Discrete Ideas

The Gift of Years

The Count — Sat, 30 Jan 2010 21:30:26 +0000

What?! Another birthday so soon!? No, you haven’t lost 6 months of your life, and The Count isn’t 41… yet! Due to the overwhelming popularity of my Birthday article, it became clear to me that even those people that weren’t born during the awesome year of 1969 would like to be able to see their own age in huge-numbered detail.

After many hours of toil, The Count has made available to you, his loyal reader, a page designed to give you all the joy he felt in seeing his age in Earth years converted to galactically-large numbers of other units. Simply click the link below, and enjoy!

The Count’s Age Converter

The Count

A Hard Lesson Learned

The Count — Sun, 13 Dec 2009 06:52:39 +0000

Have you ever had a problem stuck in your head, and you couldn’t find the answer? I was recently reminded of a problem I first came up with while doing a statistics workbook the summer of my 3rd grade year (yes, my math-teacher mother gave us workbooks to do during summer break… hey, it got results). The book dealt with dice and the probabilities of a 1 showing on a 6-sider, or the sum of 2 rolled dice being 7, etc. but my question had a twist I couldn’t quite solve. Now it’s easy to see that the probability of rolling a 1 on a 6-sided die is 1 in 6, but what probability exists, in rolling 2 dice, of seeing at least one 1 (on either die, or both)?

So there I was, 7 years old and stuck with a math problem I couldn’t solve. Well, what would you do? That’s right, I asked people. Over the next 9 years, I asked math teachers and other adults I thought might be able to help, but no one seemed able to explain it to me. Of course, I could brute force the answer for 2 or even 3 dice quite easily as well (and did), but by that time I wasn’t interested in just the single answer to my problem, but a more general solution for N dice. The difficulty of the problem comes from the “at least one” phrasing. Using discrete math, you could end up having to compute the probability for each of the numbers of dice smaller than your requested count and use alternating subtraction and addition to account for subset solutions and overlap.

Of course, I did come up with (or was given) attempts at a quicker solution. They usually fell into 2 types. The first type was the simplest; since we were rolling two dice, and the probability with 1 die was 1 in 6, the new probability must be 2 in 6. Unfortunately, this easily extrapolates to say that when rolling 6 dice, you absolutely must get a 1, 2, 3, 4, 5, and 6. Anyone who plays Yahtzee can certainly see the flaw there. The second type of false solution came from, at some point, someone recalling some remnant of a college statistics course (the part where previous rolls shouldn’t affect subsequent rolls) and concluding that the answer must be still 1 in 6. This answer also fails to satisfy quickly when you consider more than just 2 dice being rolled. How could the probability of getting at least one 1 when rolling 10 or even 100 dice still be just 1 in 6?

It wasn’t until I reached college and took an actual course in statistics that I found the answer. Fortunately, there’s an easy method. If you ever find a statistics problem that uses the “at least one” phrase, the best bet is to turn it around. What are the odds of not getting any 1s on those dice? As it happens, that’s merely 5 in 6 for each die rolled, multiplied together. In my problem, with only 2 dice, the probability of not getting a 1 is 5/6 * 5/6 = or 25 in 36. Now, since all the other possible results must contain a 1, we’re left with a solution of (36-25) = 11 in 36! This method works for N dice as well, with the probability of getting at least one 1 out of N dice rolled being:

P(N) = 1 – (5/6)^N.

Imagine that, all this time I’d been asking relatively smart people for the answer to a question I’d found a long time ago and the trick wasn’t really in finding the solution, but in re-wording the question so the solution became obvious. This is a lesson every math student will learn along the way. I wish I could go back and tell my 7-year-old self how easy it was to solve, without him having to see how hard it was to find the answer.

The Count

Streaking surprise

The Count — Thu, 26 Nov 2009 18:44:38 +0000

I’m a football fan… well, more specifically the NFL. I hear that colleges other than the one I went to actually have their own teams and that quite a lot of people think that the outcomes of these “college football games” are important, but I never saw the appeal. Anyway, as I said, I follow the NFL. This year, as the season has proceeded, I’ve found myself noticing quite a few long winning and losing streaks occuring this year. With 2 teams reaching 10-0, and a few teams having streaks of 6 and 7 wins (or losses), I decided to track the actual numbers and determine just how abnormal this season is.

