Where have all the factors gone? Finding the LCD through factor trees

I've read your blogs. I've asked in class (a couple times, even). But I still can't figure out why.

Why do I not have to repeat all the factors to find the least common denominator (LCD)?

I know some of us have also been stumped by this question.

But for those of us who haven't, let me explain:

                               

 Let's pretend we're adding

We need to make a common denominator. We could just multiply 12 x 15 and use a denominator of 180, but that's embarrassingly huge. We could list out multiples and see what the first common one is (12, 24, 48, 60  and 15, 30, 45, 60 shows us 60 is the first one in common)--boooring. FACTOR TREE!

                                        

So here's my problem: How do I know which of these factors--2 x 2 x 3  and 3 x 5--to include in the least common denominator? Well, I could use them all, of course:

2 x 2 x 3 x 3 x 5

Which gives me what? 180. That's a common denominator, sure, because all we're doing is multiplying all the factors together, which is the same as multiplying the numbers themselves together, which we already tried (see "embarrassingly huge," above).

We're looking for the least common denominator. What if we only use the common factors, maybe? Well, the only factor (besides 1) that 12 and 15 have in common is 3, so clearly that doesn't work.

Maybe the factors that repeat cancel each other out? 

2 x 2 x 3 x 3 x 5 = 2 x 2 x 5 = 20

20 is not a common factor of 12 and 15. So there went that idea.

I'm stumped. Which means, here at Miss B's Math Trip, that it's time to head over to Khan Academy!!

Let's take a look at this video on finding common denominators:

And just like that, there it is, starting at 1:50. Seriously. Take a look. I'll wait.

The video really does explain this concept perfectly (for me, anyway). But I'll have to learn to paraphrase it for my students anyway, so here goes:

 

Let's go back to our example of finding the LCD for 5/12 and 4/15. We recall we used factor trees to get the factors of each number:

                                       

And we know we have to find a common multiple of both 12 and 15 for our denominator:

• In order to be divisible by 12, the denominator has to have factors of 2, 2, and 3.
  

• In order to be divisible by 15, the denominator has to have the factors of 3 and 5.
  

• So as long as the denominator has a 2, a 2, a 3 (both the 12 and the 15 have a 3), and a 5 in its factors, it will be divisible by both 12 and 15, because each of the factors will be there.

Remember, we want to find the lowest common denominator. And the way to do that? Just multiply the factors we know have to be there:

2 x 2 x 3x 5 = 60.

Ssssssnap!

Just to be sure I can do this in practice, though, let me do a practice set (and yes, get that sweet Level Up!)

I did have to switch up the questions a little, since the set practices other ways of finding the LCD besides using factor trees. But I got some good practice in. For example:

Which I solved like this:

And now I have officially Leveled Up!

NICE.

Thanks for stopping by!