Monday, February 17th, 2020

Circle Equations that Make Sense

Drawn circle

After working with students for 10 years, the same mistakes surfaced every time we introduced area and perimeter formulas for circles:

Messing up the order of operations. Take the traditional area formula, A=πr². Let’s say the radius is 3. Then students multiply π by 3, square that result, and get an incorrect answer.

Confusion over the meaning of π. “It’s a constant,” we tell them, or, “It’s just a number.” But consider: it’s a symbol, a Greek letter, that represents a ratio of two measures of a circle — in other words, it’s far from intuitive. It looks more like a variable than a number.

When calculating circumference, students will probably be successful since multiplying left to right works if we’re approximating π. But if we want students to give an exact answer in terms of π, they need to pluck π out of the middle of C=2πr, then take care of their multiplication, then throw π back on.

Let’s fix the formulas

What if we just put π at the end?

Behold! The new circular area formula!

Now if you put a value in for r, just square it and you have an exact answer! The circumference formula gets the same treatment:

Or C=dπ if you prefer

The only objection from teachers I’ve heard is this: But π is a number, and numbers are coefficients, and coefficients must go the left of the variable! Is it worth it to adhere to this convention if it creates confusion for students?

I say: Try it. See what your students think.

If removing ambiguity is an important principle of mathematics, then let’s teach students formulas that do just that.

Saturday, February 15th, 2020

License Plates and Hidden Harshads

Hey PMR 322, are you hiding a harshad number?

Stuck in traffic? Might be a good time to stare at license plates and test for harshad numbers! It could be a great antidote to road rage.

  • First, find the digital sum of the numbers. We’ll use that PMR 322 plate as an example. Add up the digits: 3+2+2 = 7.
  • Next, see if your number is divisible by the digital sum. Is 322 divisible by 7? That’s the fun part! You might already know divisibility tricks, or you could download my Divisibility Toolkit.
  • If it’s divisible, then you found a Harshad Number! If it didn’t work, there’s still hope; read on….

Where did harshad numbers come from?

The word “harshad” is Sanskrit (हर्षदान), meaning “giving joy”. This name was coined by one of the most interesting mathematicians of the 20th century, D. L. Kaprekar. He was a schoolteacher in India and a fan of recreational math. A self-confessed number-theory addict, he once said, “A drunkard wants to go on drinking wine to remain in that pleasurable state. The same is the case with me in so far as numbers are concerned.”

What if it’s not a harshad number?

So there you are, stuck on the freeway behind a car that lamentably has a license plate like this:

BJV7821, not part of the harshad club — or is it?

At first look, we see that 7+8+2+1 = 18, and 7821 isn’t divisible by 18. A shame. But hang on! You have a couple options:

  • Use the digital root! It’s true that 18 is the digital sum, but keep on adding those digits until you end up with a single digit — that’s the digital root. 1+8 = 9, and 7821 is divisible by 9. These aren’t harshad numbers by definition, but maybe I’ll call them “harshavistya” (हर्षाविष्ट or “joy inside”) numbers.
  • Use a different base! Living in the world of base 10 can make us forget the infinite realms of other bases. In base 7, 7821 is actually 31542. Is that divisible by the digital sum, 18? Or maybe we should convert that 18 into base 7 as well (24). Does it work? Traffic hasn’t moved, so you might as well try.

Are these license plates with harshad numbers?

See if these license plates might be hiding some harshad (or harshavistya) numbers:

333 XDB — harshad or not?
BIC 9434 — are you hiding a harshad?
IA4 497 — after trying 497, should we tackle IA4 in base 26?
If XCV 0296 has a harshad I might just unhitch that trailer and steal it

Take it a little further….

Some questions you might want to investigate:

  • Are there numbers that will always be harshad numbers in any base?
  • How are factorials and harshad numbers related?
  • How many consecutive harshads can you find? What are they?

Saturday, June 18th, 2016

24: Making a Good Game Great

Solutions to making 24 using 1, 9, 5, 2I love the game of 24. Two marks of a good game or math problem are that it doesn’t take long to explain the rules and you can jump in right away. 24 has both of those. If you haven’t played here are the rules:

    1. Take 4 digits
    2. Make 24 using arithmetic operations
    3. Use all the numbers, but only once

With the card game you get to claim the card if you’re the first to solve it, and then you grab another card. But we play it in our classroom more like this: students walk in, start finding solutions on their whiteboards, then we progressively share them out on the classroom whiteboard. We play every Tuesday morning for grades 6–8.

As with all teaching the beauty is in the execution. So what’s the best way to play 24?

At a recent math conference I heard Judith Montgomery from MBAMP explain how she flattens the slope for entering into 24. First things first: a number talk! Put the number 24 on the board and let students respond. Why 24? 24 hours in a day? 24 pickles? And eventually someone will say something like 20+4 and then you’re off.

Interestingly enough she does not do what I want to do and jump right into the rules of the game. No, the number talk winds down and the game is never brought up. And then you return to 24 later — maybe the next day, maybe the next week — and start another number talk. I love this. Because not everyone sees factor pairs when they see 24. Let the students lead you there (as you refine where they’re going).