Using one of the many available NFL statistic sites, I was able to compile the number of streaks that occured at each of the 10 lengths available at week 11 (which was just completed). Note that only 10 games have been played by each team at this point, and that I ignored the bye week, allowing streaks to continue through the bye uninterrupted. The following are my results:

Streak Length	1	2	3	4	5	6	7	8	9	10
Streak Count	79	30	22	8	5	4	2	0	0	2

Now, to determine how abnormal this year is (so far), I first have to determine what the expected count for each of the streak lengths is. I must admit, I had to seek outside help determining what this should be, as my first few attempts bore rotten fruit. Luckily, there are some helpful folks at MathOverflow, one of which was able to find a solution:

Streak Length	1	2	3	4	5	6	7	8	9	10
Streak Count	96	44	20	9	4	1.75	0.75	0.31	0.13	0.06
I know, these numbers seem small, given that I’d expect HALF of the 2-game sequences to be a 2-game streak. The trick here is know that every 3-game streak has two 2-game streaks inside it which don’t count!

Now, let’s compare the two sets of values. I’ve found a great little charting tool for small sets of data over at Google. Here’s what it comes up with:

Hmmm, these don’t seem to be very different, do they? As you can see, it seems this season isn’t very abnormal at all; with only moderate drops in the number of 1- and 2-game streaks to account for the obvious outlier of having 2 undefeated teams this late in the season. Each streak length is very close to the expected count for that length.

There would seem to be several reasons one could come up with for why long streaks should occur in football: the literally outstanding talent on certain teams compared to others, the non-random nature of scheduling often pitting teams against obviously “unfair” set of opponents, even the difference between having to play certain teams at home or at the away park with largely varying weather. Yet it seems that the schedule is “fair”, that the teams are well matched. The math we’ve done here has shown that my anecdotal musings of unusually large streaks are without basis. How cool is that?

The Count

Discretely simple

The Count — Sun, 25 Oct 2009 14:00:43 +0000

I’ve said it before, and I’ll say it again – I like math. Now, don’t get me wrong; I don’t run around solving Fermat’s Last Theorem for fun on Sunday nights, or anything like that. Perhaps it would be more clear to say I enjoy math, for many reasons, not the least of which is, when you prove something in math, it’s true every time!

Since merely saying, “I can’t find a contradictory example” is never accepted as “it’s always true”, mathematical proofs are often filled with a multitude of complex concepts and references to other also very complex proofs. It’s very refreshing, I think, to find a few that are so simple and elegant, they make you wonder if there aren’t very simple proofs for those other theorems, that we just haven’t discovered yet. Here I present two of my favorite math proofs. I hope you’ll enjoy them as much as I do.

It’s well known that the sum of the integers from 1 to N is N*(N+1)/2, and this can easily be verified for any number you care to choose. However, it’s the proof for this statement that’s makes it so I can say that it’s true for ALL positive integers. To find the sum for an unknown N, see the image below.

First, write the numbers from 1 to N across the top, then write the same numbers underneath those, only in reverse order. This gives us N columns of numbers which we can now sum individually; as it happens, each column totals N+1. The sum of all these N columns must be N*(N+1) and, since we used all the numbers 1 to N twice each in the rows, we need only divide that sum by 2 to find the solution to our problem… N*(N+1)/2. This elegant little exercise proves that the sum will always work out to that formula. How cool is that?

The second proof is a little more esoteric. While it may not be entirely as useful in your life as the previous proof (if you found that one so), it’s another example of just how simple and elegant some math proofs can be (though most aren’t). Look at the image on the right. The blue shape is a semicircle where A-B is a diameter. What’s not entirely intuitive and/or noticeable from the picture is that all 3 of the angles (marked 1, 2, and 3) are 90-degree or right angles. In fact, the angle formed by the segments between any point on the semicircle and the 2 diametric points must be a right angle. Useful information? Maybe not, but let’s look at how we know it’s always true.

In the picture on the left, we take a look at a single arbitrary example of the angle. By treating it in a completely generic fashion, what we do will be applicable to all possible angles. Note that an additional segment has been drawn from the point D to the center of the diameter (and of the entire circle, were it shown) at point C. The angle we’re trying to prove is a right angle is the sum of the two angles a and ß. Now, since AC and CD are both radii of the semicircle, they must have the same length, and their corresponding angles in the smaller triangle ACD must be equivalent (we’ll call that a). The same thing goes for the BCD triangle, only the angle there is probably different than the ACD pair, so we’ll call it ß. Now, since ABD is a triangle too, its angles must total 180 degrees, yet from our previous statements, it must also total 2*a + 2*ß. Basic algebra will reveal that (a+ß) must equal 90 degrees. Simple and elegant, just as I promised.

Well, these are just two of my favorite math proofs. While neither is exactly mind-shattering, I hope they show that it is possible to prove something mathematically without obscure references to complex transformational theorems and multitudinous graphs and charts. The next time you see a numeric pattern or shortcut, make a try at proving it. You may just come up with a short method that’s eluded us all.

