Mathematics K–6 · Years 5–6
Rectangles from counters: factors, primes and square numbers
Number and algebra
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The idea
The rectangles that can be made from a number of counters show its factors, a prime makes only a single row, and a square number makes a square.
What you need
- 50 counters or square tiles per pair
- 1 cm grid paper
- a hundred chart and coloured pencils
- a recording table with columns number, rectangles, factors, prime or composite or square
How to do it
- Take 12 counters and make every rectangle you can with all of them. Draw each on the grid and write its dimensions. List the factors of 12 from the dimensions.
- Repeat for 7, 16, 18 and 24 counters.
- On the hundred chart, cross out 1, then the multiples of 2 except 2 itself, then the multiples of 3, 5 and 7 except 3, 5 and 7 themselves. Circle the numbers left and compare them with the numbers whose counters made only one rectangle.
- Make 1, 4, 9, 16 and 25 with counters and record the shape each makes. Predict the next square number and check it.
- Use a rectangle for 24 to write 24 as a product of its factors in more than one way.
What you should see
12 counters make 1 by 12, 2 by 6 and 3 by 4, giving the factors 1, 2, 3, 4, 6, 12. Seven counters make only 1 by 7, so 7 is prime. 16 makes 1 by 16, 2 by 8 and 4 by 4, the last a square, so 16 is a square number with factors 1, 2, 4, 8, 16. 24 makes 1 by 24, 2 by 12, 3 by 8 and 4 by 6. The sieve on the hundred chart keeps 25 primes below 100 uncrossed, and 36 is the next square after 25.
What changes
This activity lists no variables to change, measure and keep the same.
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- All odd numbers are prime.
- A number with more counters always has more factors.
Safety card
Hazards
- none beyond ordinary classroom materials
Controls
- none needed
Note
No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package is not needed; ordinary classroom supervision under the school's usual risk management.
Curriculum references
The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.
Sources
The pages the author read to write this activity.