of each number, and identify the LCM. Review their lists to check for accuracy in identifying factors and multiples.

Exit Ticket

Pose a problem: 'Sarah has 12 stickers and wants to share them equally among her friends. What are the possible numbers of friends she can share with?' Students write down the factors of 12. Then, ask: 'If John has 2 chores to do every 3 days and Mike has 2 chores to do every 4 days, when will they next have the same number of chores due on the same day?' Students calculate the LCM of 3 and 4.

Discussion Prompt

Ask students to explain in their own words why a number like 7 has only two factors (1 and 7) but an infinite number of multiples. Facilitate a discussion comparing the number of factors and multiples for prime versus composite numbers.

Frequently Asked Questions

How do you differentiate multiples from factors in Year 6?

Use visuals: multiples as skips on a number line, factors as divisor pairs in arrays. Start with concrete examples like grouping 20 pencils (factors 1x20, 2x10) versus listing multiples (20,40). Progress to abstract problems, ensuring students justify with drawings before numbers. This scaffolds from concrete to formal reasoning over 50 words.

What real-world problems teach LCM effectively?

Apply LCM to bus timetables (every 15 and 20 minutes), recipe scaling, or playground rotations. Students solve: when do bells ring together? Group work with timers simulates urgency, linking math to routines. Extensions include calendars for events, building problem-solving across contexts in 60 words.

How does active learning help students master multiples and factors?

Active methods like tile arrays and relay races make abstract concepts visible and engaging. Students discover factor pairs by building shapes, compute LCM through competitive timelines, and correct misconceptions via peer feedback. This kinesthetic approach boosts retention by 30-50% over passive drills, fosters collaboration, and connects math to decisions like fair sharing.

How to address students struggling with infinite multiples?

Contrast finite factors (list all pairs) with endless multiples (extend number line). Use loops: keep multiplying by 1,2,3... without end. Games where teams race to largest multiple reinforce infinity. Pair with visuals like infinite paths, gradually shifting to justification statements for deeper understanding.