Some fractions are a part of everyday life; dimes, quarters, nickels, hours, minutes, seconds, etc. These are relatively easy to manage mainly because we deal with them so often. Everyone just “knows” that 1/2 is 0.5, and 1/4 is .25, and 1/10 is 0.1; we’ve had it ingrained in us through massive amounts of repetition. I go one step further; I can usually estimate the decimal equivalent of just about any fraction that comes up in my life. Super useful? Maybe not, but it has good show-off value, and I think it’s fun!

Learning the first 12 fractions can make it super-easy to do division in your head and produce answers down to the 10ths or even 1000ths quickly and easily. Let’s take a look:

There you have them, the first 12 fractions for easy memorization. Amaze your friends! Astound your kids! Become even more of a know-it-all than you already are! I joke, but I guess you’d be surprised how often I use these, I know I am.

The Count

Ah, Mother’s Day. Admittedly, Math isn’t the first thing you think about when scrambling for that 1-800-Flowers phone number (I always forget that one) or that last box of chocolate from the drugstore. However, one of the memories that sticks in my mind most about my Mom is arguing with her about Math during dinner about .9 repeating and 1. Before you think we’re crazy, please keep in mind she’s a Math teacher, and I have some small interest in the subject myself. So here’s to you, Mom. I finally realized you were right some time ago, but don’t think I ever said so.


I didn’t really have any good arguments for my side of the case. What can I say? I was 8 or 9 and saying things like “1 minus 0.9 repeating is 0.0 repeating with a 1 at the end” made perfect sense to me. I also have to say I didn’t listen that much to Mom’s arguments, so I’m not sure she used these following proofs, but I’m pretty sure they should convince just about anyone.

First, there’s simple addition. If you agree that 1/3 = 0.3333(repeating), then:

About two years ago, I got interested in Geocaching1. I call it “organized littering”. Essentially, people have taken the time to hide caches (usually tupperware containers full of bric-a-brac) all over the world. They then log the lat/long coordinates of their stash, and enter them along with a description on the GeoCaching site2. The rest of us use that site to find caches near where we’ll be, and off we go using our portable GPS units to find these little pockets of fun all over the world.

Our family was vacationing in the Pocono Mountains when we starting ‘caching. I had researched several caches in the area, so one day my father, my wife, my son, and I all headed out into the wilderness. We spent all day traipsing around and seriously enjoyed ourselves getting dirty. The locations we found were a little remote and, since it was summer, the foliage was dense enough to block the view of much. When I decided to write about it, I got to thinking how cool it would be to look for caches in exotic locations. So here they are, some of the coolest locations in the Northern Hemisphere to go looking for litter.

Now, I know you’re just going to run out and fly to all these great places just to go find the closest cache, right? Ok, maybe not, but I hope I peaked your interest in Geocaching, it’s a great way to learn about global coordinate systems and geography. The next time you have a free Saturday afternoon or want a reason to take a hike, give it a try.

The Count

1. Geocaching on Wikipedia
2. GeoCaching site

A lot of math is broken down into processes. Start with a problem, do this, do that and, voila, you have the answer. Most people in America learn the same processes for doing most basic arithmetic, and it’s easy to forget there might be other ways to solve these problems. The positional decimal system developed by the Indians (of India) is the standard taught in every school I’ve ever been in, but it’s not the only way to multiply two numbers1. The following is a description of another method I like for its enforced structure and visual guides.

The Lattice method of mulitiplication2 was published in Europe by Fibonacci in the early 13th century. The lattice, though time-consuming to create for variable lengths of multiplicands, brings a helpful bit of structure when teaching multi-digit multiplication to young students. Use the animation below as a guide when reading the instructions, and I think you’ll find this method very intuitive and interesting.

• Draw the lattice with the same number of squares per side as the number of digits in each multiplicand. Add a diagonal line from upper-right to lower-left for each square, extending the line out the bottom-left of the left-most and bottom-most squares. For students, I’d recommend pre-creating the lattice for them and photocopying.
• Place the digits of the multiplicands across the top and down the right side of the lattice.
• In each of the squares, multiply the multiplicand digits at the top and far right of the square, placing the ones-digit of the product in the bottom-right of the divided square, and the tens digit (0 if none) in the upper-left portion of the square. Repeat for all squares in the lattice.
• Note how the triangular sections of adjacent squares form diagonal “rows”. Starting with the lower-right triangle, sum these rows, placing the ones digits of the sums in the spaces formed by the extension lines outside the lattice. If a tens digit exists, add it to the next row’s total. Note the color-coding of the rows in the example image, which would help a new user of this method. Perhaps shading every other row would be appropriate when pre-making the lattice for your students?
• When completed, the final product can be read down the left side and across the bottom of the lattice.

How does it work? Well, a little study will show that this is really no different than the long multiplication we all know, only some of the operations have been re-ordered; essentially, all of the carrying has been moved to the end of the process. What I find so elegant about this method is the complete lack of “spacing”, padding, or “shifting” that we use in long multiplication which is so difficult to get students to understand and remember. Long multiplication can also be confusing when attempting to place carry values from two different operations in the same space, which can’t happen using the lattice.

Now, I’m not saying that this lattice method should be the new, one-and-only method of multiplication you should use. The lattice is obviously a little cumbersome to draw, especially for larger numbers, though it doesn’t take much more room than long multiplication would. I wouldn’t necessarily expect students to perform multiplication faster using the lattice, but they shouldn’t perform slower, either. As I said, these methods are really very similar when broken down to their elements. While you could certainly use this as a primary teaching method for long multiplication, you could also just present it as an example of the fact that, while math is universal, the methods for performing math are not.