The Count

Litter By Numbers

The Count — Fri, 09 Oct 2009 04:31:20 +0000

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.

Location	Lat	Long
Pyramids of Egypt	N 29° 58′ 34.00″	E 31° 07′ 52.00″	link
Stonehenge	N 51° 10′ 43.00″	W 01° 49′ 52.85″	link
Mall of America	N 44° 51′ 13.64″	W 93° 14′ 32.43″	link
Waimea Canyon, Kaua’i	N 22° 02′ 55.00″	W 159° 39′ 29.49″	link
Goat Island, Niagra Falls	N 43° 04′ 50.15″	W 79° 04′ 07.92″	link
Eiffel Tower	N 48° 51′ 21.46″	E 02° 17′ 27.75″	link
Checkpoint Charlie, Berlin	N 52° 30′ 23.33″	E 13° 23′ 24.69″	link
Tarifa, Spain	N 36° 00′ 35.10″	W 05° 36′ 24.01″	link

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

Alternative Math Methods

The Count — Thu, 17 Sep 2009 04:04:30 +0000

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

Significance

The Count — Mon, 07 Sep 2009 02:46:26 +0000

Someone over at Stetson University1 has compiled a list of numbers2 with the reasons they might be termed “special”. Some of these reasons are out there, and maybe a little contrived, but I’ve found a few that I like. Take some time to browse this list, and I’m sure you’ll find something of interest. If not, check out the the entire list.

0 is the additive identity.
1 is the multiplicative identity.
2 is the only even prime.
3 is the number of spatial dimensions we live in.
4 is the smallest number of colors sufficient to color all planar maps.
5 is the number of Platonic solids.
6 is the smallest perfect number.
7 is the smallest number of faces of a regular polygon that is not constructible by straightedge and compass.
8 is the largest cube in the Fibonacci sequence.
12 is the smallest abundant number.
13 is the number of Archimedian solids.
18 is the only number (other than 0) that is twice the sum of its digits.
25 is the smallest square that can be written as a sum of 2 squares.
26 is the only positive number to be directly between a square and a cube.
27 is the largest number that is the sum of the digits of its cube.
31 is a Mersenne prime.
38 is the last Roman numeral when written lexicographically.
40 is the only number whose letters are in alphabetical order.
42 is the 5th Catalan number.3
46 is the number of different arrangements (up to rotation and reflection) of 9 non-attacking queens on a 9×9 chessboard.
53 is the only two digit number that is reversed in hexadecimal.
55 is the largest triangular number in the Fibonacci sequence.
65 is the smallest number that becomes square if its reverse is either added to or subtracted from it.
70 is the smallest weird number.
109 has a 5th root that starts 2.555555….
110 is the smallest number that is the product of two different substrings.
128 is the largest number which is not the sum of distinct squares.
132 is the smallest number which is the sum of all of the 2-digit numbers that can be formed with its digits.
135 = 11 + 32 + 53.
145 = 1! + 4! + 5! (a factorion).
151 is a palindromic prime.
153 = 13 + 53 + 33.
198 = 11 + 99 + 88.
200 is the smallest number which can not be made prime by changing one of its digits.
210 is the product of the first 4 primes.
257 is a Fermat prime.
536 is the number of solutions of the stomachion puzzle.
540 is divisible by its reverse.
668 is the number of legal pawn moves in Chess.
762 is the starting location of 999999 in the decimal expansion of p.
873 = 1! + 2! + 3! + 4! + 5! + 6!
901 is the sum of the digits of the first 100 positive integers.
976 has a square formed by inserting a block of digits inside itself.
1229 is the number of primes less than 10000.
1233 = 122 + 332.
1369 is a square whose digits are non-decreasing.
1620 is a highly abundant number.
1933 is a prime factor of 111111111111111111111.
2239 is a prime that remains prime if any digit is deleted.
2997 = 222 + 999 + 999 + 777.
3094 = 21658 / 7, and each digit is contained in the equation exactly once.
3313 is the smallest prime number where every digit d occurs d times.
4013 is a prime factor of 1111111111111111111111111111111111.
4725 is an odd abundant number.
4913 is the cube of the sum of its digits.
5471 contains no 0’s in base 3 through base 10.
5689 is the largest 4-digit prime with strictly increasing digits.

What an awesome list, thanks!

The Count

Stetson University
What’s special about this number?
Besides being … you know … the answer to Life, the Universe, and Everything.

Counting time

The Count — Tue, 01 Sep 2009 14:00:20 +0000

Happy Birthday to me. Yes, that’s right, you’re friendly Count was born 40 years ago today, which just happened to be Labor Day in 1969. In light of this special event, I’ve prepared a little set of numbers just for fun. None of this is particularly important, just a bit of mind-candy for my birthday. So let’s go!

