The #PowerMath classes and I were taken with the videos done by Westerville South High School in Westerville, Ohio. Especially the polynomial factoring one called “Teach Me How to Factor.”
The students asked that I put together some optional homework for them on the videos. No sense in watching something that fun and not getting to practice it!
Check out the video. Below it, there’s a free downloadable collection of “homework” problems that match each of the polynomial factoring examples in the video.
And right below that, parents and teachers can get the teacher cheat sheet I created to get a whole bunch of fairly easy polynomial factoring examples.
I’ve been dying to check out the book You Can Count on Monsters for a while. My sweet niece, without knowing how much I wanted it, gave it to me for Christmas!
It’s a book about numbers.
You Can Count on Monsters is a book that illustrates the numbers 1-100 in a very special way. Each of the prime numbers has an original “monster” that has a link to it. For instance, 2 has two eyes, 7 has a body with seven sides, and the edges of 19’s eyes – they total nineteen.
Each of the composite numbers is a blend of mosters from the prime numbers of its factorization.
So 14 is the 2-monster hanging around in the mouth of the 7-monster. The other composite numbers range from cute to crazy-hard-to-recognize (like the 72-monster).
For each number, you can examine the monster and see how it was created – what aspects of the monster correspond to the number.
Starting at the beginning seems to be the best bet. Jumping into the middle got me lost. When I began at 1 and then allowed the composite numbers to “grow” on each other, things made much more sense. And as the prime numbers get larger, it’s harder to tell what piece of the image to count.
Walking through it with a child will be very interesting too. Daughter is two, so she’s pretty much not interested in numbers higher than five. Although I can start using the terminology “composite numbers” and “prime numbers.”
You can use it as a template.
Making your own monsters is a fun, crafty way to explain prime numbers and composite numbers. Children would notice quickly that there aren’t many ways to represent 2 – drawing two sided figures is a bit of a challenge.
The obvious 2 monster has two circles, while the 3 monster has either three circles or is a triangle. But the bigger the prime numbers get, the more creative (and different from the book) you can get.
And how you put them together – well, that’s where the real fun is. There are so many different ways to create the composite numbers!
There is a drawback.
I was showing You Can Count on Monsters to my Ma. When I showed her the 6-monster she said, “That’s not counting.”
Indeed the images represent factorizations, not summations. Which means the image that involves 2 and 3 “officially” represents 6 but there are only 5 things “going on” in the picture.
She’s got a point.
What can you do?
Have you seen the book? Are your kids open to drawing number critters? And do you, or your kids, see it like Ma does?
Alright, you might not win friends with this, but the more you can do arithmetic in your head… well… okay, you can’t influence people with it either. Regardless, it’s handy to know and helps with multiplication.
Here’s one that’s a little more challenging. The key is that you can combine the bits to something you’re more comfortable doing. I don’t like multiplying nines, so I avoid those.
Can you use this? Why or why not? Ha ha, just kidding – no essay questions here! – but if you do find a use for this, or if this annoys you, let me know in the comments.
Factoring is traditionally considered an “f-word” for students of math. But it sometimes has its good sides (which I’ll show over the next couple of days).
In the meantime, I’ll show the two ways (that I know) to factor numbers.
This method’s called a factor tree:
You don’t have to put the primes in order for it to be “right.” But it helps for later things.
Here’s another tree:
Here’s another way to factor using an upside division bar-type thing:
Remember, we don’t factor for the sake of factoring. This is only a tool to use when we do other stuff. You’ll see some of this in tomorrow’s post. Here’s one last example:
The way you factor is personal preference. Pick the one you like, or flip-flop. There’re no rules!
Share your preference (or a different way) in the comments!
This is a trick that everyone should learn in the 3rd grade. If not, then at least by the 9th grade.
If you’re in the dark on this one, no big deal. Now’s the time to learn. If you’re a parent – this trick will win you points like crazy. Especially if you get to it before the teacher does.
So here’s how it works. For both 3 and 9, start by adding up the digits in the number. If the result is divisible by 3, then so’s the original number. If the result’s divisible also by 9, then… you guessed it… so’s the original number.
If your result is way to big to tell, do it again. Check out the videos for both:
Questions or comments? Leave them in the comment section below.
We learn to factor polynomials so we can solve for x. Stuff like
3x2 + 2x – 1
can be easily factored into
(3x – 1)(x + 1)
But some things aren’t so easy. And some things are just down right a pain in the bottom. Like this one:
18x2 + 189x + 490
“Just shoot me,” you might be thinking. But cool your jets, cowboy. Here’s the first tool you need to never factor polynomials again:
I know – who wants to memorize that formula? But wait. If you use that one formula that you can memorize, you never have to factor polynomials again. Watch:
So the factoring isn’t bad on that one, right? How about this one:
And how about if you come across one of these. Wouldn’t it be nice to get rid of all guesswork:
This has some disadvantages, of course. You see that there’s a lot more arithmetic. And there’s some things to look out for. But if you hate trial and error (like I do), then you might be willing to take the bad with that good.
