Choose Two Numbers to Inspect
Explore how HCF is discovered through both Listing and Short Division.
Concept: What is the Highest Common Factor (HCF)?
A factor divides a number completely with zero remainder. It represents the equal whole pieces you can split the number into.
A Common Factor is a rod or tile size that can measure both lengths 12 and 18 exactly without any leftover fragment.
The HCF is the single longest rod possible. In fractions, dividing both numerator and denominator by the HCF reduces it to lowest terms in exactly one step!
Listing Method (列舉法)
Teacher Note: Highlight to students how the common factors are divisors shared by both numbers. The highest element in the shared list is the HCF.
Short Division / Ladder (短除法)
Divide both numbers simultaneously by shared prime divisors until the bottom quotients are coprime (no common factors other than 1).
For the HCF, take only the divisors on the vertical left side that divided both numbers:
Key Distinction: Remind students that HCF only multiplies the vertical left divisors, whereas LCM continues around the bottom quotients.
When simplifying a fraction formed by A and B, dividing both by their HCF directly reduces it to simplest form without multiple reduction steps!
Choose Two Numbers to Inspect
Explore how LCM is discovered through both Listing and Short Division.
Concept: What is the Least Common Multiple (LCM)?
A multiple is obtained by multiplying the number by integers $1, 2, 3\dots$ It is the sequence of steps landing forward on a number track.
If Runner A leaps in steps of 4 and Runner B leaps in steps of 6, the LCM is the very first mile marker where both land simultaneously.
In fractions, denominators cannot be compared or added unless they are cut into the same unit size. The LCM provides the smallest possible common denominator.
Listing Method (列舉法)
Teacher Note: While any common multiple can be used as a common denominator, the Least Common Multiple keeps intermediate numbers and calculations simplest.
Short Division / Ladder (短除法)
Extract shared prime factors on the left until the bottom quotients are coprime. Then multiply down the left column and across the bottom row.
Multiply all divisors in the left column by the bottom remaining quotients:
Visual Memory Trick: Notice the factors form the shape of a capital letter "L" (for LCM)!
When given two fractions with denominators 4 and 6, their LCM (12) is the primary target for re-slicing both wholes!
1. Find Common Denominator via LCM
"Notice the denominators are different. To compare or combine them, we must divide both wholes into identical unit slices using a common multiple."
Synchronized Fraction Strips
Circular Pie Models
Original 4 slices divided into 12 slices.
Original 6 slices divided into 12 slices.
LCM Calculation for Denominators
Multiply all common divisors in the left column together with bottom quotients: