Factors of Whole NumbersActivities & Teaching Strategies
Active learning works well for factors because students need to see and touch the idea of pairing numbers. When they build arrays or group objects, they move from abstract symbols to concrete evidence of how factors multiply to form a whole. This hands-on approach builds confidence and accuracy, especially for learners who struggle with rote recall.
Learning Objectives
- 1Identify all factor pairs for whole numbers up to 100.
- 2Classify numbers as prime or composite based on their factors.
- 3Explain the relationship between factors and the construction of arrays.
- 4Apply divisibility rules for 2, 3, 5, and 10 to determine factors.
- 5Compare the number of factors for square numbers versus non-square numbers.
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Array Builder: Factor Grids
Provide counters and grid paper. Students select a number like 24 and build rectangular arrays, recording factor pairs such as 3x8 or 4x6. They test if arrays fit perfectly and discuss why some dimensions fail. Extend by finding all pairs systematically.
Prepare & details
Differentiate between a factor and a multiple of a number.
Facilitation Tip: During Array Builder, circulate and ask students to explain how their grid shows both factors of the number and not multiples.
Setup: Flat table or floor space for arranging hexagons
Materials: Pre-printed hexagon cards (15-25 per group), Large paper for final arrangement
Divisibility Dash: Rule Relay
Divide class into teams. Place number cards around the room. Students run to a card, apply one divisibility rule to identify a factor, and tag the next teammate. Teams compare lists at the end and verify with multiplication.
Prepare & details
Construct a method to find all factors of a given number.
Facilitation Tip: For Divisibility Dash, stand near the rule cards so you can gently correct misapplied rules in real time.
Setup: Flat table or floor space for arranging hexagons
Materials: Pre-printed hexagon cards (15-25 per group), Large paper for final arrangement
Factor Pair Match-Up: Card Game
Create cards with numbers and possible pairs. In pairs, students match factor pairs to products, then justify using arrays or rules. Shuffle for multiple rounds, timing for speed and accuracy.
Prepare & details
Explain how factors are used in real-world situations like grouping.
Facilitation Tip: In Factor Pair Match-Up, listen for students using divisibility rules to justify their matches before confirming correctness.
Setup: Flat table or floor space for arranging hexagons
Materials: Pre-printed hexagon cards (15-25 per group), Large paper for final arrangement
Grouping Challenge: Real-World Scenarios
Present problems like dividing 36 cookies among friends. Students draw arrays or list factors to find sharing options. Groups present solutions and vote on the fairest method.
Prepare & details
Differentiate between a factor and a multiple of a number.
Facilitation Tip: During Grouping Challenge, ask guiding questions like 'How many ways can you divide 24 cookies fairly?' to prompt systematic thinking.
Setup: Flat table or floor space for arranging hexagons
Materials: Pre-printed hexagon cards (15-25 per group), Large paper for final arrangement
Teaching This Topic
Teachers approach this topic by starting with concrete manipulatives before moving to abstract symbols, aligning with research showing that visual and tactile experiences strengthen number sense. Avoid rushing to formulas; instead, guide students to discover patterns through repeated exposure to arrays and grouping. Emphasise that factors are not just about listing but about understanding the structure of numbers, which prepares them for later work with prime factorisation and greatest common factors.
What to Expect
Successful learning looks like students confidently listing factor pairs without missing numbers and explaining why a number is prime or composite. They should use divisibility rules to speed up their work and describe the relationship between factors and multiples clearly.
These activities are a starting point. A full mission is the experience.
- Complete facilitation script with teacher dialogue
- Printable student materials, ready for class
- Differentiation strategies for every learner
Watch Out for These Misconceptions
Common MisconceptionDuring Array Builder, watch for students confusing factors with multiples by counting rows as multiples and columns as factors.
What to Teach Instead
Prompt students to label their arrays clearly: write the total number of counters at the top and the factor pairs along the sides to reinforce the difference.
Common MisconceptionDuring Factor Pair Match-Up, watch for students assuming all numbers have exactly two factors.
What to Teach Instead
Have students count the number of cards in each match and discuss why some numbers (like 7) have only one pair while others (like 12) have multiple.
Common MisconceptionDuring Grouping Challenge, watch for students excluding 1 as a factor because it feels too simple.
What to Teach Instead
Ask groups to start with 1xN arrays and discuss why 1 is always a factor, using examples like 1x5=5 to normalise its inclusion.
Assessment Ideas
After Array Builder, give students 24. Ask them to: 1. List all factor pairs. 2. Draw an array for 24. 3. Circle the pair that shows a square number.
After Divisibility Dash, write 14, 27, and 40 on the board. Ask students to hold up fingers for the number of factors each has, then write the factors for one number and explain if it is prime or composite.
During Grouping Challenge, pose: 'If 18 is a multiple of 3, what do you know about 18’s factors?' Guide students to explain that 3 must be a factor and to explore other factors of 18.
Extensions & Scaffolding
- Challenge: Create a number scroll to 100, marking each number as prime or composite and listing its factor pairs.
- Scaffolding: Provide partially completed arrays for students to finish, or let them use counters to build smaller numbers first.
- Deeper exploration: Explore square numbers and their unique factor pairs (e.g., 36 has 6 paired with itself).
Key Vocabulary
| Factor | A factor is a whole number that divides exactly into another whole number without leaving a remainder. For example, 3 and 4 are factors of 12. |
| Multiple | A multiple is the result of multiplying a whole number by another whole number. For example, 12 is a multiple of 3 and 4. |
| Array | An array is a visual arrangement of objects in rows and columns, representing multiplication. For example, 3 rows of 4 counters form an array for 12. |
| Divisibility Rule | A divisibility rule is a shortcut to determine if a number can be divided by another number without a remainder. Rules exist for numbers like 2, 3, 5, and 10. |
| 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. |
Suggested Methodologies
Planning templates for Mathematics
5E Model
The 5E Model structures lessons through five phases (Engage, Explore, Explain, Elaborate, and Evaluate), guiding students from curiosity to deep understanding through inquiry-based learning.
Unit PlannerMath Unit
Plan a multi-week math unit with conceptual coherence: from building number sense and procedural fluency to applying skills in context and developing mathematical reasoning across a connected sequence of lessons.
RubricMath Rubric
Build a math rubric that assesses problem-solving, mathematical reasoning, and communication alongside procedural accuracy, giving students feedback on how they think, not just whether they got the right answer.
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