## Posts tagged ‘counting’

### Grid with 100 Paths

Due to current circumstances, the National Council of Teachers of Mathematics (NCTM) had to cancel their Centennial Annual Meeting and Exposition, which was to be held April 1-4 in Chicago. As a replacement, though, they presented an amazing gift to the math education community — 100 free webinars led by selected speakers from the Chicago program. Dubbed **100 Days of Professional Learning**, these webinars are to be held on select days from April through October.

As part of the 100 Days, I presented “100 Problems Involving the Number 100” on May 14. To celebrate NCTM’s 100th Anniversary, I collected or created 100 problems, each of which included the number 100. One of my favorites, Dog Days, looked like this:

At the end of the webinar, the conversation continued “backstage” with several members of NCTM staff, NCTM President Trena Wilkerson, and me. During that conversation, Trena made the outlandish suggestion,

Now we need a collection of 100 problems for which the answer is always 100.

I had just finished preparing a webinar with 100 problems, and now she was asking for *another* 100 problems. But never one to shy away from a challenge, I began to think about what kinds of problems might be included in such a collection. One type that came to mind was path-counting problems, like this one:

**Moving only north or east on the segments in the diagram, how many distinct paths are possible from A to B?**

That particular grid, measuring just 3 × 4, has fewer than 100 distinct paths from A to B. (How many paths, exactly? That’s left as an exercise for the reader.) What got me excited, though, was wondering if there were any grids that have exactly 100 paths — and hence providing 1% of the content for the collection that Trena requested.

As it turns out, there are no unmodified *m* × *n* grids that have 100 distinct paths. But what if some segments were removed? For instance, what if one of the middle vertical segments were discarded from a 4 × 6 grid, as shown below? How many distinct paths from A to B would there be?

As it turns out, a lot more than 100. (Again, finding the exact number is an exercise I’ll leave for you. If you need help, Richard Rusczyk from Art of Problem Solving has a video showing how to count paths on a grid.)

So, this is where I leave you:

**Can you create a grid with some segments removed that will have exactly 100 distinct paths?**

Have fun! Good luck!

As for the webinar, which lasted only 60 minutes, there wasn’t nearly enough time to cover 100 problems, but we had fun with 5 of them.

If you missed the webinar, you can hear the discussion about the Dog Days problem and 4 others, as well as get a PDF of all 100 problems, via the links below.

Enjoy!

### Four, or F**k You?

If you asked a student, “How many sides does a quadrilaterals have?” and you received the following response…

…well, you might be upset.

But perhaps the student learned to count in binary on her fingers, where the right thumb is the register for 1, the right index finger is the register for 2, the right middle finger is the register for 4, and so on. Then the response above would be appropriate, despite appearances.

If you then asked, “Into how many regions will a circle be divided if 6 points are placed randomly on a circle, and each point is connected to every other point?” the student might appear to wave at you — or, she may just be telling you (correctly) that 31 regions would be created by holding up all 5 fingers. (In binary counting, all five fingers add up to 1 + 2 + 4 + 8 + 16 = 31.)

My sons learned to count in binary when, at age 5, they asserted that the highest you can count on your fingers is 10. “Actually,” I told them, “You can count as high as 1,023 on your fingers. If you want, I can show you how.”

Of course, they wanted to learn, and I was happy to teach them. There are at least four good reasons for teaching students to count in other bases, and “Dr. Peterson” at the Math Forum had this to say:

I taught my son to multiply in binary before he really learned it in decimal, because it’s easier; you have only the algorithm (method) with no multiplication tables to learn.

Knowing how bases work helps to develop number sense while clarifying the concept of place value. And not understanding place value leads to things like this…

My former boss shared this video with me on Facebook recently, and he asked,

Does this work with other numbers?

I had a fun time playing with that question, so let me now give you a chance to think about it. Can you find another pair of numbers that produce analogous incorrect results when multiplying and dividing? And if you’re feeling really ambitious, can you generalize to determine what types of number pairs will always give these kinds of incorrect results?

### Never Good at Math

In *The Lexicographer’s Dilemma*, author Jack Lynch describes the reaction of strangers when they learn of his vocation:

When I’m introduced at a party as an English professor, people immediately turn apologetic about their grammar and shuffle uncomfortably, fearful of offending me and embarassing themselves. No one feels compelled to confess to engineers that they never got the knack of building bridges, or to doctors that they don’t understand the lymphatic system — but nearly everyone feels a strange obligation to come clean to someone who is supposed to be an expert in “grammar.”

This passage struck me, because I often get a similar reaction when people learn that I’m a math professional. I cannot begin to count the number of times I’ve heard a taxi driver, a real estate agent, a waiter, a waitress, a flight attendant, and even a local newscaster utter the following line:

Math folks often get upset by this statement. I often hear them lament, “Illiterate people would never tell you that they can’t read, but no one has a problem telling you that they’re not good at math.”

That’s a bad analogy. When someone claims to be bad at math, what they usually mean is that they never figured out how to factor a trinomial in Algebra I or that they were tripped up by two-column proofs in Geometry. While I don’t have data to prove it, I suspect that they are able to count, and I would further guess that they have little trouble with the four basic operations. So while you may hear someone admit, “I was never very good at math,” you will likely never hear, “I can’t count.”

By comparison, when someone admits, “I can’t read,” they are admitting that they never acquired the most basic skill associated with letters and words. Reading is the linguistic equivalent of counting. Were it the case that someone was unable to count, she would likely be as embarrassed about it as an illiterate person would be about her inability to read.

Ever the cynic, when someone tells me, “I was never very good at math,” my first thought is usually, “Your teachers were not very good at teaching math.” Every student has the ability to shine mathematically, but it usually takes a teacher who is willing to pull back the curtain and show them what we mathy folks already know — that the beauty of mathematics lies far beyond rote computation, in a realm where exploration, failure and epiphany provide an infinity of pleasure.

Stepping down from my soapbox, here is a passage about the beauty of mathematics from *Dirk Gently’s Holistic Detective Agency* by Douglas Adams:

The things by which our emotions can be moved — the shape of a flower or a Grecian urn, the way a baby grows, the way the wind brushes across your face, the way clouds move, their shapes, the way light dances on the water, or daffodils flutter in the breeze, the way in which the person you love moves their head, the way their hair follows that movement, the curve described by the dying fall of the last chord of a piece of music — all these things can be described by the complex flow of numbers.

That’s not a reduction of it, that’s the beauty of it.

Ask Newton.

Ask Einstein.

Ask the poet (Keats) who said that what the imagination seizes as beauty must be truth.

He might also have said that what the hand seizes as a ball must be truth, but he didn’t, because he was a poet and preferred loafing about under trees with a bottle of laudanum and a notebook to playing cricket, but it would have been equally true.