# Finding the GCF and LCM using the Cake Method

Today’s lesson in my 7th grade math class was on finding the greatest common factor and least common multiple of a pair of numbers.

I started by gathering background information about what the students already knew about the GCF and LCM.  With a little bit of help the students were able to come up with definitions for each of them and described the method they had used in the past to find the greatest common factor and least common multiple of two numbers.

Previously the students had learned to list all the factors of 2 numbers and find the largest one they had in common.  Similarly, to find the least common multiple, the class had learned to list multiples of the numbers until they found one in common.

We went through examples of finding the GCF and LCM the “old school” way.

Then we discussed the limitations of this method for finding the GCF and LCM, mainly the fact that if I gave them much larger numbers it would be very time-consuming to list all the factors or list multiples until they found one in common.  So…I then introduced the class to the “Cake Method”.  (Alot of people refer to it as the ladder method or division ladder, but I first saw it called the cake method in the book Math Doesn’t Suck by Danika McKellar, and I thought it was cute, so that is what I go by, since it resembles an upside-down cake).

The cake method is great for a couple of different reasons…you can make one “cake” and use it to find both the GCF and the LCM and it saves you alot of time over the more traditional method.

For anyone who hasn’t seen this method before, you start by writing the numbers for which you are trying to find the GCF and LCM next to each other and draw a “layer of cake” around them.  You then find ANY common factor of the two numbers and write that factor to the left of them.  You then divide the numbers by their common factor and write the quotients underneath the numbers.  This process repeats then with the new 2 numbers you have inside the cake.  You do this as many times as necessary until the 2 numbers at the bottom are relatively prime.  To find the greatest common factor, you multiply all the numbers to the left of the cake.  To find the LCM, I tell the students to draw a big “L” around the cake, and multiply all the numbers in the “L” together.

Here is the example we did in class to find the GCF and LCM of 24 and 36.

We had a discussion about how different people could make different “cakes” by choosing different factors, but that the final answer would be the same.  We also discussed how the bigger the factor, the less layers of cake there would be.

Overall, the lesson went great!  The students loved this “new” method of finding the GCF and LCM!

I am looking forward to showing them how the cake method can be used to simplify fractions, too.  (Make a cake for the numerator and denominator and the 2 relatively prime numbers at the bottom will be the simplified fraction.  For example, the cake example above could have been used to simplify the fraction 24/36 to 2/3.)