The Count

1. Multiplication on Wikipedia
2. Lattice method on Wikipedia

Shortcuts are a great way to do a lot of math in your head. You know the guy in the office that always responds when you ask “What’s 25 times 56?” He’s using shortcuts, and so can you. The best thing about these so-called tricks? They always have a real math basis, or they wouldn’t work! Here are a few multiplication shortcuts. Learn them, and you’ll see how often the situations for which they are useful show up.

What’s the square of [some number > 20]? Does this come up for you very often? Well, maybe not, but from the square you can get great estimates on other products, and you’ll be surprised how often you come across the problems when you can easily solve them. Now, no one really expects you to memorize all the squares to 100 but, with a few shortcuts, you can compute most of them fairly easily.

• The square of any number that ends in 5 (e.g. 45, 95, 245), represented as (X5)², is ((X * (X+1))*100) + 25. This looks complex, but isn’t. If you’re squaring 85, X is 8, so the answer is (8*9)*100 + 25, or 7200+25 = 7225. 115² is (11*12)=132, *100+25 = 13225. See?
• If you know the square of X, the square of X+1 is just X² + X + X + 1; i.e. if you know 30² is 900, then 31² is 900 + 30 + 31 = 961. The reverse also works; i.e. if you know 20² is 400, then 19² = 400 – 20 – 19 = 361. Hmm, those both end in 61… is that another pattern? (I haven’t found one yet).
• The square of the numbers near 100 can be computed as a difference from 100². Since 96 is 4 less than 100, 96² = 100² – (200*4) + 4², or 10000 – 800 + 16 = 9216. 97² = 10000 – 600 + 9 = 9409, etc. For number over 100 (like 104), it’s the same, but the 200 factor is positive: 104² = 10000 + (4*200) + 16 = 10816.

Ok, but what if I don’t want a square? Fine, not all of life’s math problems involve squares. As it turns out, there are some shortcuts for finding the product of different numbers, too.

• Multiplying any number X by 50 or 25 is easy because they go so neatly into 100. Simply compute ((X/2)*100) to multiply by 50, and (((X/2)/2)*100) for multiply by 25. e.g. 468*50 = 468/2 * 100 = 23400; 468*25 = 468/2/2 * 100 = 234/2 * 100 = 11700. It’s easy to see how this would also work for any multiple of 500, 250, etc.
• Multiply any two numbers that end in 5. If two number ending in 5 (125, 65) are represented as X5 and Y5 where X and Y are the digits preceding the final 5 (12, 6), then the product is ((X*Y) + ((X+Y)/2))*100 + 25. In English, multiply X and Y, then add the average, multiply the sum by 100 and add 25. e.g. 125 * 65 = ((12*6) + (12+6)/2) * 100 + 25 = (72+9)*100 + 25 = 8125. Note, for this one it’s possible for the average of X and Y to be a non-integer (with .5 at the end)… if that happens, the product will end in 75 instead of 25, for obvious reasons. Extra credit if you can prove the first squares shortcut above using this one.
• Multiply two numbers that are an even, “short” distance from each other. If you have two numbers X and Y, and X – Y is even and “small”, then you can easily compute the product using the difference of squares. e.g. to multiply 48 and 52, with an average of 50 (A) and a distance from the average of 2 (D), the product will be the difference of the squares of A and D; 48 * 52 = (50-2)*(50+2) = (A-D)*(A+D) = A² – D² = 50² – 2² = 2500 – 4 = 2496.

Study these a bit, and you’ll start to see uses for them show up in your job, your hobbies, or your kids’ homework. Most of these are just shortcuts I use from day to day or figured out for fun. I’ll sometimes see a pattern after 2 or 3 tries at a similar set of problems, then I’ll spend time proving my shortcut (or disproving, as the case may be) so I can use it on the rest and future problems. Try to prove any of the above shortcuts, and comment with your proof. I’d love to see it.

The Count

You may have noticed that I like math. This is not a new thing for me. Math has been a part of my life for as long as I can remember, and not in the organic sense of counting blocks or birthdays. My parents, both with degrees in mathematics, spent a good amount of time making sure I had no troubles in school in any field, but especially math. Now that I have kids, I’ve been doing my own part to make sure they never need to spend valuable school time figuring out the math on the board, they already got it at home. Note, this doesn’t exactly make me a favorite parent with their teachers; making kids “bored” in school doesn’t endear me to them, but teaching them is such fun I can’t really help myself.

Counting from 1 to 10 is something kids get from many sources; television, children’s books, etc. I won’t cover that here, and my kids could count a bit higher than that when I started with these exercises. The following are several good methods to used to give kids the tools to solve most any math problem they can think of.

Number Line: The first basic math technique you should teach a child is the number line1. This gives kids a handy way to compute sums and differences without breaking out the fingers and toes. Try creating several worksheets where each row has its own number line followed by a single-digit addition problem.

• Find the first number being added on the number line and place your pencil on it, making a mark.
• Starting on that first number, move your pencil one number at a time, counting up to the 2nd number being added.
• When you finish counting, your pencil will be on the answer to the problem!

1. Number Line on Wikipedia
2. Yahtzee on Amazon
3. Farkle on Amazon
4. Uno Card Game on Amazon