Practice 1
List all unordered positive factor pairs of 18. How many distinct pairs are there?
Answer and teaching response
3 · The pairs are 1 × 18, 2 × 9 and 3 × 6.
Find every factor pair without counting reversals twice
Organize whole-number rectangle pairs systematically, identify when the search repeats and connect the complete list to prime or composite numbers.
Get the student and teacher materialsPrepare the group
Learning goal: A learner can produce a complete factor-pair list for a number up to 100 and explain why the list stops.
You will need: 24 equal counters, squared paper and a pencil. A factor pair here is an unordered pair of positive whole numbers, so 3 × 8 and 8 × 3 count as one pair.
Start with this question
What are all the positive whole-number factor pairs of 24, counting each reversed pair only once?
1 × 24, 2 × 12, 3 × 8 and 4 × 6. Each product is 24, and there are four distinct pairs.
- Begin with 1 as a possible row count and ask whether 24 counters form equal whole-number rows. Record 1 × 24.
- Try row counts in order: 2, 3 and 4 give partners 12, 8 and 6. Five does not divide 24 into equal whole-number rows, so do not record a pair for it.
- At 6, the partner is 4, reversing a pair already listed. Larger possible first factors can only produce partners below the ones already considered; the ordered search has crossed over.
- Check each recorded product and explain completeness. Since 24 has factors other than 1 and itself, it is composite. Reversing a rectangle does not create a new unordered pair.
Run the 15-minute workshop
- 3 minutes. Build a rectangle with 24 counters and ask whether turning it creates a new unordered pair.
- 4 minutes. Record possible first factors in order, including the unsuccessful trial of 5, and explain the crossover at 6.
- 5 minutes. List the pairs in each practice item before reporting the requested count.
- 3 minutes. Use the independent check; ask how the complete list supports calling a number prime or composite.
A useful error to discuss
A learner might say: Stop after finding two pairs, or count each rotated rectangle as a new pair.
Use an ordered first-factor column and keep a record of unsuccessful trials. A systematic search shows both that every pair was considered and why a reversal adds no new pair.
Practise, then check independently
Use the first three tasks for guided practice. Try the final task without help. Record a drawing or equation as well as the answer.
Practice 2
List all unordered positive factor pairs of 16. How many pairs are there? Count 4 × 4 once.
Answer and teaching response
3 · The pairs are 1 × 16, 2 × 8 and 4 × 4.
Practice 3
List the unordered positive factor pairs of 13. How many pairs are there, and what does that show?
Answer and teaching response
1 · Only 1 × 13 works; 13 is prime because its only positive factors are 1 and itself.
Independent check
Independently list every unordered positive factor pair of 30. Report the number of pairs and explain how you know the list is complete.
Answer and teaching response
4 · The pairs are 1 × 30, 2 × 15, 3 × 10 and 5 × 6. Trials of 4 do not divide evenly, and 6 reverses 5 × 6.
Transfer the idea
The number 1 also has just one unordered pair, 1 × 1. Why does that not make it prime?
Discuss a possible response
A prime number has exactly two distinct positive factors, 1 and itself. The number 1 has only one distinct positive factor, so it is neither prime nor composite.
Choose the next teaching step
Look for:
- Lists actual pairs before reporting a count.
- Explains reversals and distinguishes pair count from number of distinct factors.
If the idea is not secure: Use counters to test each first-factor row count and record whether a remainder is left.
If the learner is ready: Compare 16 and 18: both have three unordered pairs, but 16 has five distinct positive factors and 18 has six. Explain the role of the equal-factor pair.
Materials for your group
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