40 years on this Earth has some easy conversions into smaller time periods. 480 months makes 2,080 weeks pretty easily. Days is more of a problem, as there are some irregularities in the Leap Day counts. As it happens, those 40 years turned into 14,600 days at 365/year plus 10 leap days is 14,610 total days. This computes to 350,400 hours, times 60 for 21,024,000 minutes, and again for 1,261,440,000 seconds. That’s right, over a billion seconds I’ve been alive!

Just because I lived all those 40 years on Earth, doesn’t mean that time can’t be measured by the years of the other planets in our system. Check out how long my life has been on those planets!1

Mercury	Venus	Earth	Mars	Jupiter	Saturn	Uranus	Neptune	Pluto*
166 years	64.9 years	40 years	21.25 years	3.37 years	1.36 years	0.476 years	0.242 years	0.1615 years
*Hey, for most of my life, Pluto counted as a planet!

I’ve done a fair bit of travelling in my life; as an Army brat, you’re sent across the country or across the world every 3 or 4 years. Then again, merely moving around on the face of this planet is chump change compared to how far I (and you) have really travelled during my life. Just sitting on the surface of Earth means I travelled around 40,076 km/day (the planetary circumference2), for a total of 585,109,600 km travelled in 14,600 Earth-sized circles. With an orbital path of 240,800,000 km/year3, I’ve also travelled 9,632,000,000 km (1/10th of 1 percent of a light year) in larger circles around the Sun. Finally, the Sun’s moves along its orbital path in the galaxy at about 251 km/s. At this rate, I’ve traveled over 316.6 billion km, over 1/30th of a light year through the universe during my life… so far.

Hey, this was fun! I hope you’ve enjoyed my numerical recounting (get it?) of my life. I feel a lot better about just how small a number 40 is. Maybe I’m not so old, after all.

The Count

Convert Earth years to other planets
Earth on Wikipedia
Orbital lengths
The Sun on Wikipedia

The shortest path

The Count — Fri, 14 Aug 2009 16:39:50 +0000

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

Making more math geeks

The Count — Sat, 01 Aug 2009 22:41:40 +0000

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.

Go through several of the problems on the sheet with your child, explaining the process of using the number line to solve the sum:

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!

This technique will have your kids doing easy sums very quickly. You can start this before your kids can write their number; just have them circle the answer on the first few. Progression from this most basic sheet can be done by making the number lines longer (to 20 or more), as well as using a single line for multiple problems (which helps disconnect the line from the specific problem). Even farther out, this makes a great tool for teaching subtraction, too!

Verbal quizzing: In the car running errands, or at dinner in a restaurant? Here’s a handy technique that will have your kids learning instead of babbling at each other or staring blankly at SpongeBob: Doubles and Halves. Simply put, just ask your children “What’s the double of [insert number here]?”. Start small with single digits (0 to 5). Your kids will be happy you’re involved with them during what could be a boring time, and they’ll be less inclined to act up. Progression for this technique can be the obvious use of larger numbers, though keep in mind that double-10 is probably easier than double-8 for your kid. You can also switch it up with “What’s half of [insert number here]“, starting with small even numbers and moving on to larger or odd numbers.

Number-based Games: Games are an easy way to keep kids interested in math, especially ones where they have to do math to see who won! Dice games like Yahtzee2 and Farkle3 or card games that have scores, like Uno4, keep children amused while they play, and give them great practice adding larger numbers when computing the score. Some of these games also teach basic multiplication (such as Yahtzee’s 3 dice showing 5’s gives a point score of 15).

Mistakes: Mistakes are a part of life, and a big part of learning math. How you deal with these errors can have a strong impact on how long your kids maintain interest and how often they’ll make the same mistakes. Don’t be quick to correct errors. It’s important to let the child know that their answer was incorrect, but do so in a manner that both isn’t triumphant and doesn’t give away the correct answer. Kids take much more pride from an answer they got on their own, even if it takes a few tries. If they get frustrated or start to “guess”, be ready to divert them to another problem saying, “We’ll come back to that one,” or offer a better path to the solution (such as breaking the larger numbers into smaller ones).

So there they are, a good set of tools and techniques for turning your kids into hellions at school giving your kids a head start in math. I hope you use them well. Of course, these aren’t the only math instruction techniques out there; they’re just a few that I’ve found worked with my own kids. Please feel free to comment with ideas and stories of your own, I may just use them!

The Count

Number Line on Wikipedia
Yahtzee on Amazon
Farkle on Amazon
Uno Card Game on Amazon