Problem
For 4 and 6 the GCF is 2 and the LCM is 12, and 4 x 6 = 2 x 12 = 24.
Working a tiny example by hand is something a learner can fully see and check, and it makes the big idea visible without any abstract symbols.
6.NS.B.4Solve An Easier Related ProblemThat balance holds for any two whole numbers: their product equals GCF times LCM.
Spotting that the same balance shows up every time lets a learner trust the rule for the harder numbers in this problem.
6.NS.B.4Look For A PatternFor any two whole numbers, the two numbers multiplied together equal their greatest common factor multiplied by their least common multiple.
The greatest common factor collects the factors the two numbers share, and the least common multiple collects those shared factors together with whatever each number holds on its own.
Every factor in the pair sits either in the shared part or in one number's own part, with none left out and none counted twice.
Multiplying the two numbers uses each shared factor twice, once from each number, and that is the same collection of factors as the greatest common factor times the least common multiple.
The same factors may be regrouped in any order without changing what they multiply to.
Substitute the GCF of 12 and the LCM of 180. The product of the two numbers must equal 12 times 180, which is 2160.
Once the rule is known, the problem shrinks to one easy multiplication a sixth grader can do.
6.NS.B.4Solve An Easier Related ProblemSince the unknown number times 36 equals 2160, divide 2160 by 36 to get the number. 2160 divided by 36 is 60.
Dividing to undo a multiplication is the natural way to recover the hidden factor.
6.NS.B.2Solve An Easier Related Problem60 = 2 x 2 x 3 x 5 and 36 = 2 x 2 x 3 x 3 give GCF 12 and LCM 180, just as required.
Testing the answer against both the GCF and the LCM is a quick, concrete way to be sure it is right rather than just hoping.
6.NS.B.4Guess And CheckTwo numbers multiplied together always equal their GCF times their LCM, so you can use that balance to find a missing number.