Tips & Tricks

As students play 24 the tricks for finding them will come out and you should highlight them as often as possible. I urge you to not just give these away. Let them arise naturally. Introduce them over time, not in the same day or week. A few to look out for:

  • Make factor pairs: 8×3 and 6×4 will appear quite often
  • Find points that are n units away from 24: If you have a 6, try to make 18 or 30 with the other numbers (And be sure to bring this up when you are working on absolute values, like |x-6|=24)
  • Use perfect squares to yield other numbers: If you have a 4, you have a 2, since √4=2 — likewise with 9 and 3

The rules of the game will establish themselves. In our classroom students want to go beyond the basic four arithmetic operations. “Can we use exponents?” “Can we use square roots?” (hence that last trick above). When you allow them, they will take off. If some students don’t understand right away how to use these other operators, that’s okay! I promise they will want to once they see how advantageous it is to have those extra tools. As the months go on in the year I might drop a hint on a new method, especially factorials.

Using! factorials! in! 24!

Here’s where it gets really interesting. Eventually when playing 24 a student will get really excited about their solution and write something like this:


And I can say, “I appreciate your enthusiasm, but did you know that ! is a math symbol? In fact, 24! is a really, REALLY big number.” [fun fact for chem nerds: 24! is really close to Avogadro’s constant]

Once the students know that 24! = 24 x 23 x 22 x … x 1, then you are about 30 seconds away from someone saying, “Hey, 4! is 24. Cool!” And now they have a new trick:

  • If I can make 4, I can make 24

And pair that up with square roots and you have:

  • If I can make 16, I can make 24 since (√16)! = 24

My favorite 24s

Check out some of the results from our class. I’ll add more as time goes on. Please note that there might be mistakes in here (or worse, missing numbers from my thumbs wiping the boards accidentally!). There are also some tricks in here I haven’t mentioned yet.  Bonus points when you find them!

IMG_20141104_120959285 IMG_20141216_113725476 IMG_20150203_154808195

Saturday, June 11th, 2016

Why do we start at Step 2?

In the West it’s common for adults to skip breakfast. This is the case even though everyone from food scientists to intuitive moms agree that breakfast is the most important meal of the day.

Perhaps this is why, in the math classroom, we often begin a problem at Step 2. Let me offer an example:


You can’t resist. You want to solve it. You want to find out what x is. But solving for x is Step 2. If that’s so, then what’s Step 1?

Step 1 is the most important step of the process! It’s what comes before Step 2. During Step 1, we should go through some process that yields the statement, “Therefore we must solve for x.”

Perhaps Step 1 in this problem is creating the equation from some verbal context, like, “Tito makes $12 per hour and gets a $25 bonus every Friday. Last Friday he made $115. How many hours did he work that day?” There has to be a reason the equation is there.

There doesn’t even have to be a real-world context. There’s nothing wrong with practicing techniques to solving an equation. But what’s essential is remembering why we’re doing it. I know the danger of using a sports analogy with mathematics, but I can’t resist the obvious! There’s a difference between idly kicking a ball at a goal and doing it while imagining you’re Ronaldo. My baseball coach would insist that during batting practice we always followed through on our swing, dropped the bat, and started sprinting towards first base. After all, what’s the point of hitting the ball if you’re not going to run? And we’re not just running anywhere! We’re going to first base. We’re intending to score.

It needs to be the same with math. Step 1 is deciding where we’re going. Are we going to graph this? Do we know it’s linear? Do we understand it’s not proportional? Is it clear that there is only one solution?

Instead of jumping headlong into Step 2, let’s make sure we instill in students — and ourselves! — a good and healthy Step 1.

Tuesday, April 12th, 2016

I Have… Who Has? Algebraic Match Game

I Have, Who Has?

I’m not sure who invented the game “I Have, Who Has?” but it’s a great one. The idea is that every student has a card with something on it and a question for the class. Once the question is asked, someone in the class will have the card with the answer and respond. Then they’ll pose another question to the class.

One way you can use it is with translation into algebraic expressions. The catch is you have to make a bunch of cards, and that’s time consuming. So why not have the computer do it for you?

Make Your Own!

I programmed this “I Have, Who Has?” card generator. There are a few options built in but let me know if you think of any more that are needed!

Sunday, March 20th, 2016

Follow the Number: Generate Your Own!

Follow the Number

Years ago I played a game with some other teachers called “I Have, Who Has?” The gist of it is to mentally follow along as each person in the room described an algebraic expression, and you had to see if the card you were holding was the one they were describing.

When I wanted to find a fun way for my students to practice arithmetic operations that game was still in the back of my head. And thus “Follow the Number” was born!

I’m sure this game must exist in some other form by some other name. Either way, the obstacle is that you are either having to reinvent a series of cards or recycle the same ones. So why not have a computer generate it for you?

Create Your Own “Follow the Number”!

That’s exactly what I did. Now it’s our Monday morning warmup game. You can try “Follow the Number” for yourself here. You’ll need to print 3 or 4 sheets for each class, and a paper cutter helps. I’m hoping someone at the 1st or 2nd grade level can try this and let me know how it goes (I’ve used it for 6th through 12th grades).

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