Now natural numbers hide patterns: primes, divisors, multiples and figures. Warm up with a multiplication tables and division training:
A visual map to your discoveries
Your mission: investigate the dividers
Discoveries itinerary
- Review multiplication and division; differentiate between exact division and division with remainder.
- Investigate divisors and prime numbers in the sieve.
- Use MDC to form equal groups and MMC to find cycle encounters.
- Explore figurative numbers. Triangles and Pythagoras appear as connections to geometry.
Start with prime numbers, then explore MDC and MMC.
Rectangles laboratory
Choose 12 objects and try 1, 2, 3, 4, 5 and 6 rows. What choices leave the rows with the same number? Then investigate 13: what changes?
Check out the investigation
The divisors of 12 are 1, 2, 3, 4, 6, and 12. 13 only has 1 and 13 as positive divisors: it is prime. Control goes up to 12 rows; draw the 13 rows with one object each to complete the investigation.
Cousins’ laboratory
Before moving: 1 is prime? And 9? Notice which numbers have exactly two positive divisors.
Kit challenge
There are 24 pencils and 36 erasers. We want to assemble as many identical kits as possible, using all the material. How many kits can we assemble? What goes in each kit?
Check the answer and strategy
The MDC of 24 and 36 is 12. Therefore, we can assemble 12 kits, each with 2 pencils and 3 erasers. MDC gives the largest number of kits that divides the two collections without any leftovers.
One more investigation: two alarms ring together now. One rings every 6 minutes and another every 8. In how many minutes will you play together again?
Check the location of alarms
In 24 minutes: the MMC of 6 and 8 is 24. List the positive multiples for each period and look for the first one in common.
Ready to move forward? Explain why 1 is not prime and when a problem calls for GCD or MMC.