I've been using Keynote this semester as an experiment, to see how it could work in my Numeracy class. So far, it's gone pretty well - especially after I bought a remote mouse so I could control it from anywhere in the room. Combined with the mini-whiteboards, it's been a really efficient way of getting students to do work. After presenting a concept, I can have them practice a few problems right away by showing the next slide, and having them work on their boards. There is no time wasted passing out worksheets. Also, I can make sure all students are focusing on a specific set of problems (versus on a worksheet, where they tend to start jumping around right away, based on what seems easiest). Then, I can show work/answers on the slide without having to pull out a transparency.

Since I've got the projector reserved and set up now, I can easily insert fun and interesting images, sounds, and video clips. I've recorded myself and other teachers singing little ditties (like the infamous "Don't add across"). I've started scouring YouTube for interesting stuff... though the ratio of total crap to interesting stuff is quite high, I've found a couple of gems. I even unearthed my old calculus professor from college, who recorded a "top ten algebra mistakes hit parade" as well as "all of calculus in 20 minutes".

So I'm in my fraction adding unit now, and we've been working with fraction circles to understand adding. Now, we're taking a break from that to do some work on prime factorization, reducing fractions by canceling common prime factors, and finding LCM. Once they get all this mastered, we can go back to adding fractions using common denominators. I hope they don't forget it all over spring break... I've always found it difficult to teach factors and multiples, and GCF and LCM because students confuse these concepts very easily. Part of the problem is their difficulty with the language of division. Just about every student I have says "divide 6 by 40" when they mean 40÷6. If I ask "does 3 go into 12?", they'll say yes. But they'll also say yes if I ask "does 12 go into 3?". (Aside: I think I'm going to devote an entire lesson to this issue - along with the whole "subtracted from"/"subtracted to" issue.)

In any case, I YouTubed LCM and GCF to see if there was anything interesting out there. I was surprised to find a method for finding both LCM and GCF at the same time using Venn Diagrams that I'd never seen before. It's mathematically equivalent to looking at the prime factorizations and picking the right factors, but it provides a nice structure for students to remember which is which. So I designed a lesson to practice finding factors and multiples, and then using this model to find LCM and GCF. It went quite well. I don't know how much will be retained over the weekend, but we'll practice more on Monday/Tuesday because I want them to have LCM down solid. Here are two of my slides, and then the original video I got the idea from.

A beautiful combinatorics argument

11 hours ago

## 10 comments:

How cool! We were just talking about these same two videos (top 10 & calc in 20 minutes) last week in the math workroom. Small world.

Thats an amazing technique using the venn diagram, I have never seen that before. Thank you so much for sharing, I am off to try that out.:)

No problem... but props have got to go to YouTube on this one (and the teacher who posted).

My students have been using the method with some pretty good successes so far.

I have seen that before, and it was in a math for elementary ed textbook. You'd be surprised what good stuff there is in those books for the pre-algebra level

As a teacher of General Math, Pre-Algebra, and Algebra 1, I see the benefit of even introducing to General Math students the terms "intersection" for GCF and "union" for LCM as it relates to the Venn diagrams. My Algebra 1 students struggle with such logical terminology when we cover compound inequalities because it's usually their first encounter with both the mathematical concept AND the logic it hinges on.

Thanks for sharing this method. It's my first time seeing it!

This is great, but you need to be careful about just multiplying all the factors to get the LCM. It works in the example, but if you did that with 36 and 48 the LCM would come out to be 288 -- and that's twice the actual LCM.

Actually, it will always work (with two numbers) as long as only 1 of each repeated factor is put in the overlapping area. That's the hard part to get some students to see. But that prevents you from double counting factors.

Dan,

I understand that you don't repeat the factors in the overlap. But the way you have LCM represented in the graphic and the way it's explained in the video, one gets the LCM by multiplying all the factors once you make sure not to repeat any shared factors, which go in the overlap. But that still won't work with, for example, 36 and 48. Let's see if I can do this in text:

36 - 2, 2, 3, 3

48 - 2, 2, 2, 2, 3

Factors in overlap - 2, 3

So, remove a shared 2 and 3 in the above factor list:

36 - 2, 3

48 - 2, 2, 2

Overlap 2, 3

Now multiply these, as suggested in your graph 2x3x2x2x2x2x3 = 288

But this is actually twice the LCM, which is 144. So, you have to clarify in your graphic and on the video that to get the LCM, you use the factors in the overlap and, when similar factors remain, use the factors from the circle that has the most. So, in this example, the factors to use are the 2 and 3 in the overlap, the 3 in the 36 circle (but not the 2) and the twos in the 48 circle (2, 2, 2). Multiply these out, 2x2x2x2x3x3 and you get 144, the actual LCM.

D'oh, I see my error. I didn't count the additional 2 from each in the overlap. Sorry to bother you about this! Please delete my comments so they don't cause further confusion. Thanks for you blog.

A method I learned from student that I'm sharing despite a fear that it won't show clearly in the comments.

List the prime factors from the first term:

12: 2, 2, 3

Below, list the prime factors from the second term. As you do so, line up any factors they have in common directly below the factors from the first term. (Some students liked using graph paper to keep their columns straight.)

In our example we'd have:

12: 2, 2, 3, _

18: 2, _, 3, 3

(This is where spacing issues for commenting get complicated. I put in an underscore where we'd leave a blank.)

To find the GCF, look for columns that are filled all the way down the column. Multiply them.

To find the LCM, multiply all the columns together.

12: 2, 2, 3, _

18: 2, _, 3, 3

GCF: 2 • 3 _ = 6

LCM: 2 • 2 • 3 • 3 =36

The advantage I found with this method over the Venn Diagram was for when we looked for LCM or GCF of more than two numbers.

12: 2, 2, 3, _, _, _

18: 2, _, 3, 3, _, _

48: 2, 2, 3, _, 2, 2

GCF: 2 • 3 _ = 6

LCM: 2 • 2 • 3 • 3 •2 • 2 = 144

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