Finding GCF and LCM with the Ladder (or Cake) Method

When I first came across the ladder method (ie: the upside-down cake method) for finding greatest common factors and lowest common multiples, I thought it was nothing short of complete genius. This was pretty recent, too! I love learning new methods for teaching math concepts. Prime factoring is super cool and extremely useful in building number sense, but if the goal is to find GCF and especially LCM, this cake method makes the process so much easier.

Here is a simple video explaining the process:

On the outside of the upside down cake are all the factors the two numbers have in common. On the bottom are the "leftover" numbers.

The GCF is the product of all the common factors on the left. The LCM is the product of the GCF and the "leftovers" on the bottom.

I wanted to make a math word wall reference for this method that students could refer to whenever they are finding GCF, LCM or even simplifying fractions (another neat use for this method brought up in our Visual MathFacebook group).

For practice, I made this set of self-checking GCF & LCM solve 'n check task cards. Students find GCF and LCM on the card then add, subtract, multiply or divide their answers and compare against the check number. Changes are if the numbers match, GCF and LCM were found correctly.

Another cool discussion that came up in the FB group was about relative prime numbers. In the word wall reference above, LCM= 2x2x2x3x10. While 10 is not prime, it is relatively prime because it and 3 have no more factors in common.

So we have the cake method for finding GCF and LCM and we have good old prime factoring. Any other methods?

When I was in grad school we learned about the Euclidean algorithm for finding GCF. Above is an example of how this algorithm works for finding that the GCF of 81 and 57 is 3.

I still like the cake method better :)

I made this GCF, LCM and Prime Factors math pennant before learning about the cake method. It was that recent! I think using the cake method would be a great way to complete the problems on the pennants.

You may say I'm a touch obsessed with prime numbers. We learned how they are used in PIN codes and other super cool spy stuff in grad school (though I only remember shadows of all this now).

What has stuck with me is that every number has its own, unique "prime number fingerprint". No two numbers share the same string of factors. Super cool.

It seems so obvious now, but when I realized this, numbers started feeling a lot like building blocks, almost like chemical elements.

A few months ago I added this prime factorization reference to my 4th grade math word wall. I also pulled these pieces out for my blog (you can get them as part of my free math resource library).

How do you teach finding GCF and LCM? Do you have kids prime factor or do you use the cake method? Something else?

The cake method is a cool way, however it could become problematic if we try to use it to find LCM for 3 (or more) numbers. Example: LCM of 8, 12, 16 - the cake method will show GCF of 4 with remaining 2, 3, 4 and student will think the LCM is 4x2x3x4=96, but in reality the LCM is 48. Because of that, the prime factorization/factor tree might be safer method. But the cake method will be good if only work with 2 numbers for GCF & LCM. :D

The cake method is a cool way, however it could become problematic if we try to use it to find LCM for 3 (or more) numbers. Example: LCM of 8, 12, 16 - the cake method will show GCF of 4 with remaining 2, 3, 4 and student will think the LCM is 4x2x3x4=96, but in reality the LCM is 48. Because of that, the prime factorization/factor tree might be safer method. But the cake method will be good if only work with 2 numbers for GCF & LCM. :D

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