For this back-to-school edition of Works-for-Me Wednesday, I’d like to share some strategies for learning and doing arithmetic.  Some of these I learned in elementary school, and others I picked up later but wish I had known in elementary school!  I’m now the data manager of a large social science research study, so I use a lot of arithmetic and algebra in my work: I have to figure out what algorithm to type into the computer to get it to do the right thing with the data, I have to check the computer’s results (to make sure that what I told it to do is what I wanted it to do!), and when I see problems in the data I have to figure out where they came from by adding and subtracting the numbers of participants who gave various answers to various questions.  I also use some of these strategies for tasks like figuring out which sale price is a better value, calculating a tip, adding up the boxes and dollars on Girl Scout Cookie order forms, and figuring out how old someone was on a certain date.

Chisanbop is a system for counting to 99 on your fingers.  It’s much more useful than counting your fingers because it not only goes higher than 10 but also represents the tens column and ones column using separate hands and enables you to see what’s happening when you carry and borrow numbers between columns.  I learned Chisanbop in second grade (thank you, Mrs. Boone!), and it was the thing that finally got all that carrying and borrowing business to make sense to me.  I still use it when I’m having trouble keeping track of addition or subtraction in my head and when I need to “hold” a number while using my short-term memory for something else.

Use the buddy system to form tens when adding a long column of numbers.  Each number between 1 and 9 has another number who is its buddy, and when they get together they make 10.  Take your pencil in one hand, and use the other hand to Chisanbop the number you’ll be “holding.”  Look at the ones column, find a pair of buddies, strike out those two numbers, and add 1 on your Chisanbop hand.  When you reach 10 on your hand, write a 1 in the hundreds column, “clear” your hand, and continue.  When you’ve found all the buddies, write the number you’re holding in the tens column, then add the remaining numbers in the ones column in the usual way and write the result.  Move on to the tens column.  Here’s an example:


In the ones column, we have 6 and 4 (count 1 on your hand), 8 and 2 (2), 2 and 8 (3), 9 and 1 (4).  Write “4” at the top of the tens column.  The only number left in the ones column is 6.  Write “6” under the ones column.
In the tens column, we have 4 and 6 (1), 2 and 8 (2) . . . and that’s all the buddy pairs . . . oh, but 4+1 makes 5 to go with that other 5 (3).  Write “3” at the top of the hundreds column.  Still in the tens column are 2, 1, and 2, so write “5” under the tens column.
In the hundreds column, we have 3 and 7 (1) and another 1 and 4 to go with a 5 (2).  Write “2” under the thousands column.  The only number left in the hundreds column is 2.  Write “2” under the hundreds column.
The answer is 2,256.

Now, wasn’t that easier than “6+8=14+6=20+2=22+4=6, I mean 36, or was it 26 or [sigh] 6+8=14…”?
I had figured this out as a strategy for extreme addition years before my uncle Ken told me the “buddy system” terminology, which had worked well with his kids.  He said that they drew some pictures of 2 and 8 playing together, etc., and hung them up as visual aids to help them memorize which numbers are buddies.

Pennies and dimes are great for visually representing carrying and borrowing problems.  Set out a pile of dimes for the tens digit and a pile of pennies for the ones digit in each number.
For addition: Combine the pennies, count out sets of ten, “take them to the bank” and trade them for dimes, and then count the remaining pennies and write that answer in the ones place.  Now count the dimes and write that answer in the tens place.
For subtraction: Look, you can’t take away that many pennies because you don’t have that many pennies.  Take a dime to the bank and trade it for ten pennies.  Now do your subtraction of pennies and write that answer in the ones place.  Now do your subtraction of dimes and write that answer in the tens place.
You can add dollar coins when you’re ready for 3-digit numbers!

Working from the left is sometimes easier.  When I’m doing arithmetic in my head (or on paper where the numbers aren’t aligned vertically, like in my checkbook), it’s often easier to start from the big end of the number, like this: “526+28?  Okay, 526+20=546; 546+8=two less than 556=554.”  It sounds weird to say it’s sometimes easier to add 10 and subtract 2 than to add 8, but somehow it is!  I’ve read that these two strategies are commonly applied by Japanese students who beat American students in arithmetical speed and accuracy, so even though I can’t explain why it works, I feel that I “have permission” to do these problems a different way than my teachers taught me.

Having trouble remembering the steps for solving a long-division problem?  “Dad, Mom, Sister, Brother” is a mnemonic to help you remember to Divide, Multiply, Subtract, and Bring down.  My fifth-grade math teacher taught me this one (thank you, Mrs. Goforth!), which was particularly easy for me to remember because those were the members of my own household in age order.  It’s really helpful for a kid who is confused about why we’re multiplying and subtracting when this is a division problem–put aside the “why” for a while and pretend you’re a machine doing these processes in your factory, and after you’ve got it running smoothly and have filled a page with these neat-looking tapering structures you now know how to make, then you can think about “why” again and probably find that you understand more than you did.
Look for patterns.  Schools tend to teach this sort of skill as a boring search for the Least Common Denominator–terminology which threw me into despair because “least common” means “hardest to find,” doesn’t it? And I was having a lot of trouble remembering which end of the fraction was the denominator, and when this new skill turned out to be about division rather than any visible fractions, oh man, I was really lost!–but it’s a lot more fun and useful to do informally.

For example, Brand A is on sale at 3 for $5 and has 20 in a package; Brand B is on sale at 5 for $10 and has 25 in a package.  Well, $10 is twice as much as $5, so we could say the Least Common Denominator is $5 (or $1 or 1c!), but really the easier way is to look at how many we can get for $10.  Double the amount of Brand A, and you get 6 packs of 20 which is 120.  $10 worth of Brand B is 5 packs of 25 which is [5×20=100, 5×5=25] 125.  Brand B is the better value.

RELATED ARTICLE: Early Encounters with Variables

6 thoughts on “Arithmetricks

  1. This is a great round up of some lesser known tricks! I like the mnemonic for division, I hadn’t heard that one before! By coincidence, I’ve got a “Works for me Wednesday” post this week about tricks for the multiplication tables ( e.g. x4 is doubling then doubling again) It was guest written by Lynne Gregorio of Apex Learning. She’s presented some “tricks” so in the end students need only “memorize” 6×7 and 7×7.

  2. Pingback: My Toyota Prius works for me…but not driving works even better! « The Earthling's Handbook

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  4. Hey Becky! I browsed around your site for a bit. Will comment more, but I am intrigued .. you seem to know a lot about math – no wonder, you manage data. I too have some cool (I think) methods for teaching my kids math that I will be documenting in the future. You might also want to check out my wife’s page She is a high school math teacher with a math degree, and I’m a computer science grad (practically a math degree) so we love the math and science.

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