# Finding the GCF & LCM of 3 or More Numbers using the Cake Method

I wrote a blog post 2 years ago about using the cake method to find the GCF & LCM of 2 numbers.  (I absolutely LOVE this method and my students have had alot of success with it!)  I have been meaning to follow up on that post by sharing how to use this method for more than 2 numbers, and since I just got a question about this last week, I figured now is the time to write that post. 🙂

For this example, I will find the greatest common factor & least common multiple of 12, 15, & 18.

Start by writing the 3 numbers next to each other and drawing a “layer of cake” around them.  Take out any number that you can divide all three numbers by (in this case 3).  Then divide the numbers by 3 and write the quotients under the original numbers, drawing another layer of cake around them.  If there is something else you can divide all 3 numbers by, repeat this process.  In this case, though, there is nothing (besides 1) that I can divide 4, 5, & 6 by so at this point I like to CHANGE COLORS.  Now you need to find if there is any number you can divide some of the numbers by.  In this case, I can divide 4 and 6 by 2, so I write a 2 on the side (in a different color) and then divide the 4 and 6 by 2.  Since I am not dividing the 5 by anything, I just bring it down.  Repeat if you can divide some of the numbers by another factor.  In this case, I can’t, so I am done.

Find the GCF by multiplying all the numbers on the left that were in the ORIGINAL color ONLY!  (In this case, it is only “3”, so my GCF = 3).  Find the LCM by multiplying all of the numbers in the “big L” around the cake.  (That includes the numbers in both colors).  (For this example the LCM = 3 x 2 x 2 x 5 x 3 = 180).

In case my explanation makes no sense, I have included a second example showing this process.

As I said, I absolutely LOVE the cake method and have found that the students love it too and find it MUCH easier than other methods for finding the GCF & LCM.  I hope this post helped explain how to use it for more than 2 numbers!

Christina

You also may be interested in my GCF & LCM Halloween Riddle Sheet – available in my TpT store for \$1.

# 5 Ways to Use Factor Trees in Middle School Math

I think factor trees are one of the coolest “tools” for middle school students!  Students tend to like them and they have several different uses in middle school math.

Here are the 5 ways I use factor trees with my students:

1.  Prime Factorization
This is the most obvious use of factor trees and it’s typically the first way students learn to use them.

When I teach prime factorization I introduce the idea of the Fundamental Theorem of Arithmetic (no matter how many different factor trees students make for a number, they will always get the same prime factorization).  I also emphasize the two things they can/should do to check their answer:
– make sure all numbers in their prime factorization are PRIME numbers
– make sure the product of the numbers in their prime factorization is the starting number

2 & 3.  GCF and LCM
Using factor trees is my second favorite way to teach the greatest common factor and least common multiple of numbers and/or monomials.  (The cake method is my favorite method).

Factor trees make simplifying radicals so easy for my pre-algebra and algebra students!  Students understand that finding the square root of a number means figuring out what number times itself equals the starting number, so in simplifying radicals they are simply pulling out one of each prime factor that is listed twice in the factor tree.

5.  Factoring Trinomials
The idea behind factoring trinomials is pretty simple.  Students need to come up with two numbers that have a certain product and sum.  Students are typically pretty good at factoring simple trinomials like this, since coming up with the two numbers is a breeze:

But, many students struggle with problems like the one below, not because they don’t understand the factoring process, but because they can’t come up with the two numbers with the given sum and product.   That’s where factor trees come in handy!

For students who have a tough time coming up with the numbers, I have them make a factor tree for 126.  Once they have it broken into its prime factors, they just need to break up the prime factors into 2 numbers every possible way until they find the ones with a sum of 23.

While this method does take some trial and error, it gives “stuck” students a starting place and a set number of possibilities to try.

Do you use factor trees in any other ways in your math classes?  If so, please share!

Christina

# 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.)