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Fraction Classroom · 2.3.0 · Teacher answers & responses

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.

Each row is one unordered factor pair of 24. The partners move toward one another; reversing a row does not add a new pair.Factor pairs of 241 × 24 = 242 × 12 = 243 × 8 = 244 × 6 = 24
Each row is one unordered factor pair of 24. The partners move toward one another; reversing a row does not add a new pair.

Build the idea

model. What are all the positive whole-number factor pairs of 24, counting each reversed pair only once?

  1. Begin with 1 as a possible row count and ask whether 24 counters form equal whole-number rows. Record 1 × 24.
  2. 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.
  3. 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.
  4. 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.

Worked example: 1 × 24, 2 × 12, 3 × 8 and 4 × 6. Each product is 24, and there are four distinct pairs.

Practise and explain

Try p1–p3 with a partner. Use a drawing and label your units. Try e1 independently before comparing responses.

p1. List all unordered positive factor pairs of 18. How many distinct pairs are there?

Answer: 3

Teacher response: The pairs are 1 × 18, 2 × 9 and 3 × 6.

p2. List all unordered positive factor pairs of 16. How many pairs are there? Count 4 × 4 once.

Answer: 3

Teacher response: The pairs are 1 × 16, 2 × 8 and 4 × 4.

p3. List the unordered positive factor pairs of 13. How many pairs are there, and what does that show?

Answer: 1

Teacher response: Only 1 × 13 works; 13 is prime because its only positive factors are 1 and itself.

e1. Independently list every unordered positive factor pair of 30. Report the number of pairs and explain how you know the list is complete.

Answer: 4

Look for:

  • Lists actual pairs before reporting a count.
  • Explains reversals and distinguishes pair count from number of distinct factors.

Teacher response: 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

transfer. The number 1 also has just one unordered pair, 1 × 1. Why does that not make it prime?

Answer: 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.

Run the 15-minute workshop

teach-1. 3 minutes: Build a rectangle with 24 counters and ask whether turning it creates a new unordered pair.

teach-2. 4 minutes: Record possible first factors in order, including the unsuccessful trial of 5, and explain the crossover at 6.

teach-3. 5 minutes: List the pairs in each practice item before reporting the requested count.

teach-4. 3 minutes: Use the independent check; ask how the complete list supports calling a number prime or composite.

Respond to the learner

misconception. Stop after finding two pairs, or count each rotated rectangle as a new pair.

Teacher response: 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.

support. Use counters to test each first-factor row count and record whether a remainder is left.

extend. 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.

Teaching notes