Time-saving tips are great, right? So I’d like to share a time-saving tip for math.
This tip SIMPLIFIES the process of finding the greatest common factor (GCF) for two numbers, a good thing to know when simplifying fractions, reducing proportions, etc.
First, have you ever noticed that when students search for a GCF, they sometimes don’t know when to stop searching? This tip alleviates that problem, for it tell students exactly when they can stop testing numbers.
It turns out that students can stop testing when they reach the DIFFERENCE between the two numbers whose GCF they’re trying to find.
As an easy example, let’s say you need to find the GCF for 16 and 20.
All you do is subtract 16 from 20, to get the difference, 4, and this number — 4 — is the largest number that could POSSIBLY go into both 16 and 20 evenly.
Once you know that, just test 2, 3, and 4 to find the highest one that goes into 16 and 20. Of course that would be 4, so you got the GCF right off the bat, in this case.
Keep in mind that that greatest possible greatest common factor is not necessarily the true, greatest common factor. But it does set an upper limit for GCFs, and having that upper limit really reduces kids’ stress.
Another example: find the GCF for 25 and 35.
35 – 25 = 10, so 10 is the greatest possible GCF. But of course 10 does not go into 25 and 35, so 10 is not the GCF. Check the numbers less than 10, and you’ll see that 5 is the GCF. But no more checking above 10, as kids are likely to do, unless you tell them when to stop.
I have dubbed this mathematical object the GPGCF, for Greatest Possible Greatest Common Factor, and I’ve found that students really appreciate learning it’s there — to alert them when it’s “quitting time.”
Try it out yourself, whenever it next flows with your lesson. Let me know what kind of reaction you get from the kids, and good luck.
By the way, if you’d like to explain to your students why this trick works, here’s a way to look at it. If you think about this situation via the number line, the GPGCF is simply the distance between the two numbers whose GCF you’re trying to find. Let’s go back to our first example: searching for the GCF for 16 and 20. The difference between 20 and 16, 4, is the distance between 16 and 20 on the number line. So if any number does go into both 16 and 20, it cannot be larger than 4, since that’s the space between the numbers.
To see this clearly, imagine that you wonder for a moment if 8 might be the GCF for 16 and 20. Well it is true that 8 does go into the first of these numbers, 16. But the next number that 8 goes into evenly must be 8 greater than 16, or 24. In other words, 8 is going to “leap past” 20, by hitting 24, when it goes into its next multiple. So the space between the numbers — 4 in this case — gives you the biggest number that could possibly fit into both numbers.
Now, to help your students get used to this tip, here are some problems.
DIRECTIONSs: Given each pair of numbers, first find the GPGCF. Then use the GPGCF to help you find the GCF.
a) 6, 10
b) 8, 12
c) 12, 15
d) 12, 20
e) 14, 28
f) 18, 26
g) 27, 36
h) 36, 48
i) 42, 60
j) 72, 80
a) 6, 10 GPGCF = 4 GCF = 2
b) 8, 12 GPGCF = 4 GCF = 4
c) 12, 15 GPGCF = 3 GCF = 3
d) 12, 20 GPGCF = 8 GCF = 4
e) 14, 28 GPGCF = 14 GCF = 14
f) 18, 26 GPGCF = 8 GCF = 2
g) 27, 36 GPGCF = 9 GCF = 9
h) 36, 48 GPGCF = 12 GCF = 12
i) 42, 60 GPGCF = 18 GCF = 2
j) 72, 80 GPGCF = 8 GCF = 8