Mathematical Mondays

Confessions of a Math Geek

I love calculus.

If there’s anything that cements and seals my “I’m a Geek” badge, that statement is it. And I’m okay with that.

Do you know what you can do with calculus? I’ve found most people who never took calculus have no idea what it involves, so here are some highlights.

You can find the slope of curves. Remember slope? You learned about it in algebra, probably with some variety of “rise-over-run.” It tells you how steep a straight line is, the rate at which it increases or decreases.

Well, with calculus, you can find that rate at a specific point on a curvy line. It’s starts off a little ugly and scary, with a formula that looks like this (or a variation on it):

After laboring through several problems with this not-so-fun process, your teacher reveals that there are ridiculously easy shortcuts.

You want to kill your teacher. (I warn my students ahead of time that this will happen.) Then you get over it and get to work.

This may not sound that useful, but think about all the things that involve rates. Velocity, acceleration, how quickly something is growing or shrinking, etc.

Later on, you learn how to find the area under curves. Again, it starts with a slightly complex process that you soon simplify (until it gets harder again). This concept extends to taking the graph of an equation, imagining that you’re spinning it around an axis, and finding the volume of that 3-D shape.

There are ways this is useful, too. But from the time I learned it, I thought it was just insanely cool all by itself.

Yup. A geek, for sure.

You know what’s even stranger? That slope-finding process and the area-finding process turn out to be inverses of each other. Totally unexpected, but it’s part of what makes it easier in the long run.

Anyone else have a topic they learned about in school that just makes them geek out to an irrational degree? A certain period in history, or a concept in science? It’s safe to share. All geeks are welcome here.

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Dressed to the Nines

We have a wonderful base-10 number system. It makes a lot of things easy, and would make things even easier if the U.S. would get with it and switch to the metric system. Think about the poor Romans. Have you seen those years noted at the end of movies made in the twentieth century? Yuck.

The nature of the base-10 system makes for some interesting things with the number that’s just one shy of ten—nine. While learning your times tables, you may have noticed these properties of nine.

Up to 9 × 10, the digits of the products add to nine.

Again up to 10, there’s a cool bookend-reversing thing going on, as the first digits go up and the second digits go down:

09
18
27
36
45
54
63
72
81
90

A side-effect of this, along with the fact we have ten fingers, is a little trick I use with kids who still struggle to remember multiplication facts with nine. (I thought everyone knew this, but have found several adults who’ve never seen it, so I figured I’d share it here.)

Hold your hands in front of you, fingers spread. Whatever number you want to multiply nine by, count that many fingers from the left and put down the finger you land on. (So if you’re doing 3 × 9, count three fingers from the left, and put down your left middle finger.)

How many fingers are up to the left of the lowered finger? (In the example, two.) How many fingers are up to the right? (Seven.) Put those together, and you have the answer. (Two and seven … 3 × 9 = 27.)

And now, I have to go do some research on the title of this post, because I’ve always kind of wondered about that phrase.

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Impress Your Friends With Mental Math

You’ll have to take my word for it that I’m going to do this entire post without touching a calculator or scribbling calculations anywhere. I also don’t know how useful these tricks will be for you, but hey, it’s fun.

Divisibility by 3, 6 or 9

Have a large-ish number and need to know whether you can divide it by one of the above numbers? Easy. Just add up the digits. If the result is divisible by 3 (or 9), then so is the original number. If it’s divisible by 9, it’s automatically divisible by 3. If it’s divisible by 3 and is even, then it’s divisible by 6.

Example: 4,374

4 + 3 + 7 + 4 = 18

18 is divisible by both 3 and 9, so 4,374 is divisible by both. Since it’s even, it’s also divisible by 6. Go ahead and check it while I try a larger number.

Example: 5,660,193

5 + 6 + 6 + 0 + 1 + 9 + 3 = 30

30 is divisible by 3, but not 9. The original number is odd, so it’s only divisible by 3 (not 9 or 6).

Multiplying by 11

We all know that multiplying a single-digit number by 11 is easy—just repeat the number. 11 times 7 is 77, 11 times 3 is 33, etc. Multiplying by larger numbers is pretty easy, too.

The first and last digits stay the same. For the middle number(s), add adjacent numbers together.

Example: 11 × 35 = 3_5.
Since 3 + 5 = 8, that’s the middle digit.
So 11 × 35 = 385.

Bigger Example: 11 × 724 = 7_ _4.
7 + 2 = 9, and 2 + 4 = 6.
So 11 × 724 = 7,964.

What if one of those middle number sums results in a 2-digit number? Still works, you’ll just have to do a little carrying over to the next column to the left.

Example: 11 × 3852 = 3_ _ _2
3 + 8 = 11. Oops, carry that 1 over to the left, so the first digit is 4.
8 + 5 = 13. Oops again. Carry that 1 to the 1 in the 11 above. (Confusing, yeah.) Second digit is 2, third is 3.
5 + 2 = 7. Okay, nothing fancy here.
So 11 × 3,852 = 42,372.

Now go and impress your non-mathematical friends.

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But … I WAS Teaching!

Funny thing happened the other day. Less “funny-ha-ha” and more “funny-huh?”

My school has this thing where a collection of administrators and specialists rotate around, observing different classrooms each week. Some look at how we’re using ASL, curriculum, or technology while others just look at the general classroom experience.

A few weeks ago, I had one such observation. I had a great little activity for my class. We briefly reviewed what we knew about three different types of functions, and I explained the activity. They’d be making predictions about a list of equations, checking those predictions, and then forming some generalizations. I circulated as they worked, dialoguing with them about what they were noticing. It all went really well.

I didn’t think about it again until I got an email from the observer a while later. She apologized for not getting any notes to me sooner, but she’d had a hard time writing up anything because she “really had not observed a ‘lesson’ so to speak.” She wanted to schedule another observation when I was teaching a new concept.

Excuse me, what?

I had taught a new concept. I’d taught my class how to recognize linear, quadratic, and exponential functions by their equations and without graphing them. The students were actively engaged in learning the whole time, doing something, rather than sitting there in a lecture-coma as I told them everything from the board.

Clearly, though, she wants to see something that looks more like a traditional “lesson.” So she’s coming back this week to observe a lesson in my physics class.

Le sigh. I don’t mind being observed again. I do mind the fact that I’m trying to follow nationally recognized “best practices” is being discounted as “not a lesson.” If it wasn’t a lesson, what did she think it was—busy work? That, I definitely mind.

Is it just me? If you were back in math class, would you rather take notes on a lecture or work on an activity that helped you figure out a concept for yourself?

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Mental Math Tips for Your Kids (that Might Help You, Too)

Every once in a while, I’ll try to post some math tips that may or may not be helpful to people. We’ll start with a couple of simple ones this time, and you’ll have to trust me when I say I’m not using a calculator for any of the examples.

Multiplying By 12

I never memorized the multiplication facts for 12 when I was a kid. Why bother? It’s easy to multiply by 11, then just add one more of whatever number your multiplying by. For example, for 12 × 7:

11 × 7 = 77
77 + 7 = 84
so 12 × 7 = 84

In third grade, I was able to do that process quickly enough to still complete the timed multiplication tests, so my teachers never knew I hadn’t actually “memorized” those facts. (Do they still do those timed tests in elementary?) Of course, that assumes you can add a single-digit number to a larger number without counting it out.

What if you need to multiply something by 12 that’s beyond the typically memorized math facts? It’s a little trickier, but in a pinch, you can still do it in your head. Multiplying by 10 is even easier than 11. Then you just double whatever you’re multiplying by, and add the two together. For example, 12 × 15:

15 × 10 = 150
15 × 2 = 30
150 + 30 = 180

Maybe that one was too easy. Let’s try 12 × 43:

43 × 10 = 430
43 × 2 = 86
430 + 86 = 516

Again, this requires some mental addition ability. So let’s look at one of the simplest cases of this to start with.

Adding (or Subtracting) 9

As crazy as it makes me to see teenagers in advanced math classes add by counting on their fingers, it’s the worst when they do it for something as easy as adding or subtracting 9. Our base-10 number system is a marvelous thing. It makes adding/subtracting 10 super-easy. To add/subtract 9, just do 10, then move one space back the other direction.

For 27 + 9:

27 + 10 = 37
37 – 1 = 36

For 82 – 9:
82 – 10 = 72
72 + 1 = 73

Yes, pretty much all of us have cell phones with calculators these days. But seriously, for simple calculations, I bet you can do it in your head more quickly than you can take out your phone and punch it in. 😉

I’ll see if I can come up with more tips and tricks another week.

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Ooh, Look at the Pretty Numbers!

I’m a little OCD. Have I mentioned that before? Not to a degree that it interferes with my life, just noticeable in a few areas. Like when I leave my class with a substitute and everything’s out of place the next day—and that can be something as little as the books on a not-quite-full shelf being shoved to the right instead of the left. *shudder* Annoying.

As a math teacher, it’s only appropriate that one of my little quirks relates to numbers. Some are prettier than others. It’s not that I can’t function when “ugly” numbers come up. I just feel a little warm fuzzy when they’re pretty instead.

So, what are some of my “pretty” numbers? Palindromes are definitely way up there. Those are numbers that read the same forward and backward. When I look at a digital clock right when it’s 12:21 or 8:18? Love it. Catching when my odometer hits one? Love that, too.

Numbers that fall in order or in a pattern are nice, too. Speaking of my odometer, it recently passed 123,456 miles. (My car is well-loved.) That was awesome.

Then there’s my car stereo. The volume increments are pretty small, but anything much over 40 is usually permanent-damage-to-the-hearing range. Within the “safe” range, I get a little weird with settings that are and are not okay … and it has little to do with whether it’s loud or soft enough. In general, prime numbers = yuck. That means even numbers are mostly good, but something like 38 (a prime times two) isn’t as pretty as 35. Multiples of 5 are very pretty, as a rule. Multiples of 3 aren’t bad, either, which means 39 is slightly better than 38, but why not go the extra notch to 40, which is prettier than both combined?

I’m nuts. I know this.

Funny thing is, none of this matches what I mean by “pretty” and “ugly” numbers in my classroom. Rational numbers are pretty. Irrational numbers are ugly. Simple as that. If my students get a pretty answer, they know they should either write the exact decimal or the equivalent fraction. If they get an ugly answer, they should either round it appropriately, or leave it in square-root form (or as a multiple of pi, whatever applies).

I imagine that definition makes a lot more sense. But it doesn’t mean I’m not very much looking forward to twelve minutes after noon on December 12th of this year.

C’mon, guys, ‘fess up. What weird little quirks do you have that make you look just a little bit crazy?

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